/compile/output.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 7, No. 3, 2014, 343-368

ISSN 1307-5543 – www.ejpam.com

Spectrum of Periodically Correlated Fields

Dominique Dehay1,!, Harry Hurd2, Andrzej Makagon3

1 Institut de Recherche Mathématique de Rennes, CNRS umr 6625, Université Rennes 2, 35043
Rennes, France
2 Department of Statistics, University of North Carolina, Chapel Hill, NC 27599-2630, USA
3 Department of Mathematics, Hampton University, Hampton, VA 26668, USA

Abstract. The paper deals with Hilbert space valued fields over any locally compact Abelian group G, in
particular over G = !n ""m, which are periodically correlated (PC) with respect to a closed subgroup
of G. PC fields can be regarded as multi-parameter extensions of PC processes. We study structure,
covariance function, and an analogue of the spectrum for such fields. As an example a weakly PC field
over !2 is thoroughly examined.

2010 Mathematics Subject Classifications: 60G12

Key Words and Phrases: Periodically correlated processes, Stochastic processes and fields, Harmoniz-
able processes, Spectrum, Shift operator, LCA group, Fourier transform

1. Introduction

Periodically correlated (PC) processes and sequences have been studied for almost half of

the century and at present they are very well understood mainly due to works of Gladyshev [12,

13], Hurd [18–22] and other authors [5, 16, 27–31]. A summary of the theory of PC sequences

can be found in [24]. Surprisingly, there are only several publications [2–4, 6, 7, 10, 11, 23, 38]

dealing with PC fields, and each one concentrates on a particular type, namely coordinate-

wise strong periodicity. An intention of this paper is to sketch a unified theory of fields over

any locally compact Abelian (LCA) group G which are periodically correlated with respect to

an arbitrary closed subgroup K of G. We emphasize the case of G = !n " "m to illustrate

the results. This work includes stationary fields as well the weakly periodically correlated

fields, that is the fields whose covariance function exhibits periodicity (or stationarity) in fewer

directions than the dimension of the group. In the latter case we assume a certain integrability

condition (see Definition 3) in order to develop some simple spectral analysis of those fields.

A work in progress treats the case where this condition is not satisfied.

!Corresponding author.

Email addresses: dominique.dehay@univ-rennes2.fr (D. Dehay), hhurd@stat.unc.edu (H. Hurd),
andrzej.makagon@hamptonu.edu (A. Makagon)

http://www.ejpam.com 343 c# 2014 EJPAM All rights reserved.



D. Dehay, H. Hurd, A. Makagon / Eur. J. Pure Appl. Math, 7 (2014), 343-368 344

The paper is organized as follows. In the remaining part of this section we introduce

notation and vocabulary used in the paper, review needed facts from harmonic analysis on LCA

groups, and outline the theory of one-parameter PC processes. In the next three sections we

study the covariance function, the notion of the spectrum, and the structure of a K-periodically

correlated field. These sections include the main results of the paper (Theorems 1, 2 and 5).

The last section contains examples that illustrate the theory developed. In particular Example 2

gives a complete analysis of the weakly periodically correlated fields over !2, introduced in

[23].

Background

To avoid confusion and to set the notations of the paper we recall some features of group

theory, Haar measures, Fourier transform, and periodic functions. For more information on

these subjects the authors refer to [14, 35, 37].

1. Quotient groups, cross-sections, Haar measure, and Fourier transform. Let G be an addi-

tive locally compact Abelian (LCA) group, !G be its dual (group of continuous characters), and

let
"
! , t
#

denote the value of a character ! $ !G at t $ G. The dual !G can be given a topol-

ogy that makes it an LCA group such that$(%)G = G. Let K be a closed subgroup of G. The

symbol G/K will stand for the quotient group and&(G/K) for its dual. Let ı denote the natural

homomorphism of G onto G/K , ı(t) := t + K , and ı! be its dual map ı! :'G/K % !G, defined as"
ı!("), t
#
=
"
", (t + K)
#

for " $'G/K and t $ G. The mapping ı! is injective and continuous,

and for each " $'G/K ,
"
ı!("), ·
#

is a K-periodic function on G (see below). Consequently'G/K
can be identified with a closed subgroup !K of !G consisting of the elements # $ !G such that

&#, t' = 1 for any t $ K . In the sequel we use the notation &#, t' := &#, ı(t)', for all # $ !K

and t $ G. By ((G) we denote the $-algebra of Borel sets on G. A cross-section % for G/K is

a mapping % : G/K % G such that

(i) % is Borel,

(ii) %(G/K) is a measurable subset of G,

(iii) %(0) = 0 and % ) ı(t) $ t + K for all t $ G, where t + K := {t + k :k $ K}.

