International Journal of Analysis and Applications
ISSN 2291-8639
Volume 14, Number 2 (2017), 107-133
http://www.etamaths.com

INTEGRAL REPRESENTATIONS OF SEMI-INNER

PRODUCTS IN FUNCTION SPACES

FLORIAN-HORIA VASILESCU∗

Abstract. Various spaces of measurable functions are usually endowed with semi-inner products

expressed in terms of positive measures. Trying to give answers to the inverse problem, we present

integral representations for some semi-inner products on function spaces of measurable functions,
obtained either directly or by adapting and extending techniques from the theory of moment problems.

1. Introduction

Let (Ω,S) be a measurable space, that is, Ω is an arbitrary (nonempty) set and S is a σ-algabra
of subsets of Ω. Giving a positive measure µ on S, we denote by Lp(Ω,µ) (p = 1, 2) the set of those
S-measurable functions f : Ω 7→ C such that |f|p is µ-integrable. (Note that we do not identify the
functions µ-equal almost everywhere.)

Let S be a vector space consisting of S-measurable complex-valued functions on Ω, invariant under
complex conjugation. Assume that S is endowed with a semi-inner product (in the sense of [7], page
96), denoted by 〈∗,∗〉0, having real values when applied to real-valued functions.

A natural question is to find a positive measure µ on Ω such that S ⊂L2(Ω,µ) and

〈f,g〉0 =
∫

Ω

f(ω)g(ω)dµ(ω), f,g ∈S. (1.1)

One possible approach to a solution of this problem is to connect it with a moment problem of a
general type, that is, not necessarily directly related to spaces of polynomials (see for instance [19]).
Of course, when (1.1) is fulfilled, the space S(2), spanned by all products fg with f,g ∈ S, lies in
L1(Ω,µ). As it is reasonable and useful to look for a probability measure µ, we should suppose that
1 ∈ S and 〈1, 1〉0 = 1. Setting Λ0(f) = 〈f, 1〉0, f ∈ S, formula (1.1) leads to a linear extension of Λ0
to the space S(2). An arbitrary element in S(2) may have various representations as a sum of products
of two functions from S but we clearly have∑

j∈J
|fj|2 =

∑
k∈K

|gk|2 =⇒
∑
j∈J
‖fj‖20 =

∑
k∈K

‖gk‖20 (1.2)

in the presence of (1.1) for all fj,gk ∈ S, j ∈ J,k ∈ K, J,K finite, where ‖ ∗ ‖0 is the semi-norm
asociated to the inner-product 〈∗,∗〉0. Condition (1.2) becomes a necessary one when trying to extend
Λ0 from S to S(2), in order to look for a unknown measure µ satisfying (1.1) (see Section 3).

Thanks to an argument originating in [23], in many situations of interest we may restrict ourselves
to the case when the space S is finite dimensional (see Theorem 3). The finite dimensionality of the
space S leads to the possibility to replace an existing measure µ by another one consisting of a finite
number of atoms, via an argument going back to [25] (see also [4]). Hence, under appropriate conditions
(see Theorem 4), there exist a finite subset {ω1, . . . ,ωd} ⊂ Ω, and positive numbers λ1, . . . ,λd with
λ1 + · · · + λd = 1, such that

〈f,g〉0 =
d∑
j=1

λjf(ωj)g(ωj), f,g ∈S.

Received 7th April, 2017; accepted 1st June, 2017; published 3rd July, 2017.

2010 Mathematics Subject Classification. 47A57, 44A60, 46C05, 46E22, 47B15.
Key words and phrases. semi-inner product; integral representation; point evaluation; moment problem; relative

idempotent; relative multiplicativity; dimensional stability.

c©2017 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

107



108 VASILESCU

As already hinted above, when trying the represent semi-inner products under the form (1.1), we
either obtain some direct statements (as in Theorem 1 and Theorem 6) or adapt some methods and
results coming from the theory moment problems (as in Theorem 8 and Theorem 9). Because the
classical moment problems are well reflected in the literature, we shall mainly mention some more
recent contributions, as for instance [5], [9], [10], [12], [14], [17], [20], [29], [31], etc.

Let us briefly describe the contents of this work. The next chapter, devided into five subsections,
contains the basic definitions, some examples, and some proofs as well. We introduce a general concept
of function space, consisting of measurable complex-valued functions, endowed with a compatible
semi-inner product (Definition 1), which extends the concept of unital square positive functional (see
[21, 29–31]). Such a pair will be called a quasi-Hilbert function space (Definition 2), and will always
be associated with a canonical Hilbert space (see Remark 1(2)). The formal general question, whose
various particular cases are discussed in this work, is stated as Problem 2. A first abstract partial
solution to Problem 2 is provided by Theorem 1, when the cardinal of the set of continuous point
evaluations equals the dimension of the associated Hilbert space.

Although the framework of general function spaces leads to some interesting results (as for instance
Theorem 1), to recapture as many as possible results known for the spaces of polynomials we restrict
ourselves to spaces of functions having a finite numbers of generators (see Subsection 2.3). A use-
ful consequence of such a hypothesis is the possibility of approaching Problem 2, using only finite
dimensional function spaces (see Theorem 3, extending a result which goes back to [23]).

The concept of idempotent element with respect to a given semi-inner product (see Definition 4)
extends the homonymous one, defined in [30]. This concept plays a central role in our development:
two of our main results (Theorem 8 and Theorem 9), and other results as well, depend on the existence
of orthogonal bases consisting of idempotent elements, with special properties.

Many of the results of this work are stated in terms of semi-inner products. Nevertheless, in some
cases (see Theorem 9) we need the slightly stronger concept of unital square positive functional (see
Subsection 2.1). The connection between these two concepts is given by Proposition 1.

A strong relation between Problem 2 and an interpolation-type problem is presented in Theorem 6,
which uses an extreme situation, that is, when the cardinal of the representing measure is supposed
to be equal to the dimension of the associated Hilbert space. In fact, such a hypothesis is present in
most of our results. A relaxation of this assumption shows the necessity of working with projections
of idempotent elements rather than with idempotents, as shown by Theorem 7.

In the function spaces having a finite number of generators, the existence of representing measures
is characterized by the existence of a orthogonal bases consisting of idempotents, satisfying a certain
”multiplicativity condition“ given by (6). The necessary condition (6) is essential for the proof of
Theorem 8, which is one of the main results of this work.

The concept of dimensional stability (see Definition 7) goes back the concept of flatness, introduced
in [9] in the context of spaces of polynomials (see also [29]). This property is used to prove Theorem
9, which is another main result of this paper. We note that the proof of Theorem 9 is an application
of Theorem 8, which makes it shorter and more transparent than those of its predecessors from [9]
or [29].

An example, originating in [14] and [16], treated in our spirit, concludes this work.

2. Preliminaries

2.1. Function Spaces and Compatible Semi-Inner Products. Let (Ω,S) be a measurable space,
and let also MS(Ω) be the algebra of all complex-valued S-measurable functions on Ω (that is,
f−1(B) ∈ S for each f ∈MS(Ω) and all Borel sets B ⊂ C).

Among some other reasons, the choice of this framework (see also [19]) is related to the use, in the
proof of Theorem 4, of an abstract version of Tchakaloff’s theorem giving a quadrature formula, due
in the actual form to C. Bayer and J. Teichmann (see [4]). Nevertheless, except for such particular but
important situations, we may restrict ouselves to the case when Ω is a Hausdorff space and S is the
σ-algabra of all Borel subsets of Ω and, when working with finite atomic measures, even to the case
when Ω is an arbitrary (nonempty) set and S is family of all subsets of Ω.



INTEGRAL REPRESENTATIONS 109

For convenience, and following [29–31], a vector subspace S ⊂ MS(Ω) such that 1 ∈ S and if
f ∈S, then f̄ ∈S, is said to be a function space (we specify “on (Ω,S)” or simply “on Ω”, whenever
necessary).

Fixing a function space S, let S(2) be the vector space spanned by all products of the form fg with
f,g ∈S, which is itself a function space. We have S ⊂S(2), and S = S(2) when S is an algebra.

For any vector subspace T ⊂S invariant under complex conjugation, the symbol RT will designate
the ”real part“ of T , that is {f ∈T ; f = f̄}.

Functions spaces on a Hausdorff (topological) space consisting of Borel measurable functions were
considered in [31].

Important examples of function spaces are associated with the space Pn of all polynomials in n ≥ 1
real variables, usually denoted by t1, . . . , tn, with complex coefficients. For every integer m ≥ 0, let
Pnm be the subspace of Pn consisting of all polynomials p with deg(p) ≤ m, where deg(p) is the total
degree of p. Both Pnm and Pn are function spaces (of continuous functions) on Rn.

We occasionally use the notation Pm instead of Pnm and P instead of Pn when the number n is
given. Note that P(2)m = P2m and P(2) = P, the latter being an algebra.

Let S be a function space, endowed with a semi-inner product denoted by 〈∗,∗〉0, whose associated
semi-norm is denoted by ‖∗‖0. Note that we have the Cauchy-Schwarz inequality, that is,

|〈f,g〉0| ≤ ‖f‖0‖g‖0, f,g ∈S. (2.1)
Set

I := {f ∈S;‖f‖0 = 0} = {f ∈S,〈f,g〉0 = 0 ∀g ∈S}, (2.2)
which is a vector subspace of S, via the Cauchy-Schwarz inequality. The semi-inner product 〈∗,∗〉0
induces on the quotient space H0 := S/I an inner product given by

〈f + I,g + I〉 = 〈f,g〉0, f,g ∈S, (2.3)
because the right hand side of this equality depends only on the equivalent classes f + I,g + I, again
by the Cauchy-Schwarz inequality. The norm corresponding to (2.3) will be denoted by ‖∗‖. For the
sake of simplicity, the equivalence class f + I will be often denoted by f̂ for every f ∈S.

Definition 1. Let S be a function space. A semi-inner product 〈∗,∗〉0 of S is said to be compatible
with (the structure of) S if

‖1‖0 = 1, 〈f̄, ḡ〉0 = 〈f,g〉0 ∀f,g ∈S. (2.4)
Similarly, the associated seminorm ‖∗‖0 is said to be compatible with (the structure of) S.

Example 1. An arbitrary semi-inner product on a function space is not necessarily compatible. Here
is an example.

Let Ω := {z = x + iy ∈ C; 0 < x,y < 1}. We consider a subspace of the space L2(Ω) of square
integrable functions with respect to the planar Lebesgue measure on Ω, defined as follows:

S := {f ∈ C1(Ω); f,∂zf,∂z̄f ∈L2(Ω)},
where ∂z = 2

−1(∂/∂x− i∂/∂y),∂z̄ = 2−1(∂/∂x + i∂/∂y).
If f,∂zf,∂z̄f ∈L2(Ω), and so f̄,∂zf,∂z̄f ∈L2(Ω), we also have ∂zf̄,∂z̄f̄ ∈L2(Ω), as one can easily

see, showing that S is a function space on Ω.
Let us define

〈f,g〉0 :=
∫ 1

0

∫ 1
0

(f + i∂zf + ∂z̄f)(g + i∂zg + ∂z̄g)dxdy, f,g ∈L2(Ω),

which is a semi-inner product on S. In particular, if f(x,y) = y and g(x,y) = 1, we obtain 〈f,g〉0 =
1 + 2−1i, showing that this semi-inner product is not compatible with the structure of S.

Remark 1. Let S be a function space, endowed with a compatible semi-inner product 〈∗,∗〉0.
(1) Because 〈∗,∗〉0 is compatible with S, its restriction to RS ×RS is a real semi-inner product.

In addition, the sum of spaces RS + iRS is direct, equals S, and

‖f + ig‖20 = ‖f‖
2
0 + ‖g‖

2
0, f,g ∈RS. (2.5)



110 VASILESCU

We also have I=RI + iRI, where the sum of the spaces is direct.
It is easily seen that the R-linear map RS/RI 3 f + RI 7→ f + I ∈ H0 is injective, because

RS ∩I = RI, allowing us to identify the space RS/RI with a real vector subspace of H0, denoted
by RH0. In fact we have the direct sum H0 = RH0 + iRH0. Setting

〈f + RI,g + RI〉 = 〈f,g〉0, f,g ∈RS,
and therefore

〈f + RI,g + RI〉 = 〈f + I,g + I〉, f,g ∈RS,
we infer that the map RH0 3 f + RI 7→ f + I ∈H0 is actually an isometry, and

‖φ + iψ‖2 = ‖φ‖2 + ‖ψ‖2, φ,ψ ∈RH0. (2.6)
(2) We may complete the space H0 with respect to the inner product 〈∗,∗〉, to get a Hilbert space.

Note that a sequence (σk = φk + iψk)k≥1 in H0, with φk,ψk ∈RH0 for all k ≥ 1 is a Cauchy sequence
if and only if both (φk)k≥1, (ψk)k≥1 are Cauchy sequences. Consequently, denoting by H (resp. RH)
the completion of H0 (resp. RH0), we must have the direct sum H = RH + iRH, and equality (2.6)
is still valid for φ,ψ ∈ RH. This decomposition allows us to represent any element φ ∈ H as a sum
φ = φ1 + iφ2, with φ1,φ2 ∈RH. In addition, the ”complex conjugate“ of φ is given by φ̄ = φ1 − iφ2.
We also note that ‖1 + I‖ = 1, as well as the identity

〈φ,ψ〉 = 〈φ̄, ψ̄〉, φ,ψ ∈HΛ,
which is a consequence of (2.4).

Given a function space S endowed with a compatible semi-inner product 〈∗,∗〉0, the Hilbert space
H obtained as above will be referred to as the Hilbert space associated to (S,〈∗,∗〉0). When H0 is
finite dimensional, in particular when S is finite dimensional, we have H = H0, and if S is actually a
Hilbert space, obviously H = S.

(3) The spaces H0 and H have induced inner products with a property similar to (2.4). A natural
question is related to the possibility to organize them as function spaces. Nevertheless, this operation
is not always possible. To explain this assertion, let us assume that S is a function space on (Ω,S),
and that

Ω0 := ∩f∈IZ(f) ∈ S. (2.7)
where Z(f) := {ω ∈ Ω; f(ω) = 0}.

Condition (2.7) is automatically fulfilled if Ω is a Hausdorff space and S consists of continuous
functions or if the space I has an at most countable algebraic basis. If 〈∗,∗〉0 is actually an inner
product, which is equivalent to I = {0}, we have again Ω0 = Ω ∈ S.

Condition (2.7) allows us to regard the elements of H0 as functions on Ω0. Indeed, if f̂ = f + I is
an arbitrary element of H0, we put f̂(ω) = f(ω) for all ω ∈ Ω0, which is correctly defined. In addition,
f̂−1(B) = (f|Ω0)−1(B) for all Borel sets B ⊂ C. In fact, S0 := {f|Ω0; f ∈ S} is a function space on
(Ω0,S0), where S0 = {A∩ Ω0; A ∈ S}. However, the map H0 3 f̂ 7→ f|Ω0 ∈ S0 is not injective, so
f̂(ω) = 0 for all ω ∈ Ω0 does not necessarily imply f̂(ω) = 0 (see Example 2).

