()


@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 14, no. 1, 2013

pp. 53-60

A RAFU linear space uniformly dense in C [a, b]

E. Corbacho Cortés

Abstract

In this paper we prove that a RAFU (radical functions) linear space,
∁, is uniformly dense in C [a, b] by means of a S-separation condition of
certain subsets of [a, b] due to Blasco-Moltó. This linear space is not a
lattice or an algebra.
Given an arbitrary function f ∈ C [a, b] we will obtain easily the se-
quence (Cn)

n
of ∁ that converges uniformly to f and we will show the

degree of uniform approximation to f with (Cn)
n
.

2010 MSC: 37L65

Keywords: RAFU; Uniform density; Uniform approximation; Radical func-
tions; Approximation algorithm.

1. Introduction

Let K be a compact Hausdorff space. The Kakutani-Stone Theorem [10]
gives a necessary and sufficient condition for the density of a lattice of C (K) in
the topology of the uniform convergence on K. The Stone-Weierstrass Theorem
[7] provides a necessary and sufficient condition under which an algebra of
C (K) is uniformly dense. Nevertheless, the above conditions are not sufficient
to ensure the uniform density of a linear space of C (K). Tietze [5], Jameson
[4], Mrowka [11], Blasco-Moltó [6], Garrido-Montalvo [8] and recently Gassó-
Hernández-Rojas [9] have studied the uniform approximation for linear spaces.

In the Section 2 we will construct a RAFU (Radical functions) linear space,
∁, in C [a, b] and we will prove that ∁ is uniformly dense in C [a, b] by using
a S-separation condition according to Blasco-Moltó [6]. We will also see that
the uniform density of ∁ in C [a, b] is not a consequence of the results given by
Kakutani-Stone, Stone-Weierstrass, Tietze, Jameson, or Mrowka.



54 E. Corbacho Cortés

It is true that Blasco-Moltó showed an example of a linear space, F , uni-
formly dense in C [0, 1] by using the S-separation condition but some questions
were not studied: the linear combinations of elements belonging to F which ap-
proximate uniformly every f ∈ C [0, 1] and the degree of uniform approximation
that F provides were unknown. In the Section 3 we will solve these problems
by using the RAFU linear space ∁. Moreover, this linear space ∁ can be used as
an example of approximation by series in the work of Gassó-Hernández-Rojas.

2. A RAFU linear space uniformly dense in C [a, b]

For each n ∈ N we consider the partition P = {x0, x1, ..., xn} of [a, b] with
xj = a + j · b−an , j = 0, ..., n and we define in [a, b] the functions

(2.1) Cn(x) = k1 +

n
∑

i=2

(ki − ki−1) · Fn(xi−1, x)

where {ki}ni=1 are a family of real arbitrary numbers and

(2.2) Fn (xi−1, x) =
2n+1

√
xi−1 − x0 + 2n+1

√
x − xi−1

2n+1
√
xn − xi−1 + 2n+1

√
xi−1 − x0

, i = 2, ..., n

We designate by ∁n the subset of C [a, b] formed by the functions Cn and we
also denote by ∁ the set ∁ = ∪n∈N∁n.

Proposition 2.1. The set ∁ is a linear space included in C [a, b].

Proof. It is clear that ∁ ⊂ C [a, b]. In the first place it is easy to check that ∁n
is a linear space n-dimensional because n is fixed and hence the values {xi}ni=0
are the same points. Moreover, a basis of ∁n is {1, Fn(x1, x), ..., Fn(xn−1, x)}.

∁ is a linear space. Let Cp and Cq be two elements belonging to ∁. Then,
Cp ∈ ∁r·p, r ∈ N and Cq ∈ ∁s·q, s ∈ N by considering zero the coefficients
(ki − ki−1) of the functions Fn(xi−1, x) that do not appear on the expressions
of Cp or Cq. In particular, Cp and Cq belong to the linear space ∁p·q and, of
course, Cp + Cq ∈ ∁. Finally, it is inmediate to check that if Cp ∈ ∁ and λ ∈ R
then λ · Cp ∈ ∁. �

Definition 2.2. A RAFU linear space is a linear space whose basis is formed
by radical functions of the type (2.2). We will say that ∁ is a RAFU linear
space.

The theorems of uniform approximation in C (K) for lattices are known as
Kakutani-Stone theorems (the interested reader can see [10], [7], [12]).

The family ∁ is not a lattice. In fact, in the interval [−1, 1] the function
C(x) = 3

√
x ∈ ∁ but |C(x)| /∈ ∁ because at x = 0 its side derivatives do not have

the same sign. Therefore, the family ∁ does not satisfy the Kakutani-Stone
theorems.