For existence and other properties of a cross-section please see [25, 39]. For each cross-section

% for G/K , the sets k+%(G/K), k $ K , are disjoint and their union is G, and hence each element

t $ G has a unique representation t = k(t) + %(ı(t)), where k(t) $ K . Note that the function

% is not additive, that is %(x + y) may be different than %(x) + %(y), x , y $ G/K .

Any LCA group has a nonnegative translation-invariant measure, unique up to a multi-

plicative constant, called a Haar measure. The Haar measures on G and !G can be normalized

in such a way that the following implication holds



D. Dehay, H. Hurd, A. Makagon / Eur. J. Pure Appl. Math, 7 (2014), 343-368 345

if f $ L1(G), !f (!) :=

(

G

"
! , t
#

f (t)#hG(d t) for ! $ !G, and !f $ L1(!G)

then f (t) =

(

!G

"
! , t
#!f (!)#h!G(d!) for a.e. t $ G.

The function !f above is called the Fourier transform of f . Here and in what follows L1(G)

stands for the space of complex functions on G which are integrable with respect to #hG , and

#hG denotes the normalized Haar measure on the group indicated in the subscript. Note that the

normalization of the Haar measures of G and !G is not unique. We follow the usual convention

that if G is compact and infinite then the normalization is such that #hG(G) = 1; if G is discrete

and infinite then the normalized Haar measure of any single point is 1; if G is both compact

and finite then its dual is also and the Haar measure on G is normalized to have a mass 1

while the Haar measure on !G is counting measure. The normalized Haar measure on " is the

Lebesgue measure divided by
*

2&. Finally, if K is a closed subgroup of G then the normalized

Haar measures satisfy Weil’s formula

(

G/K

)(

K

f (k+ s)#hK(dk)

*
#hG/K(dṡ) =

(

G

f (t)#hG(d t), f $ L1(G). (1)

The inner integral above depends only on the coset ṡ := s+ K . See e.g. [35, Section III.3.3].

If f $ L1(G) then !f is a continuous bounded function on !G but not necessarily integrable.

The Fourier transform, which is customarily denoted by the integral !f (!) =
+

G

"
! , t
#

f (t)#hG(d t)

(even if f is not integrable) extends from L1(G)+ L2(G) to an isometry from L2(G) onto L2(!G)
(Plancherel Theorem [37]). If there is a danger of confusion we will recognize the difference

by writing

!f (!) L2

=

(

G

"
! , t
#

f (t)#hG(d t), f $ L2(G).

In the sequel we say that the inverse formula holds for f if the function f is the inverse

Fourier transform of !f . If both f and !f are integrable then clearly the inverse formula holds

for both. Also if G is discrete and f $ L2(G), then the inverse formula holds for f . Indeed,

in this case !f $ L1(!G) because !G is compact, and hence f (t) =
+
!G
"
! , t
#!f (!)#h!G(d!) for all

t $ G.

For a separable Hilbert space, with inner product (·, ·), and norm - · -, , let

Lp(G;, ) := Lp(G,#hG;, ), p = 1 or 2, be the space of , -valued fields on G which are p-

integrable with respect to Haar measure #hG , that is, f $ Lp(G;, ) means that f : G %, is

#hG-measurable and the real-valued function t .% - f (t)-p, is integrable with respect to #hG . It

is well known that the space L1(G;, ) is a Banach space with the norm

- f -L1 :=

(

G

- f (t)-, #hG(d t), f $ L1(G;, ),



D. Dehay, H. Hurd, A. Makagon / Eur. J. Pure Appl. Math, 7 (2014), 343-368 346

and the space L2(G;, ) is a separable Hilbert space with the inner product

,
f , g
-

L2 :=

(

G

,
f (t), g(t)
-
, #hG(d t), f , g $ L2(G;, ).