Of course, when I = {f ∈S; f|Ω0 = 0}, then H0 may be regarded as a function space on Ω0.
(4) When a function space S and a compatible inner product 〈∗,∗〉0 are given, we use the notation

I,H0,H, and so on, in the sense from this remark, if not otherwise specified.

The construction in Remark 1 has an important particular case, to be discussed in the following.
Let S be a function space and let Λ : S(2) 7→ C be a linear map with the following properties:

(1) Λ(f̄) = Λ(f) for all f ∈S(2);
(2) Λ(|f|2) ≥ 0 for all f ∈S;
(3) Λ(1) = 1.

Adapting some terminology from [21] to our context (see also [29–31]), a linear map Λ with the
properties (1)-(3) is said to be a unital square positive functional, briefly a uspf.

A simple example of a uspf is given by a probability measure µ and a functions space S on (Ω,S),
consisting of square µ-integrable functions. Then the map S(2) 3 f 7→

∫
Ω
fdµ ∈ C is a uspf, as one

can easily see.



INTEGRAL REPRESENTATIONS 111

When the linear map Λ : S(2) 7→ C is a uspf, we may define a semi-inner product by the equality

〈f,g〉0 = Λ(fḡ), f,g ∈S,

which is easily seen to be compatible with S. Then, as in (2.2), the set

IΛ := {f ∈S; Λ(|f|2) = 0} = {f ∈S; Λ(fg) = 0 ∀g ∈S}
is a vector subspace of S. Moreover, the quotient H0Λ := S/IΛ is an inner product space, with the
inner product given by

〈f̂, ĝ〉 = Λ(fḡ), (2.8)
where f̂ = f + IΛ is the equivalence class of f ∈S modulo IΛ.

The Hilbert space obtained by completion of the inner product space H0Λ will be denoted by HΛ.
Of course, if S is finite dimensional, we have HΛ = S/IΛ.

When the uspf Λ : S(2) 7→ C is given, we shall use the notation IΛ, HΛ, f̂, with the meaning from
above, if not otherwise specified.

Problem 1. The (abstract) moment problem for a given uspf Λ : S(2) 7→ C, where S is a fixed function
space on (Ω,S), means to find necessary and sufficient conditions insuring the existence of a probability
measure µ, defined on S, such that S ⊂L2(Ω,µ) and Λ(f) =

∫
Ω
fdµ, f ∈S(2). When such a measure

µ exists, it is said to be a representing measure of Λ (with support) in Ω.

In the classical moment problem on spaces of polynomials, the role of the uspf Λ is played by the
so-called Riesz functional (see for instance [17]).

In some special cases, a uspf Λ : S(2) 7→ C may have an atomic representing measure in Ω, which (in
this text) means that there exists a finite subset ΩΛ = {ω1, . . . ,ωd}⊂ Ω consisting of distinct points,
and positive numbers λ1, . . . ,λd, with λ1 + · · · + λd = 1, such that Λ(f) =

∑d
j=1 λjf(ωj).

When we want to specify the number points {ω1, . . . ,ωd}, the corresponding atomic measure will
be sometimes called a d-atomic representing measure (of Λ in Ω). Of course, in this case we can write
Λ(f) =

∫
Ω
f(ω)dµ(ω), where µ is a probability measure defined on a σ-algebra S containing ΩΛ and

its subsets, such that µ({ωj}) = λj j = 1, . . . ,d. In particular, we may take as S the family of all
subsets of Ω.

When S is finite dimensional and the uspf Λ on S(2) has an arbitrary representing measure, then
one expects that this measure may be replaced by an atomic one. Such a property, going back to
Tchakaloff (see Corollary 2 in [25]), will be also discussed in this work (see Theorem 4).

The concept of representing measure can be also defined for a compatible semi-inner product 〈∗,∗〉0
of a function space S (see Definition 2).

2.2. Continuous Point Evaluations. A discussion concerning the point evaluations in the context
of function spaces of polynomials on Rn can be found in [30], Section 4 (see also [31] for a more general
framework). Some of the assertions of interest for us can be obtained from those in [30], with minor
modifications. Nevertheless, in the following, some results will be discussed in a more general context.

Remark 2. Let S be a function space on (Ω,S), endowed with a compatible semi-inner product
〈∗,∗〉0, and let H be the Hilbert space associated to (S,〈∗,∗〉0) (see Subsection 2.1). For every point
ω ∈ Ω, we denote by δω the point evaluation at ω, that is, δω(f) = f(ω), for every function f ∈S.

Of course, a point evaluation δω is said to be continuous if there exists a constant cω > 0 such that

|δω(f)| ≤ cω‖f‖0, f ∈S. (2.9)

A point evaluation is not necessarily continuous; see Example 2.
We denote by Z the subset of those points ω ∈ Ω such that δω is continuous.

Remark 3. We continue the discussion from Remark 1(3).
Let H0 = S/I be endowed with the inner product (2.3). If ω ∈ Z, we must have h(ω) = 0 for

all functions h ∈ I, as a consequence of (2.9). Consequently, we have the inclusion Z ⊂ ∩f∈IZ(f).



112 VASILESCU

Moreover, every point ω ∈Z induces a linear and continuous map δ̂ω : H0 7→ C, given by δ̂ω(f̂) = f(ω)
for all f ∈S, with f̂ = f + I, as the value f(ω) is constant on the equivalence class of f. In addition,

|δ̂ω(f̂)| = |f(ω)| ≤ cω‖f‖0 = cω‖f̂‖,
via (2.9), where ‖∗‖ is the norm associated to the inner product (2.3).

The linear functional δ̂ω can be extended to the completion H of H0, and keeping the same notation,
we have

|δ̂ω(φ)| ≤ cω‖φ‖, φ ∈H, ω ∈Z.

The next result is a slight extension of Lemma 6 from [30].

Lemma 1. If S is finite dimensional, we have
Z = ∩f∈IZ(f),

and Z ∈ S.

Proof. We already saw in Remark 3 that always Z ⊂∩f∈IZ(f).
Assume now that S is finite dimensional, so H = H0 is a finite dimensional Hilbert space. Let B be

an algebraic basis of IΛ, which is a finite set. Clearly, Z ⊂∩b∈BZ(b) = ∩f∈IZ(f).
Conversely, if ω ∈ ∩b∈BZ(b), then δω(b) = 0 for all b ∈ B, implying δω(f) = 0 for all f ∈ IΛ.

Therefore, δω induces a linear functional on the Hilbert space H, denoted by δ̂ω. The linear functional
δ̂ω is automatically continuous, and so δω is continuous. This shows ω ∈ Z. Consequently, Z =
∩b∈BZ(b), implying Z = ∩f∈IZ(f).

Finally, Z(b) = b−1({0}) ∈ S for all b ∈B showing that Z ∈ S. �

Remark 4. (1) The previous argument shows that if H0 is finite dimensional, so H = H0, we have
the equality Z = ∩f∈IZ(f). If, in addition, the space I has an at most countable algebraic basis, then
Z ∈ S.

(2) If I = {0} and S = H0 is finite dimensional, we must have Z = Ω.
(3) The previous lemma shows that the set Z extends the concept of algebraic variety of a moment

sequence, defined in the context of finite dimensional spaces of polynomials (see for instance (1.6)
from [10]).

Remark 5. (1) Continuing the discussion from Remark 1(3), if I = {f ∈S; f|Z = 0}, and S is finite
dimensional, the space H = H0 may be regarded as a function space on Z because from f|Z = 0, so
‖f‖0 = 0, we obtain f̂ = 0.

(2) Note that for every ω ∈Z there exists a vector νω ∈H such that δ̂ω(φ) = 〈φ,νω〉 for all φ ∈H,
via a well-known theorem by Riesz. In fact, applying the Riesz theorem firstly on RH, we deduce that
νω ∈RH.

Definition 2. Let S be a function space on (Ω,S), endowed with a compatible semi-inner product
〈∗,∗〉0. From now on, such a pair (S,〈∗,∗〉0) will be designated as a quasi-Hilbert function space (briefly,
a qHfs). When the function space S on Ω is actually a Hilbert space, endowed with a compatible inner
product 〈∗,∗〉, the pair (S,〈∗,∗〉) it will be called a Hilbert function space (briefly, a Hfs).

We say that the semi-inner product 〈∗,∗〉0 has a representing measure if there exists a probability
measure µ on S such that

〈f,g〉0 =
∫

Ω

f(ω)g(ω)dµ(ω), f,g ∈S. (2.10)

We say that the semi-inner product 〈∗,∗〉0 has an atomic representing measure in Ω if there exists
a finite subset Ω0 = {ω1, . . . ,ωd} ⊂ Ω consisting of distinct points, and positive numbers λ1, . . . ,λd,
with λ1 + · · · + λd = 1, such that

〈f,g〉0 =
d∑
j=1

λjf(ωj)g(ωj), f,g ∈S. (2.11)

When the support of a given atomic representing measure consists of d points, it will be sometimes
called a d-atomic representing measure. As usually, the points {ω1, . . . ,ωd} are called the nodes, and
the numbers {λ1, . . . ,λd} are called the weights of the measure µ.



INTEGRAL REPRESENTATIONS 113

A slightly more general question than Problem 1 is the following.

Problem 2. Let (S,〈∗,∗〉0) be a qHfs on (Ω,S). Find necessary and sufficient conditions to insure
the existence of a representing measure of the semi-inner product 〈∗,∗〉0.

Using direct arguments, as well as methods from the theory of moment problems in functions spaces
of polynomials, we shall try to give some answers to Problem 2 in the next sections.

The following lemma is similar to Lemma 7 from [30] (see also [9] for a precursor of this result).

Lemma 2. Suppose that the compatible semi-inner product 〈∗,∗〉0 of the function space S on Ω has
an atomic representing measure µ. Then supp(µ) ⊂Z.
Proof. We use the notation from Definition 2. If ωk ∈ Ω0, we have:

|f(ωk)|2 ≤
1

λk

d∑
j=1

λj|f(ωj)|2 =
1

λk
‖f‖20, f ∈S,

for all k = 1, . . . ,d, showing that supp(µ) ⊂Z. �
The next definition adapts some concepts from [14] to our context.

Definition 3. Let (S,〈∗,∗〉0) be a qHfs. We say that the semi-inner product 〈∗,∗〉0 is weakly consistent
if whenever f ∈S in null on Z, it follows that 〈f, 1〉0 = 0.

We say that 〈∗,∗〉0 is consistent if whenever
∑
k∈K fkgk = 0 on Z it follows

∑
k∈K〈fk,gk〉0 = 0,

where fk,gg ∈RS, k ∈ K, K finite.
Remark 6. (1) Consider a qHfs (S,〈∗,∗〉0). It follows from Lemma 2 that a necessary condition for
the existence of an atomic representing measure for the semi-inner product 〈∗,∗〉0 on Ω is Z 6= ∅.

(2) If the semi-inner product 〈∗,∗〉0 is consistent, it is also weakly consistent.
(3) If the semi-inner product 〈∗,∗〉0 has an atomic representing measure, then it is consistent. The

converse is not true, in general (see [14]).
(4) Let S be a function space on a set Ω and let Λ : S(2) 7→ C be a uspf. Of course, the previous results

of this subsection applies to this case, when the semi-inner product is given by 〈f,g〉0 = Λ(fḡ〉, f,g ∈S.
We recall that we use the notation HΛ,H0Λ,IΛ,ZΛ instead of H,H

0,I,Z, respectively. In this case
(see also [14] for spaces of polynomials), if 〈∗,∗〉0 is (weakly) consistent, we say that Λ is (weakly)
consistent. Also note that Λ is consistent if and only if whenever f ∈S(2) satisfies f|ZΛ = 0, we have
Λ(f) = 0. A similar property characterizes the weak consistency of Λ.

If S is finite dimensional, the uspf Λ is consistent if and only if Λ ∈ span{δ(2)ω ;
ω ∈ ZΛ}, where δ

(2)
ω is the point evaluation at ω ∈ Ω, regarded as a functional on S(2). Indeed,

if Λ ∈ span{δ(2)ω ; ω ∈ ZΛ}, then Λ is clearly consistent. Conversely, because span{δ
(2)
ω ; ω ∈ ZΛ} is in

the dual of S(2), which is finite dimensional, if Λ /∈ span{δ(2)ω ; ω ∈ZΛ}, we can find a function f0 ∈S(2)
which is null on ZΛ but with Λ(f0) 6= 0, which is not possible.

Similarly, Λ is weakly consistent if and only if Λ|S ∈ span{δω; ω ∈ZΛ}.
Finally, note that ω /∈ ZΛ for some ω ∈ Ω, if and only if there exists a sequence (fn)n in S such

that Λ(|fn|2) = 1 and |fn(ω)|→∞(n →∞).
Example 2. We consider the the uspf Λ : P14 7→ C, where P14 is the space of of polynomials in one real
variable t, with complex coefficients, of degre ≤ 4, where Λ(tk) = 1, k = 0, 1, 2, 3, Λ(t4) = 2, extended
by linearity. This is a simple example showing that there are truncated moment problems with no
representing measure in R (see [30], Example 3; see also [13]). It can be also used in connection with
Remark 5(1), and with Remark 6(4) as well.

As shown in [30], we have IΛ = {p(t) = a − at; a ∈ C}. Then clearly, ZΛ = {1}, so the point
evaluation δ1(p) = p(1), p ∈P12 , must be continuous, by Lemma 1. In fact, if p(t) = a + bt + ct2 with
a,b,c ∈ R arbitrary,

p(1)2 = (a + b + c)2 ≤ Λ(p2) = ((a + b + c)2 + c2),
and thus, when a,b,c ∈ C, we deduce that |p(1)|2 ≤ Λ(|p|2), p ∈ P12 , showing, directly, that δ1 is
Λ-continuous.

On the other hand, if θ ∈ R, θ 6= 1, setting pn(t) = 1 + n(1 − t), n ≥ 1, we obtain pn(θ) =
1 + n(1 −θ) →±∞ as n →∞, while Λ(p2n) = 1 for all n ≥ 1, as in Remark 6(4).



114 VASILESCU

Note that there are nonnul elements f̂ ∈ HΛ = P12 /IΛ such that δ̂1(f̂) = 0. Indeed, as one can
easily see (see also [31], Example 3), we have HΛ = {u1̂ + vt̂2; u,v ∈ C}. Then δ̂1(1̂ − t̂2) = 0, while
1̂ − t̂2 6= 0. In other words, the Hilbert space associated to a qHfs (S,〈∗,∗〉0) is not a Hilbert function
space.

Note also that, because ZΛ = {1}, the only possible atomic representing measure for Λ would be
δ1, via Lemma 2. But this is impossible because, for instance, δ1(t

4) = 1, while Λ(t4) = 2.

As in the case of function spaces consisting of polynomials, an extremal situation associated with
the consistency insure the existence of an atomic representing measure (see [14], Theorem 1.3).

Theorem 1. Let (S,〈∗,∗〉0) be a finite dimensional qHfs on Ω. Assume that d := card(Z) = dim(H).
The semi-inner product 〈∗,∗〉0 has a d-atomic representing measure if and only if it is consistent.