The theorems of uniform approximation in C (K) for algebras are known as
Stone-Weierstrass theorems (the interested reader can see [7], [12]).



A RAFU linear space uniformly dense in C [a, b] 55

A simple count proves that ∁ is not an algebra, therefore the set ∁ does not
verify the Stone-Weierstrass theorems.

Let X be a topological space and let C∗ (X) be the set consisting of all
bounded continuous functions and let C (X) be the set consisting of all contin-
uous functions.

Definition 2.3. Let F be a family of C∗ (X). We say that
(1) A zero-set in X is a set of the form Z(f) = {x ∈ X : f(x) = 0} with

f ∈ C∗ (X).
(2) The Lebesgue-sets of f ∈ C (X) are the sets Lα(f) = {x ∈ X : f(x) ≤ α}

and Lβ(f) = {x ∈ X : f(x) ≥ β} where α and β are real numbers.
(3) F S1- separates the subsets A and B of X when there is f ∈ F,

0 ≤ f ≤ 1 such that f(x) = 0 if x ∈ A and f(x) = 1 if x ∈ B.
(4) (Blasco-Moltó [?]). F S-separates the subsets A and B of X if for each

δ > 0, there is f ∈ F such that 0 ≤ f ≤ 1 for every x ∈ X, f(A) ⊂ [0, δ]
and f(B) ⊂ [1 − δ, 1].

(5) (Garrido-Montalvo [?]). F S′-separates the subsets A and B of X if
for each δ > 0, there is f ∈ F such that −δ ≤ f ≤ 1 + δ for every
x ∈ X, f(A) ⊂ [−δ, δ] and f(B) ⊂ [1 − δ, 1 + δ].

(6) Given a series of continuous functions
∑

i∈I fi on X, the series is locally
convergent, for every x ∈ X, if there is a neighborhood U of x such
that the series converges uniformly on U. For E ⊂ C(X),

∑

(E) is the
set of all f ∈ C (X) such that f =

∑

i∈I fi with fi ∈ E for every i ∈ I
and

∑

i∈I fi is a locally convergent series.
∑

(E) denotes the uniform
closure of

∑

(E).

Theorem 2.4 (Tietze [5], Mrowka [11]). Let F be a linear space of C∗ (X).
F is uniformly dense in C∗ (X) if and only if F S1- separates every pair of
disjoint zero-sets in X.

Theorem 2.5 (Jameson [4]). Let F be a linear space of C∗ (X). F is uniformly
dense in C∗ (X) if and only if F S1- separates every pair of disjoint closed
subsets in X.

By the properties of the functions of the linear space ∁ it is possible to deduce
that we cannot apply to ∁ the results of Tietze, Mrowka or Jameson.

Theorem 2.6 (Blasco-Moltó [6]). Let X be a topological space. A linear space
F of C∗ (X) is uniformly dense in C∗ (X) if and only if F S- separates every
pair of disjoint zero-sets in X.

We go to see that we can apply this theorem to prove the uniform density
of ∁ in C [a, b]. Let us consider in [a, b] the step function defined by f(x) =
{

k1 a ≤ x ≤ x1
k2 x1 < x ≤ b

, k1, k2 ∈ R. If we calculate, for each n ∈ N, the expressions

of the radical functions cn(x) = Mn + Nn · 2n+1
√
x − x1 that are obtained by

the conditions cn(a) = k1 and cn(b) = k2, we obtain Nn =
k2−k1

2n+1
√
b−x1+ 2n+1

√
x1−a



56 E. Corbacho Cortés

and Mn = k1 +
(k2−k1)· 2n+1

√
x1−a

2n+1
√
b−x1+ 2n+1

√
x1−a

. In this case, a elementary count shows

that the sequence (cn)n satisfies limn→+∞ cn(x) =











k1 a ≤ x < x1
k1+k2

2
x = x1

k2 x < x ≤ b
Now, we will consider an arbitrary step function in [a, b]

(2.3) f(x) = k1 · χ[x0, x1] +
m
∑

i=2

ki · χ(xi−1, xi]

where ki ∈ R, i = 1, ..., m and x ∈ [x0 = a, xm = b]. By an elementary count
we can write (2.3) in the form f(x) =

∑m
i=1 fi(x) where f1(x) = k1 · χ[x0, xm]

and fp(x) = (kp − kp−1) · χ(xp−1, xm], p = 2, ..., m.
For each fp we construct its sequence of radical functions (cp,n)n. For every

n ∈ N, the corresponding sequence for f1 is c1,n(x) = k1 and for fp, with p > 1,
we obtain cp,n(x) = Mp,n +Np,n · 2n+1

√
x − xp−1 where Np,n and Mp,n are given

by Np,n =
kp−kp−1

2n+1
√

xm−xp−1+ 2n+1
√

xp−1−x0
and Mp,n =

(kp−kp−1)· 2n+1
√

xp−1−x0
2n+1

√
xm−xp−1+ 2n+1

√
xp−1−x0

.