See e.g. [9, Chapter III] (see also [8, 15, 36]). Whenever f $ L1(G;, ) then f is Bochner inte-

grable (also called strongly integrable) and its Fourier transform exits. Furthermore Plancherel

theorem applies and defines an isometry from L2(G;, ) onto L2(!G;, ) (one-to-one), so
!f $ L2(!G;, ) is also well defined for f $ L2(G;, ).

2. Periodic functions. Given G and a closed subgroup K of G, it is natural to call a function f

defined on G to be K-periodic if

f (t + k) = f (t) for all t $ G and k $ K .

In this case, the function f is constant on cosets of K . Hence a function f on G is K-periodic if

and only if f is of the form f = fK ) ı, where fK is a function on G/K . The concrete realization

!K := ı!('G/K) / !G of'G/K as a subgroup of !G will be in the sequel called the domain of the

spectrum of f . Note that !K is not determined uniquely by f , for a K-periodic function can be

at the same time periodic with respect to a larger subgroup K 0 1 K; in other words we will not

be assuming that K is the “smallest” period of f .

If f $ L1(G/K;, ),, being the set of complex numbers $ or any separable Hilbert space,

we consider the Fourier transform of fK at # $ !K

.fK(#) :=

(

G/K

&#, x' fK(x)#hG/K(d x) (2)

that will be referred to as the spectral coefficient of f at frequency # $ !K .

A couple of remarks regarding the above definition and its relation to the standard notions

of the spectrum and its domain are certainly due here. The word spectrum comes originally

from physics, operator theory, and more recently from signal processing. It is widely used in

the theory of second order stochastic processes. Intuitively, the spectrum of a scalar function

f is a Fourier transform of f in whatever sense it exists. If f is a locally integrable function on

G = !n ""m then the spectrum F of f is a Schwartz distribution on !G, which is a functional

on a certain space of functions on !G determined by the relation F
,%'
-
=
+

G
'(t) f (t)#hG(d t),

where ' runs over the set of compactly supported functions on G which are infinitely many

times differentiable in last m variables. One can show that if f is additionally K-periodic, then

the support of F (as defined in [36]) is a subset of !K . This rationalizes the name “domain of

the spectrum” that we have assigned for !K , as well the phrase “the spectrum sits on !K” which

we will use sometimes. The first task in understanding the spectrum of a K-PC field is thus to

identify the domain of its spectrum or its second order spectrum. (See below).

The coefficient .fK(#) defined in (2) represents an “amplitude” of the harmonic &#, ·' in a

spectral decomposition of f . Indeed, if fK and.fK are integrable, then

fK(x) =

(

!K

&#, x'.fK(#)#h!K
(d#), x $ G/K ,



D. Dehay, H. Hurd, A. Makagon / Eur. J. Pure Appl. Math, 7 (2014), 343-368 347

and as a consequence of Weil’s formula (1) and the fact that &#, t' = &#, ı(t)', t $ G, # $ !K ,

we conclude that

f (t) =

(

!K

&#, t'.fK(#)#h!K
(d#), t $ G. (3)

If.fK is not integrable, then equality (3) holds only for #hG-almost every t $ G or is not valid as

stated, but a# still retains its interpretation.

For illustration suppose that f is a continuous scalar function on " which is periodic with

period T > 0, that is such that f (t) = f (t + T ) for every t $ ". In this case G = ",

K = {kT : k $ !}, the quotient group G/K can be identified with [0, T ) with addition modulo

T , the mapping ı is defined as ı(t) =
/
t
0

T
, the remainder in integer division of t by T , and

the identity %(x) = x , x $ [0, T ), is the most natural cross-section for G/K . The function

fK is defined as fK(x) = f (%(x)) = f (x), x $ [0, T ). The dual of G/K is identified with the

subgroup !K = {2& j/T : j $ !} of ", and with this identification &#, ı(t)' = &#, t' = e2i#t ,

# $ !K , t $ ". The normalized Haar measures on [0, T ) and !K are the Lebesgue measure

divided by T and the counting measure, respectively. The domain of the spectrum of f is

therefore the set !K . The spectral coefficient f̂ j of f at # = 2& j/T , is given by

f̂ j =
1

T

( T

0

e2i2& j t/T f (t) d t, j $ !.