Proof. The consistency of 〈∗,∗〉0 allows us to replace the set Ω by the set Z. Indeed, defining the
function space SZ as the space {f|Z; f ∈S}, and putting

〈f|Z,g|Z〉0,Z := 〈f,g〉0, f,g ∈S,

we obtain a semi-inner product 〈∗,∗〉0,Z on SZ. Indeed, we note that if f|Z = 0 or g|Z = 0, we have
〈f,g〉0 = 0 by the consistency of 〈∗,∗〉0. This shows that the map 〈∗,∗〉0,Z is well defined, and it is
clearly a semi-inner product. In addition, if f|Z = 0, the consistency implies f ∈ I, showing that
I = {f ∈S,f|Z = 0}. Hence we have a natural map

H3 f̂ 7→ f|Z ∈ C(Z) := {h : Z 7→ C},

which is a linear isomorphism, because it is clearly injective and surjective too, thanks to the assumption
card(Z) = dim(H).

We write now Z = {ζ1, . . . ,ζd}, and put χj := χ{ζj}, that is, the characteristic function of the
set {ζj}, j = 1, . . . ,d. Let also b̂j be the unique element of H with bj|Z = χj. Because we have
bjbk|Z = χjχk = 0 if j 6= k, we must have 〈bj,bk〉0 = 0. Similarly, χ2j = χj implies 〈bj,bj〉0 = 〈 bj, 1〉0.

Note that f|Z =
∑d
j=1 f(ζj)χj, so f −

∑d
j=1 f(ζj)bj is null on Z for all f ∈S. Thus

〈f,g〉0 = 〈
d∑
j=1

f(ζj)bj,

d∑
k=1

g(ζk)bk〉 =
d∑
j=1

λjf(ζj)g(ζj), f,g ∈S,

where λj = 〈bj, 1〉 for all j, which is a representation of the semi-inner product 〈∗,∗〉0 via a d-atomic
measure.

Conversely, if the semi-inner product 〈∗,∗〉0 has a d-atomic representing measure with d := dim(H),
then 〈∗,∗〉0 is clearly consistent. �

Remark With the previous notation, we must have 〈b̂j, b̂k〉 = 0, 〈b̂j, b̂j〉 = 〈b̂j, 1̂〉, j,k = 1, . . . ,d, j 6=
k, and f̂ =

∑d
j=1 f(ζj)b̂j for all f̂ ∈H. In other words, {b̂1, . . . , b̂d} is an orthogonal basis of H, con-

sisting of idempotents (see Subsection 2.5 for details).

2.3. Generators of Function Spaces. Let S be a function space on (Ω,S). We assume that there
exist an n-tuple θ = (θ1, . . . ,θn) of elements of RS, and an integer m ≥ 1, such that such that the
family Θm := {θα; |α| ≤ m, α ∈ Zn+} spans the space S.

When such a pair (θ,m) exists, we shortly say that the function space S is m-generated by θ.
Clearly, in this case S is of finite dimension, and the family Θ2m spans the space S(2). In particular,
S is 1-generated by θ if and only if S is the span of Θ1.

Also note that if S be a function space that is m-generated by an n-tuple θ = (θ1, . . . ,θn) of elements
of RS, and if Λ : S(2) 7→ C is a uspf, then the Hilbert space HΛ must be of finite dimension less or
equal to the cardinal of the set Θm.

As a matter of fact, if S is a function space on Ω that is m-generated by an n-tuple θ = (θ1, . . . ,θn)
of elements of RS, we must have the equality, S = {p◦θ; p ∈Pnm}, where θ is regarded as a function
from Ω into Rn.

In particular, Pnm is a function space m-generated by by the n-tuple t := (t1, . . . , tn), where t1, . . . , tn
are the independent variables of Rn.



INTEGRAL REPRESENTATIONS 115

An important particular case of finitely generated function spaces is related to the so called truncated
K-moment problem (see [12]), which means, for a fixed closed set K ⊂ Rn, to look for a representing
measure of a given uspf Λ : Pn2m,K 7→ C, where P

n
m,K := {p|K; p ∈ P

n
m}. Clearly, Pnm,K is a function

space on K, m-generated by n-tuple t|K := (t1|K,.. . , tn|K).
To exhibit another example, fix an n-tuple θ = (θ1, . . . ,θn) in Pd. Then S := {p◦ θ; p ∈ Pnm} is a

function space on Rd. In fact, if g = max1≤j≤n{degθj}, we have S ⊂Pdmg.
Let again S be a function space on (Ω,S), and let n-tuple θ = (θ1, . . . ,θn) of elements of RS. When

S spanned by the set {θα; α ∈ Zn+}, we say that S is ∞-genarated by θ. In thes case, S is actually a
unital algebra. Of course, in this case we have S = {p◦θ; p ∈Pn}.

2.4. Reduction to Finite Dimensional Spaces. We exhibit, in the following, a result allowing us
to prove the existence of a representing measure for a uspf Λ, defined on a space S of continuous
functions on a locally compact metrisable space, using representing measures of restrictions of Λ on
some finite dimensional subspaces of S. Such a reduction result goes back to the paper [23], where it
is proved in the context of spaces of polynomials. It is also approached, in a more abstract framework,
in [28].

As mentioned above, in this subsection the basic space Ω is supposed to be a locally compact metric
space. We denote by Ω∞ = Ω ∪ {∞} the one-point compactification of Ω, which is compact and
metrisable. If ρ is a fixed metric on Ω, a given sequence (ωk)k≥1 is said to tend to infinity, and we
write limk→∞ωk = ∞, if limk→∞ρ(ω0,ωk) = ∞ for some (any) point ω0 ∈ Ω.

Let S be a function space on Ω, consisting of continuous functions. We suppose that there exists
an n-tuple θ = (θ1, . . . ,θn) of functions of RS, separating the points of Ω and unbounded on Ω, that
is, limω→∞θj(ω) = ∞ for all j = 1, . . . ,n. We also assume that the space S is ∞-generated by the θ,
so S is a commutative unital algebra, closed under complex conjugation.

Next, we define the functions

qk(ω) = (1 + θ1(ω)
2 + · · · + θn(ω)2)−k

where k ≥ 0 is an integer. Then we set
Tk = {f ∈S; lim

ω→∞
qk(ω)f(ω) exists},

which is a function space satisfying 1,q−1k ∈ Tk, and Tk ⊂ Tk+1, for all k. Moreover, as we have
S = {p◦θ; p ∈Pn}, and |θα11 · · ·θ

αn
n |2 ≤ qk(ω)−1 when |α1|+· · ·+|αn| ≤ k, it follows that S = ∪k≥0Tk.

We now consider the algebra C(Ω∞), consisting of continuous functions on the compact set Ω∞.
Moreover, the set Q := {qk; k ≥ 0} is a multiplicative family in C(Ω∞), provided that each function
qk (k ≥ 1) is extended with 0 at ∞ (see [28] for details). In addition, we may regard Tk as a subspace
of q−1k C(Ω∞).

Under the hypothesis on Ω and θ from above, we have the following.

Theorem 2. A uspf Λ : S 7→ C has a representing measure with support in Ω if and only if Λ(q−1k ) > 0
and

|Λ(f)| ≤ Λ(q−1k ) sup
ω∈Ω
|qk(ω)f(ω)|, f ∈Tk, k ≥ 0.

Proof. It Λ has a representing measure, say µ, because q−1k ≥ 1 it follows Λ(q
−1
k ) ≥ Λ(1) = 1 for all

k ≥ 0. Therefore
|Λ(f)| ≤

∫
Ω

|f|dµ ≤ Λ(q−1k ) sup
ω∈Ω
|qk(ω)f(ω)|,

for all f ∈Tk.
Conversely, we shall apply Theorem 3.7 from [28] to the space S ⊂ C(Ω∞)/Q and the map Λ. First

of all, we note that Λ(q−1k ) ≥ Λ(1) = 1 for all k ≥ 0 because q
−1
k is equal to 1 plus a sum of squares.

Secondly, the space C(Ω∞) is separable. In fact, the functions q1(θ),θ1q1(θ), . . . ,θnq1(θ) separate the
points of Ω∞. Then the Weierstrass-Stone theorem implies the density of the unital algebra generated
by this family in C(Ω∞).

Note that qk(∞) = 0 for all k ≥ 1, and so {∞} is the union of the zeros of all functions from Q.
We also have 1,q−1k ∈Tk ⊂ q

−1
k C(Ω∞) for all k ≥ 0. Moreover, Tk1 ⊂Tk2 whenever k1 ≤ k2, and this

is equivalent to the fact that q−1k1 divides q
−1
k2

.



116 VASILESCU

If Λk = Λ|Tk, putting ‖f‖∞,k = supt∈Ω |qk(t)f(t)|,f ∈ C(Ω∞)/qk, which is precisely the norm on
C(Ω∞)/qk (see [28]), the conditions from the statement imply the estimates ‖Λk‖≤ Λ(q−1k ). Because
q−1k ∈Tk, and its norm is one, we must have ‖Λk‖ = Λ(q

−1
k ). According to Theorem 3.7 from [28], this

implies the existence of a positive extension M of Λ to C(Ω∞)/Q. The proof of Theorem 3.7 from [28]
shows the existence of a representing measure of M, whose support is precisely in Ω (see also Remark
3.8(1) from [28]). �

Remark 7. Theorem 2 can also be applied when the spaces (Tk)k≥0 are replaced by the simpler spaces
(Sk)k≥0, with Sk = {f ∈ S; f = p ◦ θ,p ∈ Pnk}. In fact, assuming that for the uspf Λ : S 7→ C the
restriction Λk = Λ|S2k has a representing measure for each k ≥ 0, we may obtain the assertion in the
following way. The finite dimension of the involved spaces allows us to find, for every integer k ≥ 0, an
integer rk ≥ 0 such that Tk ⊂ S2rk . Then the representing measure of Λ|S2rk induces a representing
measure for Λ|Tk for all k ≥ 0, which allows the application of Theorem 2.

Using the preceding remark, we deduce easily the following:

Theorem 3. Let Ω be a locally compact metric space, let S be a function space on Ω consisting of
continuous functions, and let 〈∗,∗〉0 be a semi-inner product on S. Let also θ = (θ1, . . . ,θn) be a
tuple of functions from RS, unbounded on Ω and separating its points, let Sm be the function space
m-generated by θ and let 〈∗,∗〉0m be the restriction of the semi-inner product 〈∗,∗〉0 to Sm (m ≥ 1).
The semi-inner product 〈∗,∗〉0 has a representing measure on Ω if and only if the semi-inner product
〈∗,∗〉0m has a representing measure on Ω for every m ≥ 1.

Theorem 3 shows that solving Problem 2 on finite dimensional function space leads to solutions
of Problem 2 in a large class of infinite dimensional function spaces, including the classical ones in
spaces of polynomials. For this reason, in the next sections we shall mainly deal with finite dimensional
function spaces.

As noticed in several works (see for instance [9]), another important feature in the context of finite
dimensional function spaces is that the existence of a representing measure of a given semi-inner
product implies the existence an atomic representing measure, as presented in the following.

Theorem 4. Let (S,〈∗,∗〉0) be a qHfs on Ω, m-generated by the n-tuple θ = (θ1, . . . ,θn) Suppose that
〈∗,∗〉0 has a representing measure. Then 〈∗,∗〉0 has an atomic representing measure.

Proof. We consider the set Zn,2m+ := {α ∈ Zn+; |α| ≤ 2m}, endowed with the lexicographic order. In
addition, we assign to each integer j ∈{1, 2, . . . ,nm}, where nm is the cardinal of Z

n,2m
+ , a multi-index

α(j) ∈ Zn,2m+ with j ≤ k iff α(j) ≤ α(k), and α(1) = 0. In this way we have a (Borel measurable) map
φ : Ω 7→ Rnm given by φ(ω) = (θα(1)(ω), . . . ,θα(nm)(ω)) ∈ Rnm .

Now assume that 〈∗,∗〉0 has a representing measure, so it has the form 〈f,g〉0 =
∫

Ω
fḡdµ, f,g ∈S,

where µ is a positive Borel measure on Ω, with µ(Ω) = 1. Let ν be the measure induced by the measure
µ and the Borel map φ. Writing α(j) = α′(j) + α′′(j) with |α′(j)|, |α′′(j)| ≤ m, we have:∫

Rnm
|xj|dν(x) =

∫
Ω

|xj ◦φ|dµ ≤
∫

Ω

|θα(j)|dµ ≤‖θα
′(j)‖0‖θα

′′(j)‖0 < ∞,

for all j = 1, . . . ,nm, where x1, . . . ,xnm are the coordinate functions in R
nm . This shows that we

may apply Corollary 2 from [4] to deduce the existence of a positive integer d ≤ nm, a set of points
ω1, . . . ,ωd in the support of the measure µ, and positive numbers λ1, . . . ,λd, such that

∑d
j=1 λj = 1,

and ∫
Ω

θαdµ =

d∑
j=1

λjθ
α(ωj), α ∈ Z

n,2m
+ .

From this equality, we infer easily that 〈∗,∗〉0 has an atomic representing measure. �

2.5. Quasi-Hilbert Function Spaces and Idempotents. The concepts of quasi-Hilbert function
space (briefly, qHfs) and Hilbert function space (briefly, Hfs) are those introduced by Definition 2.

We have already noted that the Hilbert space associated to a qHfs (S,〈∗,∗〉0) is not necessarily a
Hilbert function space (see Example 2). We should mention that our concept of Hilbert function space
is slightly different from the homonymous concept from [1]. In fact, the concept of Hilbert function



INTEGRAL REPRESENTATIONS 117

space, as defined in [1], is often called a reproducing kernel Hilbert space (see for instance [2]). Unlike
in [1], the point evaluations on a Hilbert function space in our sense are not necessarily well defined
but the inner product of such a space must be compatible.

We introduce in the following the concept of element idempotent related to a given function space
S, endowed with a compatible semi-inner product 〈∗,∗〉0. This is an extension of the concept of
indempotent with respect to a uspf, introduced in [30].

Definition 4. Let (S,〈∗,∗〉0) be a qHfs, and let H be the Hilbert space aassociated to (S,〈∗,∗〉0),
whose inner product is denoted by 〈∗,∗〉, and whose norm is ‖∗‖. An element ι ∈ RH is said to be
an idempotent (associated to S) if

‖ι‖2 = 〈ι, 1̂〉. (2.12)
We set ID(H) := {ι ∈RH;〈ι, 1̂〉 6= 0}, that is, the family of all nonnull idempotents of H.
Example 3. Let µ be a probability measure on (Ω,S), and let S be a function space on (Ω,S),
consisting of square µ-integrable functions, so its semi-inner product 〈∗,∗〉0 can be obtained by re-
stricting the semi-inner product of L2(Ω,µ) to S. Clearly, the subspace I consists of those functions
from S, which are null µ-almost everywhere. As usually, let L2(Ω,µ) be the Hilbert space consisting of
equivalence classes of square µ-integrable functions on Ω. Assume that the space H0 = S/I is dense in
L2(Ω,µ). Then L2(Ω,µ) is the Hilbert space associated to (S,〈∗,∗〉0). If χB denotes the characteristic
function of a given set B ∈ S, the class of χB is clearly an idempotent in L2(Ω,µ) associated to S,
but χB does not necessarily belong to S.