Finally, if we denote by (Cm,n)n to the sequence Cm,n(x) =
∑m

i=1 ci,n(x)

then limn→+∞ Cm,n(x) =

{

f(x) x ∈ [x0, xm] − {x1, x2, , ..., xm−1}
kp+kp+1

2
x = xp, p = 1, ..., m − 1

by

elementary properties of the limits.

Proposition 2.7. Let f be the function defined by (2.3). For any β > 0 such
that (xi − β, xi + β)∩(xj − β, xj + β) = ∅ where i 6= j and i, j ∈ {1, ..., m − 1}
the limit limn→+∞ Cm,n = f is uniform on [x0, x1 − β] ∪ [x1 + β, x2 − β] ∪ ...∪
[xm−1 + β, xm].

Proof. 1st part. It verifies that limn→∞
2n+1

√
x =

{

−1 x ∈
[

−M, − 1
K

]

1 x ∈
[

1
K
, M

]

where M and K are large positive real numbers. Moreover, the limit becomes
uniform.

The function hn(x) =
2n+1

√
x is strictly increasing on R, therefore hn(−M) ≤

hn(x) ≤ hn(− 1K ) for x ∈
[

−M, − 1
K

]

and fixed ǫ > 0 it is possible to find

nM,K ∈ N such that if n ≥ nm,K then −1−ǫ < hn(−M) ≤ hn(x) ≤ hn(− 1K ) <
−1 + ǫ. In other words, |hn(x) + 1| < ǫ. Analogous, we obtain |hn(x) − 1| < ǫ
on

[

1
K
, M

]

.
2nd part. Given a partition P = {a = x0, x1, ..., xm = b} of [a, b] with

a < x1 < ... < b. For each n ∈ N and p = 2, ..., m we define in [a, b] the
function

Fn (xp−1, m, x) =
2n+1

√
xp−1 − x0 + 2n+1

√
x − xp−1

2n+1
√
xm − xp−1 + 2n+1

√
xp−1 − x0



A RAFU linear space uniformly dense in C [a, b] 57

Then, it follows that limn→∞ Fn (xp−1, m, x) =











0 x < xp−1
1
2

x = xp−1
1 x > xp−1

, p = 2, ..., m

and these limits are uniform on [x0, x1 − β]∪[x1 + β, x2 − β]∪...∪[xm−1 + β, xm]
The first assertion is consequence of the elementary properties of the limits

and the second is obtained by aplying the first part and take into acount that
for each p = 2, ..., m only a root of Fn (xp−1, m, x) depends upon x.

3rd part. By the second part and the definitions of Cm,n and f we obtain
the result which we want to prove. �

Proposition 2.8. Let β > 0 be such that (xi − β, xi + β) ∩ (xj − β, xj + β) =
∅ where i 6= j and i, j ∈ {1, ..., m − 1}. Then, for all ε > 0, there exists n0 ∈ N
such that for n > n0 it follows that

1. | Cm,n(x) − f(x) |<| kj+1 − kj | +ε
2. |Cm,n(x) − (kj · (1 − α) + kj+1 · α)| < ε
where x ∈ (xj − β, xj + β), j = 1, ..., m − 1and α ∈ (0, 1).

Proof. 1st Part. Let x ∈ (xj − β, xj + β) be, j = 1, ..., m − 1. By the
Proposition 2.7 the sequence (Fn)n converges uniformly to 1 as p − 1 < j
and to 0 as p − 1 > j.

Moreover there exists n0 ∈ N such that ∀n > n0 the function
(kj+1 − kj) Fn (xp−1, m, x) transforms the interval (xj − β, xj + β) into the in-
terval (0, (kj+1 − kj)). The rest is obtained by the elementary properties of the
limits and the definition of Cm,n(x).

2nd Part. It is analogous to the 1st part by considering ∀n > n0 the func-
tion (kj+1 − kj) Fn (xp−1, m, x) attains on (xj − β, xj + β) the values (kj+1 − kj)·
α, α ∈ (0, 1). �

Theorem 2.9. The RAFU linear space ∁ is uniformly dense in C [a, b].