Note that the sequence { f̂ j} is square-summable and consequently f (t) =
13

j=23 ei2& j t/T f̂ j ,

where the series above converges in L2[0, T], so in L2([2A,A]) for every 0 < A <3. The

spectrum F of f is defined by the relation F
,%'
-
= 1*

2&

+3
23'(t) f (t) d t. If ' is an infinitely

times differentiable with compact support then

F
,%'
-
=

1*
2&

( 3

23
'(t) f (t) d t =

(

"

%'(t)F(d t),

where F =
13

j=23 f̂ j ({2& j/T}, and (a denotes the measure of mass 1 concentrated at {a}. The

spectrum of f can be therefore identified with a $-additive complex measure F on " sitting

on !K and defined by F =
13

j=23 f̂ j ({2& j/T}. If the sequence { f̂ j} is summable, then F is a

finite measure, but it does not have to be in general.

2. Periodically Correlated Fields

Let, be a separable Hilbert space with the inner product (·, ·), . In a probabilistic context

the space, represents the space of zero-mean complex random variables with finite variance.

A (stochastic) field X = {X (t): t $ G} is a measurable function X : G%, . Let

,X := span {X (t): t $ G} be the smallest closed linear subspace of , that contains all X (t),

t $ G. The function KX (t, s) :=
,
X (t), X (s)
-
, , t, s $ G, is referred to as the covariance function

of the field X . A field X is called stationary if it is continuous and for all t, s $ G, the function

KX (t + u, s+ u) does not depend on u $ G. If X is stationary then

KX (t + s, s) = KX (t + 0,0) =: RX (t), t, s $ G,



D. Dehay, H. Hurd, A. Makagon / Eur. J. Pure Appl. Math, 7 (2014), 343-368 348

and RX has the form

RX (t) =

(

!G

"
! , t
#
"(d!),

where " is a non-negative Borel measure on !G (Bochner Theorem [35, Section IV.4.4]). A field

X is called harmonizable if there is a (complex) measure % on !G " !G such that

KX (t, s) =

((

!G2

"
! , t
# "
) , s
#
%(d! , d)), t, s $ G, (4)

see [17, 33, 34]. The measure % above is called the second order spectral (SO-spectral) measure

of the harmonizable field X . Note that every stationary field is harmonizable with the measure

% sitting on the diagonal: %(#) = "{! $ !G :(! ,!) $#},# $((!G" !G). The SO-spectrum of a

harmonizable field X is the spectral measure % associated with function KX (t,2s), s, t $ G via

relation (4). Here we adopt the terminology of “second order spectrum (SO-spectrum)” of the

field X , instead of the usual term “spectrum”, in order to avoid confusion with the spectrum

of a periodic function (or field). By this way we point at the fact that we are considering not

the field X by itself but its covariance function KX . This leads to the following definition.

Definition 1. Let X be a continuous stochastic field over G. The second order spectrum (SO-

spectrum) of the field X is the spectrum (Fourier transform, in whatever sense it may exist) of the

function G " G 4 (t, s) .% KX (t,2s). The domain of the SO-spectrum of the field X is defined as

the domain of the spectrum of this function.

Definition 2. Let K be a closed subgroup of G. A field X is called K-periodically correlated (K-

PC) if X is continuous and the function G 4 u .% KX (t+u, s+u) is K-periodic in u for all t, s $ G.

The group K will be called the period of the PC process X .

If G = " (or !) and K = {kT : k $ !} then we will use the phrase “PC process (or PC

sequence) with period T > 0”, rather than K-PC field. Note that every stationary field over G

is K-PC field with K = G. A K-PC field is labeled strongly PC if G/K is compact, and weakly PC

otherwise.