The following result is, in fact, an extension Lemma 4 from [30], with a proof in the present context.

Lemma 3. Assume H to be separable, and let {ηj}j∈J be an orthonormal family in RH such that
〈ηj, 1̂〉 6= 0 for each j ∈ J, for some J ⊂ N. Then the set {ιj}j∈J is an orthogonal family in ID(H),
where ιj = 〈ηj, 1̂〉ηj for all j ∈ J. If the set {ηj}j∈N is an orthonormal basis of H, the set {ιj}j∈N is
an orthogonal basis of H in ID(H). Consequently,

φ =

∞∑
j=1

〈ιj, 1̂〉−1〈φ,ιj〉ιj, φ ∈H,

where the series is convergent in H. In particular 1̂ =
∑∞
j=1 ιj, where the series is convergent in H.

Proof. Setting ιj = 〈ηj, 1̂〉ηj for all j ∈ J, we have

‖ιj‖2 = 〈〈ηj, 1̂〉ηj,〈ηj, 1〉ηj〉 = 〈ηj, 1̂〉2 = 〈ιj, 1̂〉 6= 0,
showing that {ιj}j∈J is a family in ID(H). In addition,

〈ιj, ιk〉 = 〈ηj, 1̂〉〈ηk, 1̂〉〈ηj,ηk〉 = 0, j,k ∈ J, j 6= k,
so the family {ηj}j∈J is orthogonal.

If {ηj}j∈N is an orthonormal basis, we must have

φ =

∞∑
j=1

〈φ,ηj〉ηj =
∞∑
j=1

〈ιj, 1̂〉−1〈φ,ιj〉ιj, f ∈S,

and the series is clearly convergent in H. In particular, we must have 1̂ =
∑∞
j=1 ιj, where the series is

convergent in H. �
Corollary 1. If H is finite dimensional and {η1, . . . ,ηd}⊂RH is an orthonormal basis with 〈ηj, 1̂〉 6=
0, j = 1, . . . ,d, the set {〈η1, 1̂〉η1, . . .〈ηd, 1̂〉ηd} is an orthogonal basis of H consisting of idempotents.
Moreover,

〈η1, 1̂〉η1 + · · · + 〈ηd, 1̂〉ηd = 1̂.
Corollary 2. Assume that H is finite dimensional. Then there are functions b1, . . . ,bd ∈ RS such
that ‖bj‖20 = 〈bj, 1〉0 > 0, 〈bj,bk〉0 = 0 for all j,k = 1, . . . ,d, j 6= k,

∑d
j=1〈bj, 1〉0 − 1 ∈ I, and every

f ∈S can be uniquely represented as

f =

d∑
j=1

〈bj, 1〉−10 〈f,bj〉0bj + f0,



118 VASILESCU

with f0 ∈I and d = dimH.

Using Lemma 3, we obtain the next result (which extends Theorem 1 from [30]).

Theorem 5. Let (S,〈∗,∗〉0) be a qHfs, and let H be the associated Hilbert space. If H is separable
and of dimension ≥ 2, it has infinitely many orthogonal bases consisting of idempotent elements.

Proof. We may work in an abstract framework. Replacing H by RH, we may assume, with no loss of
generality, that H is a separable real Hilbert space, endowed with the real inner product 〈∗,∗〉, and with
the corresponding norm ‖∗‖. Let {λj}j∈N be a sequence of positive numbers such

∑
j∈N λ

2
j = 1. Let

also {ηj}j∈N be an orthonormal basis of H, and let η =
∑
j∈N λjηj, so ‖η‖ = 1. Then 〈η,ηj〉 = λj > 0

for all j. Setting ζj := 〈η,ηj〉ηj, j ≥ 1, we obtain an orthogonal basis {ζj}j∈N such that ‖ζj‖2 = 〈ζj,η〉
for all j ≥ 1.

Next, we fix an element e ∈ H such that ‖e‖ = 1, and choose an orthogonal transformation O of
H such that Oη = e. Putting, ej = Oηj, and ιj = 〈e,ej〉ej = Oζj (j ≥ 1), the family {ιj}j∈N is an
orthogonal basis of H. Moreover, ‖ιj‖2 = 〈ιj,e〉 for all j ≥ 1.

Going back to our initial case, for e = 1̂ we obtain an orthogonal basis {ιj}j∈N of H consisting of
idempotents. The construction from above shows that there are infinitely many possibilities to obtain
such a basis {ιj}j∈N. �

Remark 8. Of course, the assertions from this subsection apply when the function space S is endowed
with a usps Λ : S(2) 7→ C, having the semi-inner product given by 〈f,g〉0 = Λ(fḡ), f,g ∈S. Note that
if S is finite dimensional, an element f̂ ∈RHΛ is an idempotent if and only if Λ(f2) = Λ(f). For this
reason, such an element may be called a Λ-idempotent, as in [30].

3. USPF’s versus Semi-Inner Products

As mentioned above, the main aim of this work is to give necessary and sufficient conditions insur-
ing the existence of integral representations of some given semi-inner products on function spaces of
measurable functions; in other words, to look for solutions to Problem 2. One possible approach to
Problem 2 is to adapt techniques from the theory of moment problems. We refer especially to [9, 10] as
well as to [29–31]. To this aim, it is necessary to clarify the connection between semi-inner products
and unital square positive functionals, which is the main concern of this section.

Remark 9. Let (S,〈∗,∗〉0) be a qHfs. We want to relate the semi-inner, product 〈∗,∗〉0 with a uspf
Λ : S(2) 7→ C in a natural and unique way. Roughly speaking, we want to have Λ(fḡ) = 〈f,g〉0, for all
f,g ∈S. In fact, we have the following.

Lemma 4. Let S be a function space on Ω. The following assertions are equivalent:
(1) the function space S has a compatible semi-inner product 〈∗,∗〉0 satisfying∑

k∈K

fkgk = 0 on Ω =⇒
∑
k∈K

〈fk,gk〉0 = 0 (3.1)

for all fk,gk ∈RS, k ∈ K, K finite;
(2) the function space S has a uspf Λ : S(2) 7→ C.

Proof. (1) =⇒ (2). Choosing fk,gk ∈S, k ∈ K, K finite, we write fk = f′k + if
′′
k , ḡk = g

′
k − ig

′′
k , with

f′k,f
′′
k ,g
′
k,g
′′
k ∈ RS, k ∈ K. Assuming

∑
k∈K fkḡk = 0 we must have

∑
k∈K(f

′
kg
′
k + f

′′
k g
′′
k ) = 0 and∑

k∈K(f
′′
k g
′
k −f

′
kg
′′
k ) = 0 According to (3.1), we deduce that∑

k∈K

(〈f′k,g
′
k〉0 + 〈f

′′
k ,g
′′
k〉0) = 0 and

∑
k∈K

(〈f′′k ,g
′
k〉0 −〈f

′
k,g
′′
k〉0) = 0.

Because the seminorm 〈∗,∗〉0 is compatible, we infer that
∑
k∈K〈fk,gk〉0 = 0. Consequently, fixing an

arbitrary element F =
∑
k∈k fkḡk ∈S

(2), we put

Λ(F) =
∑
k∈K

〈fk,gk〉0. (3.2)



INTEGRAL REPRESENTATIONS 119

The previous argument shows that this definition does not depend on the particular representation of
F , implying that the map Λ : S(2) 7→ C is linear. In addition, Λ(1) = 〈1, 1〉0 = 1,

Λ(F̄) =
∑
k∈K

〈f̄k, ḡk〉0 =
∑
k∈K

〈fk,gk〉0 = Λ(F), F =
∑
k∈K

fkḡk ∈S(2),

and Λ(|f|2) = 〈f,f〉0 ≥ 0 for all f ∈S.
Conversely, given a uspf Λ : S(2) 7→ C, the formula 〈f,g〉0 = Λ(fḡ), f,g ∈ S defines a semi-inner

product compatible with the structure of S, as noticed in Subsection 2.1. Therefore, we also have
(2) =⇒ (1). �
Remark. Note that, in a qHfs S whose semi-inner product 〈∗,∗〉0 has the property (3.1), taking
f,g ∈S and h ∈RS such that fh,gh ∈S, we must have 〈fh,g〉0 = 〈f,gh〉0.
Proposition 1. Let S be a function space on Ω. The following assertions are equivalent:

(1) the space S is a qfHs on Ω, whose compatible seminorm ‖∗‖0 has the property∑
k∈K

f2k =
∑
l∈L

g2l on Ω =⇒
∑
k∈K

‖fk‖20 =
∑
l∈L

‖gl‖20, (3.3)

for all fk,gl ∈RS, k ∈ K, l ∈ L, K,L finite;
(2) the space S has a uspf Λ : S(2) 7→ C.
Moreover, the uspf Λ is uniquely determined by the semi-inner product 〈∗,∗〉0 with the property

(3.3).

Proof. To prove the equivalence of the conditions (1) and (2) from the statement, it is enough to
verify that condition (3.3) is equivalent to condition (3.1). To show that, we simply apply the obvious
polarization formula: ∑

k∈K

ukvk =
1

4

∑
k∈K

[(uk + vk)
2 − (uk −vk)2], (∗),

and its corresponding version∑
k∈K

〈uk,vk〉0 =
1

4

∑
k∈K

‖uk + vk‖20 −‖uk −vk‖
2
0], (∗∗),

valid for all uk,vk ∈ RS, k ∈ K, K finite. We also note that in (3.3) we may always assume K = L,
with no loss of generality.

The details of this verification, and the uniqueness of Λ as well, are left to the reader. �
Remark. Going back to Problem 2, Proposition 1 gives an answer to the question how to associate
a qHfs with a uspf, in order to approach this problem as a moment problem.

Definition 5. Let (S,〈∗,∗〉0) be a qHfs space. We say that the semi-inner product 〈∗,∗〉0 is expandable
if it has the property (3.3).

If 〈∗,∗〉0 is expandable, the unique uspf Λ : S(2) 7→ C given by Theorem 3.3 is said to be associated
to 〈∗,∗〉0.
Example 4. Let µ be a probability measure on (Ω,S), and let S = L2(Ω,µ). Then S is a qHfs, whose
natural semi-norm ‖f‖0 = (

∫
Ω
|f|2dµ)1/2, f ∈S, is expandable. Moreover, it is easily seen that S(2) =

L1(Ω,µ), and the uspf Λ : S(2) 7→ C, given by Theorem 1, is precisely Λ(f) =
∫

Ω
fdµ, f ∈L1(Ω,µ).

Example 5. The multidimensional Hamburger moment problem can be also related to Problem 2 in
the following way. The basic function space is in this case Pn on Rn, consistning of all polynomilas in
t = (t1, . . . , tn). First of all, we fix a multi-sequence of real numbers γ = (γα)α∈Zn

+
with the property∑

α,β

aαāβγα+β ≥ 0 (3.4)

for all finite multi-sequences (aα)α of complex numbers. In fact, we work in this example only with
finite sums, and the multi-indices are from Zn+, with the order α ≤ β if β −α ∈ Zn+.

For any two polynomials p(t) =
∑
α cαt

α, q(t) =
∑
β dβt

β with complex coefficients we put

〈p,q〉0 =
∑
α,β

cαd̄βγα+β. (3.5)



120 VASILESCU

The choice of the multi-sequence γ shows that the assignment (3.5) is a semi-inner product on Pn. To
illustrate Lemma 4, we shall verify directly that this semi-inner product is also expandable.

Let
∑
j pjq̄j = 0, with pj(t) =

∑
α cj,αt

α, qj(t) =
∑
β dj,βt

β. Then

∑
j

pjq̄j =
∑
j

∑
α,β

cj,αd̄j,βt
α+β =

∑
σ


 ∑
j,α≤σ

cj,αd̄j,σ−α


tσ = 0.

Therefore,
∑
j,α≤σ cj,αd̄j,σ−α = 0 for all σ. On the other hand,

∑
j

〈pj,qj〉0 =
∑
j

∑
α,β

cj,αd̄j,βγα+β =
∑
σ


 ∑
j,α≤σ

cj,αd̄j,σ−α


γσ = 0.

Although condition (3.4) for n ≥ 2 does not necessarily impliy the existence of a representing
measure for the semi-inner product (3.5), it suffices to show that (3.5) is expandable.

Example 6. For some details concerning this example we cite the work [3].
Let B be the open unit ball in Rn, let S be the boundary of B, and let h2(B) be the real Hilbert space

of all harmonic functions in B, which are Poisson transforms of real-valued functions from L2(S,σ),
where σ is the unique Borel probability measure on S that is rotation invariant. We denote by H the
space h2(B) + ih2(B), which is a Hilbert space of complex-valued harmonic functions, and it is also a
function space on B. The inner product of H is given by

〈f,g〉H =
∫
S

f]g]dσ, f,g ∈ H,

where f] denotes the unique element from L2(S,σ) whose Poisson transform is f. This inner product
is clearly compatible with the function space H.

Assuming now that
∑
j∈J f

2
j =

∑
k∈K g

2
k for some fj,gk ∈ RH, j ∈ J,k ∈ K, J,K finite, we infer

that
∑
j∈J(f

]
j )

2 =
∑
k∈K(g

]
k)

2, so∑
j∈J
‖fj‖2H =

∫
S

∑
j∈J

(f
]
j )

2dσ =

∫
S

∑
k∈K

(g
]
k)

2dσ =
∑
k∈K

‖gk‖2H.

In other words, the norm of H is expandable, allowing us to define a uspf on H(2), in a natural way.

Example 7. Let W21 be the Sobolev space consisting of all complex-valued functions on the interval
[0, 1], which are absolutely continuous and whose derivatives are square integrable, endowed with the
norm

‖f‖21 =
∫ 1

0

(|f(t)|2 + |f′(t)|2)dt, f ∈W21 .

This is a reproducing kernel Hilbert space, as shown in [1], Example 2.7. It is also a function space on
[0, 1] and its inner product is compatible. Nevertheless, the norm is not expandable. Indeed, taking
f(t) = (2 + 2t2)1/2, f1(t) = 1 + t, f2(t) = 1−t, we have f2 = f21 + f22 , while ‖f‖21 6= ‖f1‖21 +‖f2‖21. To
see this, it is enough to remark that f′(t)2 = 2t2(1 + t2)−1, and so ‖f‖21 is an expression depending
explicitly on π, while ‖f1‖21 + ‖f2‖21 is a rational number. Consequently, the norm of W21 is not
expandable. In particular, Problem 2 has no solution for the Hfs (W21 ,‖∗‖1).

Example 8. We now give a more abstract example.
Let S be a Hfs on Ω endowed with a norm ‖∗‖, given by a compatible inner product 〈∗,∗〉. Assume

that there exists a family of functions {v1, . . . ,vd} in RS, and a family of points {ω1, . . . ,ωd} in Ω
such that

〈vj, 1〉 6= 0, f(ωj) = 〈vj, 1〉−1〈f,vj〉, and f =
d∑
j=1

f(ωj)vj, ∀f ∈S.