Proof. Consider the family L of all sets which are finite unions of disjoint
compact intervals. First, we will prove that ∁ S-separates every pair of disjoint
sets of L. Clearly , it suffices to prove the following fact: Given δ > 0 and
the intervals [α1, βi], 1 ≤ i ≤ m, m ≥ 2, 0 ≤ αj < βj < αj+1 < 1, there is a
function f in ∁ such that 0 ≤ f(x) ≤ 1 for every x ∈ [a, b], f ([αi, βi]) ⊂ [0, δ]
for i odd and f ([αi, βi]) ⊂ [1 − δ, 1] for i even, 1 ≤ i ≤ m.

Consider a partition P = {xi}n0 of [a, b] with xj = a + j ·
b−a
n

, j = 0, ..., n
such that βj < xp < αj+1 for every j and some xp. We also consider a step
function h defined in [a, b] from the values xj such that h(x) = 0 or h(x) = 1
for every x ∈ [a, b] but verifying that h ([xs, xt]) = 0 when [αi, βi] ⊂ [xs, xt]
and i is odd, h ([xk, xl]) = 1 when [αi, βi] ⊂ [xk, xl] and i is even, 1 ≤ i ≤ m.

Fixed an appropriate value β ≤ min
{

|xp−βj|
2

,
|αj+1−xp|

2

}

and given δ > 0

we can choose suitable partitions of [a, b] into 2kn intervals, if it was necessary
for some k ∈ N, supporting the previous conditions and, by the propositions
2.7 and 2.8, we can obtain a function C2kn ∈ ∁ such that 0 ≤ C2kn(x) ≤



58 E. Corbacho Cortés

1 and |C2kn − h| < δ, that is to say, C2kn ([αi, βi]) ⊂ [0, δ] for i odd and
C2kn ([αi, βi]) ⊂ [1 − δ, 1] for i even.

Next, we will prove that ∁ S-separates every pair of disjoint zero-sets Z1
and Z2 of [a, b]. Since L is a basis for the closed sets of [a, b] we have Z1 =
∩ {B ∈ L : Z1 ⊂ B}. As Z2 is compact the family {Z2} ∪ {B ∈ L : Z1 ⊂ B}
does not have the finite intersection property. Therefore Z2 ∩B1 ∩...∩Bp = Ø,
for some Bi ∈ L, Zi ⊂ B, 1 ≤ i ≤ p. Since L is closed under finite intersections
it follows that B′ = B1 ∩ ... ∩ Bp ∈ L, Z1 ⊂ B′ and B′ ∩ Z2 = Ø. In the same
way we find B′′ ∈ L, such that Z2 ⊂ B′′ and B′ ∩ B′′ = Ø. Since ∁ S-separates
B′ and B′′, by Blasco-Moltó’s Theorem, ∁ is uniformly dense in C [a, b]. �

The S-separation of subsets is equivalent to the S′-separation of subsets
in linear spaces containing constant functions (Garrido-Montalvo [8]). Clearly
∁ contains the constant functions, therefore we can also deduce the uniform
density of ∁ in C [a, b] by using the S′-separation condition of every pair of
disjoint zero-sets in X.

3. The degree of uniform approximation with the RAFU linear
space

Blasco-Moltó [6] proved that the linear subspace F of C [0, 1] generated by
the functions

{exp ((x + µ)n) : µ ∈ R, x ∈ [0, 1] , n = 0, 1, 3, ..., 2k + 1, ...}
is uniformly dense in C [0, 1], but the linear combinations which approximate
uniformly a function f ∈ C [0, 1] and the degree of uniform approximation that
F provides were not studied.

The following result has been proved recently in the XXII CEDYA-XII
CMA [2] and solves these two problems by considering the linear space ∁

Theorem 3.1. Let f be a continuous function defined on [a, b] and let P =
{x0 = a, x1, ..., xn = b} be a partition of [a, b] with xj = a + j · b−an , j = 0, ...,
n. Then,

‖Cn − f‖ ≤
M − m√

n
+ ω

(

b − a
n

)

, n ≥ 2

where ‖‖ denotes the uniform norm, M and m are the maximum and the min-
imum of f on [a, b] respectively, ω (δ) its modulus of continuty and Cn(x) is
defined for all x ∈ [a, b] and n ∈ N by Cn(x) = f(a) +

∑n

j=2[f(xj) − f(xj−1)] ·
Fn (xj−1, x)

Let us observe that the values {ki}ni=1of (2.1) becomes {f(xi)}
n

i=1in this
case.