If, moreover,

〈vk,vl〉 =
d∑
j=1

〈vj, 1〉−1〈vk,vj〉〈vl,vj〉, k, l = 1, . . . ,d,



INTEGRAL REPRESENTATIONS 121

the norm ‖∗‖ is expandable.
We can prove this assertion either directly or applying Theorem 7. In fact, Theorem 7 shows that,

under conditions slightly larger than those from above, there exists a measure for the inner product of
S.

4. An Interpolation Approach

The existence of an atomic representing measure for a semi-inner product may be characterized in
terms of an interpolation property. The next result is an extension of Proposition 3 in [31]. As in
Theorem 1, this is an extreme situation in the sense that the number of nodes, usually larger than
the dimension of the associated Hilbert space, is assumed to be equal to that dimension (see also [14],
Theorem 1.3).

Theorem 6. Let (S,〈∗,∗〉0) be a qHfs on Ω, and let H be the Hilbert space associated to (S,〈∗,∗〉0),
supposed to be finite dimensional. The semi-inner product 〈∗,∗〉0 has a d-atomic representing measure
in Ω with d := dimH atoms if and only if there exist an orthogonal basis of H consisting of idempotents
B = {b̂1, . . . , b̂d}, and a set Z = {ζ1, . . . ,ζd} ⊂ Z such that bj(ζj) = 1 and bk(ζj) = 0 for all
j,k = 1, . . . ,d, j 6= k. In addition, the set Z is the support of the corresponding representing measure.

Proof. The first part of the proof shares some arguments with that of Theorem 1.
To begin with, assume that the semi-inner product 〈∗,∗〉0 has a representing measure in Ω, given

by

〈f,g〉0 =
d∑
j=1

λjf(ζj)g(ζj), f,g ∈S,

with λj > 0 for all j = 1, . . . ,d, and
∑d
j=1 λj = 1, where d = dimH, and the points ζ1, . . . ,ζd are

distinct. Set Z = {ζ1, . . . ,ζd}, which is a subset of Z, via Lemma 2.
According to (2.2), we must have I = {f ∈S; f|Z}. This shows that there exists a map ρ : H 7→ C(Z)

given by f̂ 7→ f|Z, which is correctly defined, linear and injective. This map is also surjective because
we have dim(H) = dim(C(Z)).

Let χk ∈ C(Z) be the characteristic function of the set {ζk} and let b̂k ∈ H be the element with
ρ(b̂k) = χk, k = 1, . . . ,d. Note that the element bk, representing the equivalence class b̂k, may be
chosen in RS, because bk|Z = χk has real values, and we may replace, if necessary, the function bk by
its real part, for each k = 1, . . . ,d.

As 〈bj,bk〉0 =
∑d
l=1 λl(χjχk)(ζl), and so 〈bj,bk〉0 = 0, 〈bj,bj〉0 = λj = 〈bj, 1〉0 for all j,k =

1, . . . ,d, j 6= k, we deduce that the set {b̂1, . . . , b̂d} is a family of orthogonal idempotents in H = H0,
which is actually a basis. Clearly, bj(ζj) = 1 and bk(ζj) = 0 for all j,k = 1, . . . ,d, j 6= k, proving the
necessity of the condition in the statement.

Conversely, if there exist an orthogonal basis of H = H0 consisting of idempotents B = {b̂1, . . . , b̂d},
and a set Z = {ζ1, . . . ,ζd}⊂Z such that bj(ζj) = 1 and bk(ζj) = 0 for all j,k = 1, . . . ,d, j 6= k, then
〈∗,∗〉0 has a representing measure whose support is Z. Indeed, it follows from Corollary 2 that for
every f ∈S we have

f =

d∑
j=1

〈bj, 1〉−10 〈f,bj〉0bj + f0,

with f0 ∈I. Hence

f(ζk) =

d∑
j=1

〈bj, 1〉−10 〈f,bj〉0bj(ζk) = 〈bk, 1〉
−1
0 〈f,bk〉0

because f0(ζk) = 0, for all k = 1, . . . ,d. Taking another function g ∈ S and using the relations from
above, we infer that

〈f,g〉0 =
d∑

j,k=1

〈bj, 1〉−10 〈bk, 1〉
−1
0 〈f,bj〉0〈ḡ,bk〉0〈bj,bk〉0 =



122 VASILESCU

d∑
j=1

〈bj, 1〉−10 〈f,bj〉0〈ḡ,bj〉0 =
d∑
j=1

〈bj, 1〉0f(ζj)g(ζj).

Because 〈bj, 1〉0 > 0 for all j = 1, . . . ,d and
∑d
j=1〈bj, 1〉0 = 1, we have obtained the existence of a

representing measure of 〈∗,∗〉0 in Ω having d atoms, whose support is the set Z. �
Proposition 3 from [31] is now a consequence of Theorem 6:

Corollary 3. Let S be a finite dimensional function space on Ω. A uspf Λ : S(2) 7→ C has a rep-
resenting measure in Ω with d := dimHΛ atoms if and only if there exist an orthogonal basis of HΛ
consisting of idempotents B = {b̂1, . . . , b̂d}, and a set ΩΛ = {ω1, . . . ,ωd} ⊂ ZΛ such that bj(ωj) = 1
and bk(ωj) = 0 for all j,k = 1, . . . ,d, j 6= k.

Proposition 2. Let (S,〈∗,∗〉0) be a qHfs on Ω, and let H be the Hilbert space associated to (S,〈∗,∗〉0),
supposed to be finite dimensional. Then there exists at most one d-atomic representing measure of the
semi-inner product 〈∗,∗〉0 with support in Ω, having d := dimH atoms.

Proof. If the semi-inner product 〈∗,∗〉0 has a d-atomic representing measure in Ω with d := dimH
atoms, say µ, it follows from the proof of Theorem 6 that the map H3 f̂ 7→ f|Z ∈ L2(Z,µ) is a unitary
operator. Indeed, we have only to note that L2(Z,µ) can be identified with C(Z), so the map f̂ 7→ f|Z
is bijective, and ‖f̂‖2 =

∫
Z
|f|2dµ for all f̂.

Now assume that there exists another d-atomic representing measure of 〈∗,∗〉0 in Ω, with support
Ξ := {ξ1, . . . ,ξd}. As in the previus case, the map f̂ 7→ f|Ξ induces a unitary operator from H onto
L2(Ξ,ν).

We now extend µ (resp. ν) to Ω by setting µ(Ω \ Z) = 0 (resp. ν(Ω \ Ξ) = 0). If ζ ∈ Z \ Ξ, for the
characteristic function χ of the set {ζ} (defined on Ω) we must have

0 6=
∫

Ω

χdµ = 〈χ, 1〉0 =
∫

Ω

χdν = 0,

which is impossible, so Z ⊂ Ξ. A similar argument shows that Ξ ⊂ Z. Therefore, Z = Ξ. In fact, this
argument shows that the weights of both measures at a given point must be the same. �

A more general form of Theorem 6 is given by the following. Unlike in Theorem 6, the number of
nodes may be greater than the dimension of the associated Hilbert space (see also Theorem 5 from [30]).

Theorem 7. Let (S,〈∗,∗〉0) be a qHfs on Ω, and let H be the associated Hilbert space, supposed to be
finite dimensional. The semi-inner product 〈∗,∗〉0 has a d-atomic representing measure in Ω for some
integer d ≥ 1 if and only if d ≥ dimH, and there exist a family of functions {v1, . . . ,vd} in RS, and
a family of points {ζ1, . . . ,ζd} in Ω, such that

〈vj, 1〉0 6= 0, f(ζj) = 〈vj, 1〉−10 〈f,vj〉0, f −
d∑
j=1

f(ζj)vj ∈I, ∀f ∈S, (4.1)

and

〈vk,vl〉0 =
d∑
j=1

〈vj, 1〉−10 〈vk,vj〉0〈vl,vj〉0, k, l = 1, . . . ,d. (4.2)

Proof. We assume first that the semi-inner product 〈∗,∗〉0 has a d-atomic representing measure in Ω,
say µ, and so we may proceed as in the first part of the proof of Theorem 6. In other words, there
exist a set Z := {ζ1, . . . ,ζd}⊂Z so that

〈f,g〉0 =
d∑
j=1

λjf(ζj)g(ζj), f,g ∈S,

with λj > 0 for all j = 1, . . . ,d, and
∑d
j=1 λj = 1 for some integer d ≥ 1. In fact, µ({ζj}) = λj, j =

1, . . . ,d. Moreover, I = {f ∈S; f|Z = 0}. This shows that there exists a map ρ : H 7→ L2(Z,µ) given
by f̂ 7→ f|Z, which is a linear isometry. The image H0 := ρ(H) ⊂ L2(Z,µ) is a Hilbert subspace, and
the map ρ : H 7→H0 is a unitary operator.



INTEGRAL REPRESENTATIONS 123

Let χk ∈ L2(Z,µ) be the characteristic function of the set {ζk}, k = 1, . . . ,d. Clearly, the family
{χ1, . . . ,χd} is an orthogonal basis of the space L2(Z,µ), and hence

d = dim(L2(Z,µ)) ≥ dim(H0) = dim(H).
Let P0 be the orthogonal projection of L

2(Z,µ) onto H0, and let vk|Z := P0χk, k = 1, . . . ,d, with
vk ∈ S fixed. As for each f = f̄ ∈ S the number 〈f,vj〉0 = λjf(ζj) is real, the function vj may be
assumed to be real-valued for all j = 1, . . . ,d.

From the equality φ =
∑d
j=1 φ(ζj)χj, valid for all φ ∈ L

2(Z,µ), we deduce that f|Z =
∑d
j=1 f(ζj)vk|Z,

for all f ∈S. Therefore, f −
∑d
j=1 f(ζj)vj ∈I for all f ∈S. Note also that

〈vj, 1〉0 = 〈χj, 1〉 = λj > 0, j = 1, . . . ,d,
and

f(ζj) = λ
−1
j 〈f|Z,χj〉 = 〈vj, 1〉

−1
0 〈f,vj〉0, j = 1, . . . ,d,

and so equation(4.1) holds. In particular, we have the equality (vkvl)(ζj) = λ
−2
j 〈vk,vj〉0〈vl,vj〉0,

whence we infer that

〈vk,vl〉0 =
d∑
j=1

〈vj, 1〉−1〈vk,vj〉0〈vl,vj〉0, k, l = 1, . . . ,d,

which is precisely equation (4.2).
Conversely, assuming that there exists a family of functions {v1, . . . ,vd} in RS, and a family of

points {ζ1, . . . ,ζd} in Ω, such that equations (4.1) and (4.2) hold, we can write

〈f,g〉0 = 〈
d∑
j=1

f(ζj)vj,

d∑
k=1

g(ζk)vk〉0 =

d∑
j=1

d∑
k=1

f(ζj)g(ζk)〈vj,vk〉0 =
d∑
j=1

d∑
k=1

f(ζj)g(ζk)

d∑
l=1

〈vl, 1〉−10 〈vj,vl〉0〈vk,vl〉0 =

d∑
l=1

〈vl, 1〉−10
d∑
j=1

f(ζj)〈vj,vl〉0
d∑
k=1

g(ζk)〈vk,vl〉0 =
d∑
l=1

〈vl, 1〉−10 〈f,vl〉0〈ḡ,vl〉0 =

d∑
l=1

〈vl, 1〉0f(ζl)g(ζl), f,g ∈S,

showing that the inner product of S has a representing measure. �

Corollary 4. Let (S,〈∗,∗〉0) be a qHfs on Ω, and let H be the associated Hilbert space, supposed to be
finite dimensional. If the semi-inner product 〈∗,∗〉0 has an atomic representing measure µ in Ω with
support Z, then card(Z) ≥ dim(H), and the map H3 f̂ 7→ f|Z ∈ L2(Ξ,µ) is a linear isomatry.

Remark 10. Let (S,〈∗,∗〉) be a finite dimensional Hfs on Ω. Then every point evaluation is auto-
matically continuous. Consequently, the space S has a reproducing kernel denoted by K(∗,∗), that is,
f(ω) = 〈f,Kω〉 for all f ∈S and ω ∈ Ω, where Kω(∗) = K(∗,ω) (see [1, 2, 18] etc. for details). In the
present framework, the function K(∗,∗) must be real valued and therefore symmetric.

The next result is an application of Theorem 6.

Proposition 3. Let (S,〈∗,∗〉) be a finite dimensional Hilbert function space on Ω, and let K(∗,∗) be
its kernel. The inner product 〈∗,∗〉 has a d-atomic representing measure, with d = dimS, if and only
if there are d distinct points ζ1, . . . ,ζd in Ω such that K(ζj,ζk) = 0 for all j,k = 1, . . . ,d, j 6= k.

Proof. Assume first that there are d distinct points ζ1, . . . ,ζd in Ω such that K(ζj,ζk) = 0 for all

j,k = 1, . . . ,d, j 6= k. Set ej(ω) = K(ζj,ζj)−1/2K(ζj,ω),j = 1, . . . ,d, ω ∈ Ω. Since we have

〈ej,ek〉 = K(ζj,ζj)−1/2K(ζk,ζk)−1/2〈K(ζj,∗),K(ζk,∗)〉 =

K(ζj,ζj)
−1/2K(ζk,ζk)

−1/2K(ζj,ζk) = 0



124 VASILESCU

if j 6= k, and
〈ej,ej〉 = K(ζj,ζj)−1〈K(ζj,∗),K(ζj,∗)〉 = 1,

the family {e1, . . . ,ed} is an orthonormal basis of S. Moreover, 〈ej, 1〉 = K(ζj,ζj)−1/2 > 0 for all j,
and so, setting bj = K(ζj,ζj)

−1/2ej, we obtain a family {b1, . . . ,bd}, which is an orthonormal basis of
S consisting of idempotents. Clearly, bj(ζj) = 1 and bj(ζk) = 0 if j 6= k. Using Theorem 6, we infer
the existence of an d-atomic representing measure for 〈∗,∗〉.

Conversely, if the inner product 〈∗,∗〉 has a d-atomic representing measure, that is, 〈f,g〉 =∑d
j=1 λjf(ζj)g(ζj), f,g ∈ S, for some distinct points ζ1, . . . ,ζd in Ω, with λj > 0 for all j = 1, . . . ,d,∑d
j=1 λj = 1, as in the first part of the proof of Theorem 6 we find a basis {b1, . . . ,bd} of S con-

sisting of orthogonal idempotents. Therefore, f(ω) =
∑d
j=1〈bj, 1〉

−1〈f,bj〉bj(ω) for all f ∈ S and
ω ∈ Ω. Moreover, λj = 〈bj, 1〉, bj(ζj) = 1 and bk(ζj) = 0 for all j,k = 1, . . . ,d, j 6= k. Setting,
K(ζ,ω) =

∑d
j=1 λ

−1
j bj(ζ)bj(ω), ζ,ω ∈ Ω, we deduce that

〈f,Kω〉 =
d∑
k=1

λkf(ζk)

d∑
j=1

λ−1j bj(ζk)bj(ω) =

d∑
j=1

λ−1j

(
d∑
k=1

λkf(ζk)bj(ζk)

)
bj(ω) = f(ω), f ∈S, ω ∈ Ω,

because
∑d
k=1 λkf(ζk)bj(ζk) = 〈f,bj〉, showing that K(∗,∗) is the kernel of S. In addition, we clearly

have K(ζj,ζk) = 0 for all j,k = 1, . . . ,d, j 6= k. �

5. Relative Multiplicativity

As done in [30] and [31] for uspf’s, we can also characterize the existence of a representing measure
of a semi-inner product in terms of idempotents. The following is a basic concept, which generalizes a
corresponding one from [30], Definition 3.

Definition 6. Let (S,〈∗,∗〉0) be a qHfs m-generaterd by the n-tuple θ. Let also B = {b̂1, . . . , b̂d} be
an orthogonal basis consisting of idempotent elements of the associated Hilbert space. We say that
the basis B is multiplicative (with respect to θ) if

〈θα,bj〉0〈θβ,bj〉0 = 〈bj, 1〉0〈θα+β,bj〉0 (5.1)
whenever |α| + |β| ≤ m, j = 1, . . . ,d.

The next result is an extension of Theorem 3 from [31], which in turn is an extension of Theorem
2 from [30]. In addition, the present proof is simpler and more transparent.

Theorem 8. Let (S,〈∗,∗〉0) be a qHfs on Ω, m-generaterd by the n-tuple θ. Assume that the associated
Hilbert space H is finite dimensional.

The inner product of H has a representing measure on Ω consisting of d := dimH atoms if and only
if there exists an orthogonal basis B = {b̂1, . . . , b̂d} of H, consisting of idempotent elements, which is
multiplicative with respect to θ, and δ(θ̂) ∈ θ(Ω), δ ∈ ∆, where ∆ is the dual basis of B.

Proof. Let B = {b̂1, . . . , b̂d} be an orthogonal basis of H consisting of idempotent elements. Every
element f̂ ∈H has a unique representation of the form f̂ =

∑d
j=1〈b̂j, 1̂〉

−1〈f̂, b̂j〉b̂j, via Lemma 3.
We consider on H the linear functionals δj(f̂) = 〈b̂j, 1̂〉−1〈f̂, b̂j〉, j = 1, . . . ,d, so f̂ =

∑d
j=1 δj(f̂)b̂j

for all f̂ ∈ H. In particular, δj(b̂j) = 1 and δj(b̂k) = 0 for all j,k = 1, . . . ,d, j 6= k. In other words,
the set ∆ := {δ1, . . . ,δd} is the dual basis of B.

Next, we define the functions f̂∆ : ∆ 7→ C by f̂∆(δ) = δ(f̂) for all f̂ ∈ H and δ ∈ ∆. Setting
H∆ := {f̂∆; f̂ ∈ H}, we have a linear map H 3 f̂ 7→ f̂∆ ∈ H∆, which is surjective by definition, and
injective because f̂∆ = 0 implies f̂ = 0. In other words, the map H 3 f̂ 7→ f̂∆ ∈ H∆ is a linear
isomorphism. In addition, f̂∆ =

∑d
k=1 f̂∆(δj)b̂k∆ for all f̂ ∈H.

As a matter of fact, the function b̂k∆ is the characteristic function of the set {δk}, k = 1, . . . ,d. This
shows that H∆ = C(∆) and the spaces H, H∆ = C(∆), have the same dimension. (Here, as before,



INTEGRAL REPRESENTATIONS 125

C(∆) := {φ : ∆ 7→ C}, is regarded as a finite dimensional C∗-algebra.) In fact, H∆ and C(∆) are
isomorphic as C∗-algebras. Indeed, the product of two functions from H∆, say f̂∆ =

∑d
j=1 δj(f̂)b̂j∆,

ĝ∆ =
∑d
j=1 δj(ĝ)b̂j∆, is given by

f̂∆ĝ∆ =

d∑
j=1

δj(f̂)cj(ĝ)b̂j∆,

which coincides with the product of C(∆). In particular, f̂∆ĝ∆ is an element of H∆, and the C∗-algebra
structure of C(∆) is inherited by H∆.

We now assume that B is multiplicative with respect to θ, and that δ(θ̂) ∈ θ(Ω), δ ∈ ∆.
We note that the space H is spanned by the family {θ̂α; |α| ≤ m}, by hypothesis, so the vector space

H∆ is spanned by the family {θ̂α∆; |α| ≤ m}, while the C∗-algebra H∆ is generated by the family
{θ̂1∆ . . . θ̂n∆}. We need a more explicit relation between these families, obtained by using (5.2), which
will be proved in the following. Because we have

〈θα,bj〉0〈θβ,bj〉0 = 〈b̂j, 1̂〉2δj(θ̂α)δj(θ̂β) =

〈bj, 1〉0〈θα+β,bj〉0 = 〈b̂j, 1̂〉2δj(θ̂α+β)
whenever |α| + |β| ≤ m and j = 1, . . . ,d, via (5.1), we infer that

θ̂α∆θ̂β∆ = θ̂α+β∆,

whenever |α| + |β| ≤ m. Hence, by recurrence, we deduce that

θ̂α∆ = (θ̂∆)
α if |α| ≤ m. (5.2)

The hypothesis δ(θ̂) ∈ θ(Ω), δ ∈ ∆, allows us to find a point ζj ∈ Ω such that θ(ζj) = δj(θ̂) for each
j = 1, . . . ,d.

Let f ∈ S be a fixed element. As S is m-generated by θ, there exists a polynomial p ∈ Pnm such
that f = p◦θ. Then we have f̂ = p◦ θ̂, and so f̂∆ = p◦ θ̂∆, via (5.2). Hence, we must have

δj(f̂) = f̂∆(δj) = p(θ̂∆(δj)) = (p◦θ)(ζj) = f(ζj), j = 1, . . . ,d,
This equality leads to

〈f,g〉0 = 〈f̂, ĝ〉 =
d∑
j=1

〈bj, 1〉f(ζj)g(ζj), f,g ∈S, (5.3)

where 〈bj, 1〉 > 0 for all j and
∑d
j=1〈bj, 1〉 = 1, via Lemma 3. Consequently, the inner product of has

a representing measure on Ω.
Conversely, assume that there exists a finite family {ζ1, . . . ,ζd} ⊂ Ω, consisting of distinct points,

such that

〈f,g〉0 =
d∑
j=1

λjf(ζj)g(ζj), f,g ∈S,

where λj > 0 for all j,
∑d
j=1 λj = 1, and d = dimH.

We proceed as in the proof of Theorem 6. Set Z = {ζ1, . . . ,ζd}, which is a subset of Z, via Lemma
2. As we must have I = {f ∈S; f|Z = 0}, there exists a map ρ : H 7→ C(Z) given by f̂ 7→ f|Z, which
is correctly defined, linear and bijective. We denote by χk ∈ C(Z) the characteristic function of the set
{ζk}, and by b̂k ∈ RH the element with ρ(b̂k) = χk, k = 1, . . . ,d. Then the set B := {b̂1, . . . , b̂d} is a
family of orthogonal idempotents in H, which is actually a basis. Moreover, bj(ζj) = 1 and bk(ζj) = 0
for all j,k = 1, . . . ,d, j 6= k.

Setting δj(f̂) = f(ζj), f ∈S, j = 1, . . . ,d, and ∆ := {δ1, . . . ,δd}, we infer that ∆ is the dual basis
of B, and we have

δj(θ̂α) = θ
α(ζj) = (θ1(ζj)

α1 · · ·θn(ζj)αn ) = δj(θ̂α),
whenever |α| ≤ m and j = 1, . . . ,d, showing that B is a multiplicative basis (with respect to θ), as
in (5.2). In addition, the obvious equality δj(θ̂) = θ(ζj), j = 1, . . . ,d, concludes the proof of Theorem
8. �



126 VASILESCU

Corollary 5. Let (S,〈∗,∗〉0) be a qHfs on Ω, 1-generated by the n-tuple θ. The semi-inner product
〈∗,∗〉0 has a representing measure on Ω consisting of d := dimH atoms if either

(1) there exists an orthogonal basis B of H consisting of idempotent elements such that δ(θ̂) ∈
θ(Ω), δ ∈ ∆, where ∆ is the dual basis of B, or

(2) θ(Ω) = Rn.

Proof. Because (S,〈∗,∗〉0) is 1-generated, property (5.1) is automatically fulfilled. To get the assertion
(1) from the statement we need the inclusion δ(θ̂) ∈ θ(Ω), δ ∈ ∆, where ∆ is the dual basis of B, in
order to apply the previous theorem, while to get (2), such an inclusion is always true, for an arbitrary
orthogonal basis consisting of idempotents. �

6. Dimensional Stability and Consequences

In this section we intend to extend and recapture, in the present context, some results regarding
the dimensional stability, developed in [29]. We also recall that the concept of dimensional stability
in function spaces of polynomials, as approached in [29], is equivalet to that of flatness, due to Curto
and Fialkow (see [9, 10]).

Remark 11. Let S = Sm be a function space m-generated by the n-tuple θ := (θ1, . . . ,θn) for some
integer m > 0. For every positive integer k ≤ m, we denote by Sk the function space k-generated by
θ. We fix a semi-inner product 〈∗,∗〉0 = 〈∗,∗〉0m, compatible with S and let 〈∗,∗〉0k be the semi-inner
product induced by 〈∗,∗〉0m on Sk. Clearly, (Sk,〈∗,∗〉0k) is a qHfs, and we denote by (Hk,〈∗,∗〉k) its
associated Hilbert space, so Hk = Sk/Ik, where Ik = {f ∈ Sk;‖f‖0k = 0}, whenever 0 < k ≤ m. If
k < m, we have Ik ⊂Ik+1 ⊂ ···⊂ Im, and

〈f + Ik,g + Ik〉k = 〈f + Ik+1,g + Ik+1〉k+1 = · · · = 〈f + Im,g + Im〉m
for all f,g ∈ Sk. In particular, if 0 < k ≤ l ≤ m, we have a natural linear map Jk,l : Hk 7→ Hl given
by Jk,l(f + Ik) = f + Il, f ∈Sk, which is an isometry.

When l = k + 1, we write sometimes Jk instead of Jk,k+1.
We also put S0 = C, endowed with its natural inner product, so I0 = {0}, and H0 = C.
For a given qHfs (S,〈∗,∗〉0) which is m-generated by an n-tuple θ := (θ1, . . . ,θn), we keep the

notation from above, if not otherwise specified. The family (Hk)nk=0 will be designated as the sequence
of Hilbert spaces associated to (S,〈∗,∗〉0,θ).

When the semi-inner product 〈∗,∗〉0 is expandable, so there exists a uspf Λ : S(2) 7→ C 〈∗,∗〉0, the
family (Hk)nk=0 will be also called the sequence of Hilbert spaces associated to (S, Λ,θ).

Definition 7. Let (S,〈∗,∗〉0) be a qHfs m-generated by the n-tuple θ := (θ1, . . . ,θn) for some integer
m ≥ 1. Let also k ∈{0, . . . ,m−1}. We say that the sequence of Hilbert spaces (Hk)mk=0 associated to
(S,〈∗,∗〉0,θ) is stable at k if dim(Hk) = dim(Hk+1).

When the semi-inner product 〈∗,∗〉0 is expandable, so there exists a unique uspf Λ : S(2) 7→ C
associated to 〈∗,∗〉0, and if (S,〈∗,∗〉0,θ) is stable at k, we say shortly that Λ is stable at k.

Definition 7 implies that the isometry Jk : Hk 7→Hk+1 is actually a unitary operator whenever the
sequence (Hl)nl=0 is stable at k. In fact, Jk unitary means that it is surjective, so for each g ∈Sk+1 we
can find an f ∈Sk such that g −f ∈Ik+1. In particular, if g ∈RSk+1, we can find an f ∈RSk such
that g −f ∈RIk+1.

Example 9. The stability introduced by Definition 7 is a rather strong condition. Let us illustrate
it by an example. Let (S,〈∗,∗〉0) be a function space, with 〈∗,∗〉0 expandable, and m-generated by
the n-tuple θ := (θ1, . . . ,θn) for some integer m ≥ 3. Let Sk be the subspace k-generated by θ, and
let 〈∗,∗〉0k be the restriction of 〈∗,∗〉0 to the space Sk, which is a compatible semi-inner product on
for Sk for all k = 1, 2, . . . ,m. Let also H0 = {0}. Assume that the sequence (Hk)mk=0 is stable at 0, so
the space H1 is unitarily equivalent to the space H0 = C. Then S1 = I1 + C, and H1 = C1̂. Let us
show that we also have H2 = C1̂. For, note that θj = τj + hj, with τj ∈ C and hj ∈I1, j = 1, . . . ,n.
To go further, we need to show that if f ∈ S1 and h ∈ I1, then fh ∈ I2. Indeed, because 〈∗,∗〉0 is
expandable, and fhfh−hf̄fh = 0, we must have

‖fh‖202 = |〈h,f̄fh〉03| ≤ ‖h‖03‖f̄fh‖03 = ‖h‖
2
01‖f̄fh‖

2
03 = 0,



INTEGRAL REPRESENTATIONS 127

by Lemma 4 and the Cauchy-Schwarz inequality. In particular, with the notation from above,

θjθk = τjτk + τjhk + τkhj + hjhk ∈ C + I2, j,k = 1, . . . ,n,

which implies the equality H2 = C1̂.
A more general situation, and under weaker conditions, will be presented in the following (see

Theorem 10 and Remark 15).

The dimensional stability in the case of function spaces of polynomials implies the existence of
representing measures for uspf’s (see [9, 10, 29]). Nevertheless, in the present context, it is not always
the case.

Example 10. Let Ω be a nonempty set and let θ1 be a real-valued function on Ω. Let also S = {c0 +
c1θ1; c0,c1 ∈ C}, which is a function space 1-generated by {θ1}. Set Λ(1) = 1, Λ(θ1) = α, Λ(θ21 ) = α2
for some α > 0, and extend this map to S(2) by linearity. Because we have Λ(|c0+c1θ1|2) = |c0+c1α|2 ≥
0, it follows that Λ is a uspf. Moreover,

IΛ = {c0 + c1θ1; c0 + c1α = 0,c0,c1 ∈ C}.
In fact, if f = c0 + c1θ1 is arbitrary in S, it can be uniquely written as

f = (−c1α + c1θ1) + (c0 + c1α) ∈IΛ + C.