Theorem 3.2 (Gassó-Hernández-Rojas [9]). Let A be a subset of C (X) and
E a linear space of C (X) which S-separates Lebesgue-sets of A. Then the

sublattice generated by A is contained in
∑

(E).



A RAFU linear space uniformly dense in C [a, b] 59

(a) Approximation to f(x) (b) Approximation to g(x)

(c) Approximation to h(x) (d) Approximation to l(x)

Figure 1. Approximation with the RAFU linear space ∁

The RAFU linear space ∁ satifies the Theorem 3.1 when X = [a, b] because
every Lebesgue-set is also a zero-set since Lα(f) = Z((f − α) ∨ 0) and Lβ(f) =
Z((f − β) ∧ 0) and we have proved that ∁ S-separates every pair of disjoint
zero-sets Z1 and Z2 of [a, b]. In this case, if A = C (X) we can say that

C (X) is contained in
∑

(∁). In fact, given f ∈ C [a, b], we already knew that
f(x) =

∑∞
n=1 cn(x) where cn ∈ ∁, n ∈ N, and the series converges uniformly.

Example 3.3. Given the functions f(x) = e−x
2

on [−3, 3], g(x) = 3x
x2+1

on

[−5, 6], h(x) = 5(x + 8)(x + 6)(x + 2)x(x − 3)(x − 5) on [−10, 6] and l(x) =| x |
on [−10, 6], the Figure 1 shows the graphics of these functions together with
their approximations by means of its respective radical function C75 belonging
to the RAFU linear space ∁.

4. Conclusions

The RAFU method is an original and unknown procedure of uniform ap-
proximation on C [a, b]. This method improves the instability of the polynomial
interpolation and it is based in the use of radical functions to approximate any
continuous function defined in [a, b]. We have constructed a linear Space ∁
uniformly dense on C [a, b] and this linear space is not a lattice or an alge-
bra. At the moment, the proof of this result was direct but in this work we
have proved that ∁ is uniformly dense on C [a, b] by using a S-separation con-
dition due to Blasco-Moltó [6] or an equivalent S′-separation condition due
to Garrido-Montalvo [8]. We already knew another example of a linear space
uniformly dense by using these separation conditions [6] but by considering
the set ∁, we can know easily the linear combinations of elements belonging to



60 E. Corbacho Cortés

∁ which approximate uniformly every f ∈ C [a, b] and the degree of uniform
approximation that ∁ provides.

References

[1] E. Corbacho, Teoŕıa general de la aproximación mediante funciones radicales, ISBN
84-690-1149-9, (Mérida, 2006).

[2] E. Corbacho, The degree of uniform approximation by radical funtions, XXII CEDYA-
XII CMA, September 5th-9th, (Palma de Mallorca, 2011).

[3] E. Corbacho, Uniform approximation by means of radical functions, I Jaen Conference
on Approximation Theory, July 4th-9th, (Jaen, 2010).

[4] G. J. O. Jameson, Topology and normed spaces, Chapman and Hall, (London, 1974).
[5] H. Tietze, Uber functionen die anf einer abgeschlossenen menge stetig sind, Journ.

Math. 145 (1915), 9–14.
[6] J. L. Blasco and A. Moltó, On the uniform closure of a linear space of bounded real-

valued functions, Annali di Matematica Pura ed Applicata IV vol. CXXXIV (1983),
233–239.

[7] M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans.
Amer. Math. Soc. 41 (1937), 375–481.

[8] M. I. Garrido, Aproximación uniforme en espacios de funciones continuas, Publica-
ciones del Departamento de Matemáticas Universidad de Extremadura 24, ( Univ. Ex-

tremadura, Badajoz 1990).
[9] M. T. Gassó, S. Hernández and S. Rojas, Aproximación por series en espacios de fun-

ciones continuas, (Univesitat Jaume I, Departament de Matemàtiques, 2010).
[10] S. Kakutani, Concrete representation of abstract (M)-spaces, Ann. Math. 42 (1941),

994–1024.
[11] S. Mrowka, On some approximation theorems, Nieuw Archief voor Wiskunde (3) XVI

(1968), 94–111.
[12] S. Stone, A generalized Weierstrass approximation theorem, Math. Magazine 21 (1948)

167–184, 237–254.

(Received January 2012 – Accepted December 2012)

E. Corbacho Cortés (ecorcor@unex.es)
Department of Mathematics, University of Extremadura, Spain.


	A RAFU linear space uniformly dense in C[a,b]. By E. Corbacho Cortés