Therefore, the Hilbert space HΛ is isomorphic to C. In addition, θ̂1 = α1̂.
To check whether the uspf Λ has a representing measure, we shall apply Corollary 5. Clearly {1̂}

is an idempotent and B := {1̂} is a basis of HΛ, which is multiplicative with respect to θ := (θ1), by
Corollary 5. If δ1(c1̂) = c for all c ∈ C, it clear that ∆ := (δ1) is the dual basis of B. Then Corollary
5 insures the existence of a 1-atomic representing measure for Λ if and only if δ1(θ̂1) = α ∈ θ1(Ω).
Assuming θ1(ω0) = α for some ω0 ∈ Ω, we obtain Λ(h) = h(ω0) for all h ∈ S(2). In other words, the
uspf Λ is represented by the Dirac measure at ω0. Nevertheless, this representation is not necessarily
unique because there might exist several points ω with the property θ1(ω) = θ1(ω0). Moreover, if
α /∈ θ1(Ω), we have no representation measure of Λ with support in Ω.

The next result extends Lemma 2.3 from [30].

Lemma 5. Let (S,〈∗,∗〉0) be a qHfs m-generated by θ = (θ1, . . . ,θn), with 〈∗,∗〉0 expandable. If the
sequence of Hilbert spaces (Hk)mk=0 associated to (S,〈∗,∗〉0,θ) is stable at m−1, then (

∑n
j=1 θjIm)∩S ⊂

Im. In particular, θjIm−1 ⊂Im for all j = 1, . . . ,n.

Proof. Let f =
∑n
j=1 θjfj ∈S with fj ∈Im for all j = 1, . . . ,n, and let g ∈Sm−1. Then

|〈f,g〉0| ≤
n∑
j=1

|〈θjfj,g)〉0| ≤
n∑
j=1

‖fj‖0‖θjg‖0 = 0

by the Cauchy-Schwarz inequality.
Now, let h ∈ Sm−1 be such that f − h ∈ Im, which exists because of the stability of (Hk)mk=0 at

m− 1. Then
‖f‖20 = 〈f,h〉0 + 〈f,f −h〉0 = 0,

by the previous computation and the Cauchy-Schwarz inequality. Therefore f ∈Im.
The last assertion is obvious. �

Remark 12. Let (S,〈∗,∗〉0) be a qHfs on Ω, m-generated θ = (θ1, . . . ,θn), with 〈∗,∗〉0 expandable,
and let Λ : S(2) 7→ C be the uspf associated to 〈∗,∗〉0. We have S = {f = p ◦ θ,p ∈ Pnm}, and set
Sθ := {p|θ(Ω); p ∈Pnm}, which is a function space on Ωθ := θ(Ω). We define a map Λθ : S

(2)
θ 7→ C by

the equality Λθ(φ) = Λ(p◦θ), φ = p|Ωθ, p ∈ Pn2m. The definition is correct because p|Ωθ = 0 implies
p ◦ θ = 0, and so Λ(p ◦ θ) = 0. In fact, Λθ is a uspf. In addition, the space Sθ is m-generated by
τ = (τ1, . . . ,τn), with τj := tj|Ωθ, j = 1, . . . ,n.

Assume now that (S,〈∗,∗〉0,θ) is stable at m − 1. Choosing a function φ = p|Ωθ, p ∈ Pnm, so
f := p◦ θ ∈ Sm, we can find a function g ∈ Sm−1 such that h := f − g ∈ Im. As g = q ◦ θ for some
q ∈Pnm−1, and h = r◦θ for some r ∈Pnm, we obtain the equality φ = ψ + ι, where ψ := q|Ωθ ∈Sθ,m−1
and ι ∈Iθ,m because Λθ(|ι|2) = Λ(|r ◦θ|2) = 0. In other words, Λθ is stable at m− 1.



128 VASILESCU

Finding an atomic representing measure for Λθ means to solve a Ωθ-moment problem, whose so-
lution, when it exists, leads to a representing measure for Λ. For a closed K ⊂ Rn, the K-moment
problem has been approached in [12]. Because our framework is slightly larger and our results are gen-
erally not covered by the contents of [12], we present in the following a direct approach, independent
of [12], but following the lines of [29].

Remark 13. From now on, if not otherwise specified, we fix a qHfs (S,〈∗,∗〉0) m-generated by
θ = (θ1, . . . ,θn), with 〈∗,∗〉0 expandable.

Next, assume that the sequence of Hilbert spaces (Hk)mk=0 associated to (S,〈∗,∗〉0,θ) is stable at
m − 1. Lemma 5 allows us to define correctly the map Mj : Hm−1 7→ Hm by the equality Mj(f +
Im−1) = θjf + Im for all j = 1, . . . ,m. Setting J = Jm−1 (see Remark 11), we may consider on the
Hilbert space Hm the linear operators Tj = MjJ−1 for all j = 1, . . . ,n. Note that, fixing f ∈Sm and
choosing g ∈ Sm−1 such that f − g ∈ Im, we have Tj(f + Im) = θjg + Im for all j. As mentioned
after Definition 7, if f ∈ RSm we can choose g ∈ RSm−1 such that f − g ∈ RIm. Therefore,
Tj(RHm) ⊂RHm for all j = 1, . . . ,n.

With this notation, we have the following.

Proposition 4. The linear maps Tj, j = 1, . . . ,m, are self-adjoint operators, and T = (T1, . . . ,Tn) is
a commuting tuple on Hm.

Proof. Let fk ∈Sm and gk ∈Sm−1 be such that fk −gk ∈Im (k = 1, 2). Then
〈T∗j (f1 + Im),f2 + Im〉 = 〈f1 + Im,θjg2 + Im〉 = 〈f1,θjg2〉0

= 〈θjg1,f2〉0 = 〈Tj(f1 + Im),f2 + Im〉,
via Lemma 5 and Remark 13. Hence T1, . . . ,Tn are self-adjoint.

We prove now that T1, . . . ,Tn mutually commute. It suffices to show that MjJ
−1Mk = MkJ

−1Mj
for all j,k = 1, . . . ,n. To show this, fix a function f ∈ Sm−1. Thus Mj(f + Im−1) = θjf + Im.
We can choose gj ∈ Sm−1 such that θjf − gj ∈ Im. Therefore, J−1(θjf + Im) = gj + Im−1, and
Mk(gj + Im−1) = θkgj + Im.

Similarly, Mk(f + Im−1) = θkf + Im, and we can choose gk ∈ Sm−1 such that θkf − gk ∈ Im, so
Mj(gk +Im−1) = θjgk +Im. To complete the proof, it suffices to show that θkgj −θjgk ∈Im. Indeed,
note that θjθkf −θjgk ∈ θjIm and θkθjf −θkgj ∈ θkIm. Consequently,

θkgj −θjgk ∈ (θkIm + θjIm) ∩Sm ⊂Im,
via Lemma 5. Consequently, T1, . . . ,Tn mutually commute. �

Remark 14. With the notation from Proposition 4, if α,β are multi-indices with |α + β| ≤ m, then
Tα(θβ+Im) = θα+β+Im. Indeed, if |β| < m, we have Tj(θβ+Im) = (θjθβ+Im), as in Remark 13. The
general formula can be obtained by recurrence. In particular, if |α| ≤ m, then Tα(1 +Im) = θα +Im,
and so p(T)(1 +Im) = p(θ) +Im for all p ∈Pnm. Moreover, fixing p ∈Pnm, as we have p(θ) = q(θ) + h,
with q ∈Pnm−1 and h ∈Im, we obtain p(T)(1 + Im) = q(T)(1 + Im).

The following assertion is now obtained as an application of Theorem 8. See also Theorem 2.11 and
Corollary 2.13 from [29] (proved in a different way), as well as Corollary 7.11 from [9].

Theorem 9. Let (S,〈∗,∗〉0) be a qHfs on Ω, m-generated by θ = (θ1, . . . ,θn), with 〈∗,∗〉0 expandable.
We assume that the sequence of Hilbert spaces (Hk)mk=0 associated to (S,〈∗,∗〉0,θ) is stable at m −
1 (m ≥ 1). Then we have:

(1) there exists an orthogonal basis B = {b̂1, . . . , b̂d} of H := Hm consisting of idempotent elements,
which is multiplicative with respect to θ;

(2) the semi-inner product 〈∗,∗〉0 has a d-atomic representing measure with support in Ω, where
d := dimH, if and only δ(θ̂) ∈ θ(Ω), δ ∈ ∆, where ∆ is the dual basis of B;

(3) if the semi-inner product 〈∗,∗〉0 has an atomic representing measure with support in Ω, this
atomic measure is uniquely determined.

Proof. (1) First of all, note that H = {p(T)1̂; p ∈Pnm}. Indeed, if f̂ ∈H is an arbitrary element, as S is
m-generated by θ, we can find a polynomial p ∈Pnm such that f̂ = p◦θ +Im, and so f̂ = p(T)(1 +Im),
via Remark 14.



INTEGRAL REPRESENTATIONS 129

Next, we want to apply Theorem 8 to show that there exists an orthogonal basis B = {b̂1, . . . , b̂d}
of H consisting of idempotent elements, which is multiplicative with respect to θ.

We first consider the commuting n-tuple T = (T1, . . . ,Tn), consisting of self-adjoint operators,
acting in H, given by Proposition 4. The spectral theorem for n-tuples of commuting self-adjoint
operators (see for instance [6], [24], [26] etc.) implies the existence of commuting self-adjoint projections

Ej = E({ξ(j)}), j = 1, . . . ,d, such that h(T) =
∑d
j=1 h(ξ

(j))Ej for every function h : σ(T) 7→ C, where
σ(T) := {ξ(1), . . . ,ξ(d)} is the joint spectrum of T , which coincides with the support of E. Moreover,
if the function h is real-valued, the operator h(T) is self-adjoint. In addition, because the space RH
is invariant under T1, . . . ,Tn (see Remark 13), it must be also invariant under h(T), whenever h is
real-valued. In particular, RH is invariant under Ej, j = 1, . . . ,d,.

We now construct an orthogonal family {b̂1, . . . , b̂d} of H consisting of idempotents. Because∑d
j=1 Ej is the identity on H, setting b̂j = Ej1̂ ∈ RH,j = 1, . . . ,d, we obtain a decomposition

1̂ =
∑d
j=1 b̂j. As Ej 6= 0, we must have Ejĝ = ĝ 6= 0 for some ĝ = q ◦ θ + Im = q(T)(1 + Im),

with q ∈ Pnm, via Remark 14. Assuming b̂j = 0, we would obtain Ejĝ = ĝ = q(T)b̂j = 0, which is
not possible. Therefore, b̂j 6= 0 for all j = 1, . . . ,d. Note also that 〈b̂j, 1̂〉 = 〈b̂j, b̂j〉 > 0, so b̂j is an
idempotent for all j = 1, . . . ,d. In other words, {b̂1, . . . , b̂d} is an an orthogonal family in H consisting
of idempotent elements.

To show that B = {b̂1, . . . , b̂d} is a basis of H it suffices to show that dim(H) = d. For, we consider
the sub-C∗-algebra CT generated by T in the C∗-algebra of all linear (automatically bounded) operators
acting in H. Because CT is finite dimensional, we must have CT = {p(T); p ∈ Pns }, for some integer
s ≥ m. In fact, choosing an element p(T) with p ∈ Pns , we may replace p by a polynomial q ∈ Pnm,
such that p(T) = q(T). We prove this assertion by recurrence. Let pj(t) = tjp(t), with p ∈Pnm. Then
there exists q ∈ Pnm−1 such that p(T)1̂ = q(T)1̂, by Remark 14. Therefore, pj(T)1̂ = qj(T)1̂, where
qj(t) = tjq(t) ∈Pnm. Using the fact that every ĥ ∈H is of the form g(T)1̂ for some polynomial g, we
deduce the equality pj(T) = qj(T). An induction argument shows that for p(T) with p ∈Pns , s > m,
we have the equality p(T) = q(T) for some polynomial q ∈Pnm. Particularly, CT = {p(T); p ∈Pnm}.

As mentioned above, the spectral theorem allows us to write

p(T) =

d∑
j=1

p(ξ(j))Ej, p ∈Pnm.

In particular, {E1, . . . ,Ed}, which is clearly a linearly independent family of operators, is actually an
algebraic basis of (the linear space) CT . Note also that

p(T)1̂ =

d∑
j=1

p(ξ(j))b̂j, p ∈Pnm.

Consequently, using the equality H = {p(T)1̂; p ∈ Pnm} mentioned above, we deduce that dim(H) =
dim(CT ) = d. In particular, B = {b̂1, . . . , b̂d} is an orthogonal basis of H, consisting of idempotents.
In addition, considering the measure ν(∗) = 〈E(∗)1̂, 1̂〉 on σ(T), and putting λj = 〈Ej1̂, 1̂〉 = 〈b̂j, 1̂〉,
we have

〈θα,bj〉0〈θβ,bj〉0 = 〈Tα1̂,Ej1̂〉〈Tβ1̂,Ej1̂〉 =∫
{ξ(j)}

tαdν(t)

∫
{ξ(j)}

tβdν(t) = λ2j(ξ
(j))α(ξ(j))β =

λj

∫
{ξ(j)}

tα+βdν(t) = λj〈θα+β,bj〉0

whenever |α| + |β| ≤ m, j = 1, . . . ,d. In other words, the basis B is multiplicative with respect to θ,
which concludes the asserion (1) from the statement.

To obtain the assertion (2) from the statement, we recall that the dual basis ∆ := {δ1, . . . ,δd} of B
is given by δj(f̂) = 〈b̂j, 1̂〉−1〈f̂, b̂j〉, j = 1, . . . ,d. In particular,

δj(θ̂k) = λ
−1
j 〈EjTk1̂, 1̂〉 =

∫
{ξ(j)}

tkdν(t) = ξ
(j)
k , j,k = 1, . . . ,d.



130 VASILESCU

Theorem 8 shows that the inner product of H has a representing measure on Ω consisting of d := dimH
atoms if and only δ(θ̂) ∈ θ(Ω), δ ∈ ∆, which concludes the proof (2).

(3) This assertion is not a direct consequence of Proposition 2, becuase we may consider a priori
two atomic measures whose supports have different cardinals. To apply Proposition 2, we need a
supplementary argument.

An explicit form of the integral representation whose existence is given in (2) is obtained as for
equation (5.3). Specifically, choosing ζj ∈ Ω such that ξ(j) = δj(ζj), j = 1, . . . ,d, we deduce the
equality

〈f,g〉0 =
d∑
j=1

λjf(ζj)g(ζj),

providing a (d-atomic) representing measure for the semi-inner product of S.
Let µ be this representing measure of the inner product 〈∗,∗〉0), with support Z = {ζ1, . . .ζd}

and weights λj = µ(ξ
(j)), j = 1, . . . ,d. Assume that the semi-inner product 〈∗,∗〉0 has another

atomic representing measure in Ω, say ν, with support Σ := {σ1, . . . ,σg} ⊂ Ω. Then necessarily,
g ≥ d = dim(H), and the map H 3 f̂ 7→ f|Σ ∈ L2(Ξ,ν) is an isometry (see Corollary 4). Moreover,
Im = {f ∈S; f|Σ = 0}.

Let Bj be the linear operator on L
2(Σ,ν) given by Bjh = θjh for all j = 1, . . . ,n and h ∈ L2(Σ,ν).

Then B := (B1, . . . ,Bn) is an n-tuple of commuting self-adjoint operators. With T = (T1, . . . ,Tn)

as before, fixing f̂ ∈ H with f ∈ Sm, and choosing g ∈ Sm−1 with h := f − g ∈ Im, so Tjf̂ = θ̂jg,
(f − g)|Σ = 0, and (θjg)|Ξ = (θjf)|Σ = Bj(f|Σ). In other words, identifying the Hilbert space H
with the (Hilbert) subspace {f|Σ; f ∈ S}, we see that Bj is an extension of the operator Tj for all
j = 1, . . . ,n. In particular, the spectral measure E of T is the restriction of the spectral measure EB
of B to H.

We now consider the elements EB({σj})(1|Σ), which must belong to H, because H is invariant
under EB. Therefore, setting ĉj = EB({σj})(1|Σ) = E({σj})1̂, j = 1, . . . ,g, as in the second part of
the proof, {ĉ1, . . . , ĉg} is an orthogonal family of nonnull idempotent elements of H. Consequently, we
must have g = d, and so dim(L2(Ξ,ν)) = d. We may now apply Proposition 2, to get the asserton (3).

The next result is an extension of Theorem 2.6 from [29] (see also Theorem 7.8 and Corollary 7.9
from [9]).

Theorem 10. Let (S,〈∗,∗〉0) be a qHfs on Ω, m-generated by θ = (θ1, . . . ,θn), with 〈∗,∗〉0 expandable.
We assume that the sequence of Hilbert spaces (Hk)mk=0 associated to (S,〈∗,∗〉0,θ) is stable at m −
1 (m ≥ 1). Then the semi-inner product 〈∗,∗〉0 can be uniquely extended to an expandable semi-inner
product of S∞, which has a d-atomic measure in Ω, where d = dim(Hm).

Proof. Using the notation and arguments from (the proof of) Theorem 9, we have

〈f,g〉0 =
d∑
j=1

λjf(ζj)g(ζj), f,g ∈S,

which is an integral representation of 〈∗,∗〉0 by a d-atomic measure. A direct extension of this formula
allows us to define

〈f,g〉0∞ =
d∑
j=1

λjf(ζj)g(ζj), f,g ∈S∞,

which is an expandable semi-inner product on S∞. We want to show that 〈∗,∗〉0∞ is uniquely deter-
mined.

Let 〈∗,∗〉′0∞,〈∗,∗〉′′0∞ be two expandable semi-inner products on S∞, both of them extending 〈∗,∗〉0.
For k ≥ m + 1, let Sk = {p◦ θ; p ∈ Pnk}, I

′
k = {f ∈ Sk;〈f,f〉

′
0∞ = 0}, I′′k = {f ∈ Sk;〈f,f〉

′′
0∞ = 0}.

Clearly, Im ⊂I′k ∩I
′′
k for all k ≥ m + 1.

We shall show by induction that for every multi-index α, with |α| ≥ m, there exists an element
fα ∈ Sm−1, such that θα −fα ∈ I′|α| ∩I

′′
|α|. The assertion is obvious for |α| = m, via the stability at

m− 1. Assume the property true for all multi-indices of length k, for a k ≥ m, and let us prove it for



INTEGRAL REPRESENTATIONS 131

multi-indices of length k + 1. If |α| = k + 1, there exists a number j ∈ {1, . . . ,n} and a multi-index
β with |β| = k such that θα = θjθβ. By the induction hypothesis, we can find a function fβ ∈ Sm−1
such that θβ −fβ ∈I′k ∩I

′′
k . Therefore, θ

α −θjfβ ∈I′k+1 ∩I
′′
k+1, by the Cauchy-Schwarz inequality.

Further, θjfβ ∈ Sm and so we can find a function fj,β ∈ Sm−1 such that θjfβ − fj,β ∈ Im, via the
stability at m− 1. Consequently,

θα −fα = θα −θjfβ + θjfβ −fj,β ∈I′k+1 ∩I
′′
k+1 + Im = I

′
k+1 ∩I

′′
k+1,

where fα = fj,β.
Extending the property from above to arbitrary functions from S∞, we deduce, in particular, that

for every pair of function f1,f2 ∈ Sk, with k ≥ m, we can find a pair g1,g2 ∈ Sm−1 such that
fj −gj ∈I′k ∩I

′′
k , j = 1, 2. Therefore,

〈f1,f2〉′0∞ = 〈g1,g2〉0 = 〈f1,f2〉
′′
0∞,

showing the uniqueness of the natural extension 〈∗,∗〉0∞ of the semi-inner product 〈∗,∗〉0∞. �

Remark 15. From the proof of the previous theorem, we deduce that for all k ≥ m and f ∈Sk there
exists g ∈Sm−1 such that f −g ∈Ik, where Ik := {h ∈Sk;〈h,h〉0∞ = 0}. This implies that all spaces
Hk := Sk/Ik are unitarily equivalent Hilbert spaces. This assertion is true even for k = ∞.

7. An Example

Example 11. This is an example related to the paper [16](see also [14]). Specifically, we look for
atomic representing measures of a given semi-inner products of the space, P2m (m ≥ 1), whose suport
lies in the curve Γ := {(t,t3) ∈ R2; t ∈ R}. In what follows, we want to solve a Γ-moment problem
(see Subsection 2.3), trying to use some of our techniques. The basic function space will be Sm =
{P |Γ; P ∈ P2m} = P2m,Γ. Because the representation of an element P |Γ ∈ Sm is, in general, not
unique, let us characterize the subspace J := {P ∈ P2m; P |Γ = 0}. Given a polynomial P(x1,x2) =∑

0≤k+l≤m ak,lx
k
1x

l
2 in P2m, we have P |Γ = 0 if and only if P(t,t3) =

∑
0≤k+l≤m ak,lt

k+3l = 0 for all
t ∈ R. Explicitly, we must have

P(t,t3) =

3m∑
j=0


 ∑
l∈I(j)

aj−3l,l


tj = 0, t ∈ R,

with I(j) = {l ≥ 0; 3l ≤ j ≤ m + 2l}, which happens if and only if
∑
l∈I(j) aj−3l,l

= 0 whenever 0 ≤ j ≤ 3m. In other words,

J = {P(x1,x2) =
∑

0≤k+l≤m

ak,lx
k
1x

l
2;
∑
l∈I(j)

aj−3l,l = 0, 0 ≤ j ≤ 3m}.

Moreover, the space Sm is isomorphic to the space P2m/J , the elements of Sm may be regarded as
equivalence classes of P2m modulo J , and they will be denoted by P̃ = P + J , P ∈P2m.

Let θ1,θ2 : R 7→ R be given by θ1(t) = t, θ2(t) = t3, t ∈ R. We have a natural map of Sm into P13m
given by P̃ 7→ P ◦θ. Since P ∈J is equivalent to P ◦θ = 0, this map is correctly defined, linear, and
injective; it is surjective too. Indeed, we may define a linear map from P13m into P2m in the following
way. If p0(t) =

∑3m
k=0 akt

k, we use the representation

p0(t) =

m∑
l=0

a3lt
3l +

m−1∑
l=0

a3l+1t
3l+1 +

m−1∑
l=0

a3l+2t
3l+2.

We now make a certain choice, obviously not unique. Namely, we replace the monomial t3l by the
monomial xl2, the monomial t

3l+1 by the monomial x1x
l
2, and the monomial t

3l+2 by the monomial
x21x

l
2, so we obtain the polynomial

P0(x1,x2) =

m∑
l=0

a3lx
l
2 +

m−1∑
l=0

a3l+1x1x
l
2 +

m−1∑
l=0

a3l+2x
2
1x
l
2,

we clearly have p0 = P0 ◦θ, showing that the image of P̃0 is precisely p0.



132 VASILESCU

Let 〈∗,∗〉0 be an expandable semi-inner product of Sm. We want to define an expandable semi-inner
product of P13m. If p,q ∈P13m, as we have p = P ◦θ,q = Q◦θ for some P,Q ∈P2m, we set

〈p,q〉0 := 〈P̃,Q̃〉0.
The definition does not depend on the choice of P,Q ∈P2m with p = P ◦θ,q = Q◦θ because, if either
P ◦ θ = 0 or Q ◦ θ = 0, we must have 〈P̃,Q̃〉0 = 0. In addition, it clearly provides an expandable
semi-inner product of P13m.

The existence of atomic measures for expandable semi-inner products of spaces of polynomials in
one variable, in particular for the space P13m, treated as a moment problem, can be explicitly described
(see for instance [8]). We restrict ourselves to a particular case of Theorem 9, applied to P13m, endowed
with the expandable inner product 〈p,q〉0, p,q ∈P13m. The dimensional stability at 3m−1 is given by
a condition of the form

‖t3m −
3m−1∑
k=0

ckt
k‖0 = 0,

for some polynomial
∑3m−1
k=0 ckt

k, which is equivalent to the condition∥∥∥∥∥x3m2 −
m−1∑
l=0

c3lx
l
2 −

m−1∑
l=0

c3l+1x1x
l
2 −

m−1∑
l=0

c3l+2x
2
1x
l
2 + J

∥∥∥∥∥
0

,

expressed only in the given terms.
A solution of this moment problem means the existence of (distinct) points τ1, . . . ,τr ∈ R and

positive numbers λ1, . . . ,λr with λ1 + · · · + λr = 1 such that

〈p,q〉0 =
d∑
j=1

λjp(τj)q(τj), p,q ∈P13m.

Therefore, if p,q ∈Sm are written under the form p = P ◦θ,q = Q◦θ with P,Q ∈P2m, we have

〈P̃,Q̃〉0 = 〈p,q〉0 =
d∑
j=1

λjP(τj,τ
3
j )Q(τj,τ

3
j ),

which is a solution of our Γ-moment problem.

References

[1] J. Agler and J. E. McCarthy, Pick Interpolaton and Hilbert Function Spaces, AMS Graduate Studies in Mathematics,
Vol 44, Providence, Rhode Island, 2002.

[2] D. Alpay, The Schur Algorithm, Reproducing Kernel Spaces and System Theory, SMF/AMS Texts and Monographs,

Vol. 5, 2001.
[3] S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, Springer-Verlag, New York/Berlin/Heidelberg,

2001.

[4] C. Bayer and J. Teichmann, The proof of Tchakaloff’s theorem, Proc. Amer. Math. Soc., 134 (10) (2006), 3035-3040.
[5] C. Berg, J. P. R. Christensen and P. Ressel, Harmonic analysis on semigroups. Theory of positive definite and

related functions, Graduate Texts in Mathematics, 100. Springer-Verlag, New York, 1984.

[6] M. S. Birman and M.Z. Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, D. Reidel Publishing
Company, Dordrecht, 1987.

[7] J.B. Conway, A Course in Abstract Analysis, Graduate Studies in Mathematics Vol. 141, AMS, Providence, Rhode

Island, 2012.
[8] R. E. Curto and L. A. Fialkow, Recursiveness, positivity, and truncated moment problems, Huston J. Math. 17 (4)

(1991), 603-635.

[9] R. E. Curto and L. A. Fialkow, Solution of the truncated complex moment problem for flat data, Memoirs of the
AMS, Number 568, 1996.

[10] R. E. Curto and L. A. Fialkow, Flat extensions of positive moment matrices: Recursively generated relations,
Memoirs of the AMS, Number 648, 1998.

[11] R. E. Curto and L. A. Fialkow, A duality proof of Tchakaloff’s theorem, J. Math. Anal. Appl., 269 (2002), 519-532.

[12] R. E. Curto and L. A. Fialkow, Truncated K-moment problems in several variables, J. Operator Theory, 54 (1)
(2005), 189-226.

[13] R. E. Curto and L. A. Fialkow, An analogue of the Riesz-Haviland theorem for the truncated moment problem, J.

Funct. Anal. 255 (2008), 2709-2731.
[14] R. E. Curto, L. A. Fialkow and H. M. Möller, The extremal truncated moment problem, Integral Equations Oper.

Theory, 60 (2008), 177-200.



INTEGRAL REPRESENTATIONS 133

[15] N. Dunford and J.T. Schwartz, Linear Operators, Part I: General Theory, Interscience Publishers, New
York/London, 1958.

[16] L. Fialkow, Solution of the truncated moment problem with variety y = x3, Trans. Amer. Math. Soc. 363 (2011),

3133-3165.
[17] L. Fialkow and J. Nie, Positivity of Riesz functionals and solutions of quadratic and quartic moment problems, J.

Funct. Anal. 258 (2010), 328-356.

[18] E. Hille, Introduction to general theory of reproducing kernels, Rocky Mountain J. Math. 2 (1972), 321-368.
[19] J. H. B. Kemperman, The general moment problem, a geometric approach, Ann. Math. Statist. 39 (1968), 93-122.

[20] M. Laurent, Sums of squares, moment matrices and optimization over polynomials, Emerging applications of alge-

braic geometry, IMA Vol. Math. Appl., 149, 157-270, Springer, New York, 2009.
[21] H. M. Möller, On square positive extensions and cubature formulas, J. Comput. Appl. Math. 199 (2006), 80-88.

[22] M. Putinar, On Tchakaloff’s theorem, Proc. Amer. Math. Soc. 125 (1997), 2409-2414.

[23] J. Stochel, Solving the truncated moment problem solves the full moment problem, Glasg. Math. J. 43(2001),
335-341.

[24] K. Schmüdgen, Unbounded self-adjoint operators on Hilbert space, Graduate Texts in Mathematics, 265. Springer,
Dordrecht, 2012.

[25] V. Tchakaloff, Formule de cubatures mécaniques à coefficients non négatifs, Bull. Sci. Math. 81 (2), 1957, 123-134.

[26] F.-H. Vasilescu, Analytic Functional Calculus and Spectral Decompositions, D. Reidel Publishing Company, Dor-
drecht, 1982.

[27] F.-H. Vasilescu, Operator theoretic characterizations of moment functions, 17th OT Conference Proceedings, Theta,

2000, 405-415.
[28] F.-H. Vasilescu, Spaces of fractions and positive functionals, Math. Scand. 96 (2005), 257-279.

[29] F.-H. Vasilescu, Dimensional stability in truncated moment problems, J. Math. Anal. Appl. 388 (2012), 219-230

[30] F.-H. Vasilescu, An Idempotent Approach to Truncated Moment Problems, Integral Equations Oper. Theory 79 (3)
(2014), 301-335.

[31] F.-H. Vasilescu, Square Positive Functionals in an Abstract Setting, Operator Theory: the State of the Art, 145-167,

Theta, 2016.

Department of Mathematics, University of Lille 1, 59655 Villeneuve d’Ascq, France

∗Corresponding author: fhvasil@math.univ-lille1.fr


	1. Introduction
	2. Preliminaries
	2.1. Function Spaces and Compatible Semi-Inner Products
	2.2. Continuous Point Evaluations
	2.3. Generators of Function Spaces
	2.4. Reduction to Finite Dimensional Spaces
	2.5. Quasi-Hilbert Function Spaces and Idempotents

	3. USPF's versus Semi-Inner Products
	4. An Interpolation Approach
	5. Relative Multiplicativity
	6. Dimensional Stability and Consequences
	7. An Example
	References