For example if X is a field over "2 (or !2) such that for every s, t $ "2 (or !2),

KX (s, t) = KX

,
s+ (T1, 0), t+ (T1, 0)

-
= KX

,
s+ (0, T2), t+ (0, T2)

-
, 0 < T1, T2 <3, then X is

strongly K-PC with K = {(k1T1, k2, T2) :k1, k2 $ !}. Since KX is invariant under shifts from K

this leads to the existence of unitary operators U1, U2 in,X such that U1X (t) = X (t1+ T1, t2)

and U2X (t) = X (t1, t2 + T2) for every t= (t1, t2). If the field X instead satisfies

KX (s, t) = KX

,
s+ (T1, T2), t+ (T1, T2)

-
,

then X is weakly K-PC with K = {k(T1, T2) : k $ !}. This leads to a unitary operator U such

that UX (t) = X
,
t+ (T1, T2)
-
, t= (t1, t2).

Examples of PC fields on!2 can be constructed by a periodic amplitude or time deformation

of a stationary field. Suppose X (t) = f (t)Y (t), t = (t1, t2) $ !2, where Y is a stationary field

and f is a non-random periodic function such that f (t1, t2) = f (t1 + T1, t2) = f (t1, t2 + T2).



D. Dehay, H. Hurd, A. Makagon / Eur. J. Pure Appl. Math, 7 (2014), 343-368 349

Then the field X is strongly K-PC with K = {(k1T1, k2T2) : k1, k2 $ !}. If f above instead

satisfies f (t1, t2) = f (t1 + T1, t2 + T2), then X is weakly K-PC with K = {k(T1, T2) :k $ !}. If

the function f is two-dimensional integer valued, then the field X defined by X (t) = Y
,
t+ f (t)
-

will be weakly PC.

Remark that generally KX (t + u, s+ u) = KX (t 2 s+ s+ u, s+ u), so a continuous field X is

K-PC if and only if KX (t + u,u) is a K-periodic function of u for every t $ G. If X is K-PC field

then for all t, s $ G there is a unique function x .% bX (t, s; x) on G/K such that

KX (t + u, s+ u) = bX (t, s; ı(u)), t, s,u $ G.

The canonical map ı : G % G/K is continuous and open [35, Section III.1.6], so the function

x .% bX (s, t; x) is continuous. Denote

BX (t; x) := bX (t, 0; x), t $ G, x $ G/K .

Note that bX (t, s; ı(u)) = bX

,
t 2 s, 0; ı(s+ u)
-
= BX

,
t 2 s; ı(s+ u)
-

for all t, s,u $ G.

In this work we need the following notion to proceed to the spectral analysis.

Definition 3. A K-PC field X over G is called G/K-square integrable if the function BX (0; ·) is

integrable with respect to the Haar measure on G/K.

If X is a G/K-square integrable K-PC field, then from translation-invariance of the Haar

measure it follows that for every t $ G, bX (t, t; ·) is #hG/K -integrable, and

(

%(G/K)

-X (t + u)-2, (#hG/K ) %21)(du) =

(

G/K

bX (t, t; x)#hG/K(d x)

=

(

G/K

BX (0; x)#hG/K(d x) =

(

%(G/K)

-X (u)-2, (#hG/K ) %21)(du)<3,

where % is any cross-section for G/K . Also note that if bX (t, t; ·) is #hG/K -integrable for any

t $ G, then by Cauchy-Schwarz inequality bX (t, s; ·) is #hG/K -integrable for all t, s $ G.

When X = P is a K-periodic continuous field, then it is a PC field and we can readilly prove

the following equivalence

P is G/K-square integrable 56 PK $ L2(G/K;, ),

where PK is the field defined on G/K by P = PK ) ı.

When X is a PC process on " with period T > 0 (i.e. KX (t, s) = KX (t + T, s + T ) for

all t, s $ "), it is well known that the SO-spectrum of X can be described as a sequence of

complex measures * j , j $ !, on " (cf. [24]). If in addition
1

j Var(* j) <3 then the process

X is harmonizable and

KX (t, s) =

((

"2

ei(ut2vs) %(du, dv), (5)



D. Dehay, H. Hurd, A. Makagon / Eur. J. Pure Appl. Math, 7 (2014), 343-368 350

where % :=
1

j %j , and %j is the image of * j via the mapping + j(u) := (u,u 2 2& j/T ). If1
j Var(* j) =3 then %=

1
j %j can still be viewed as the SO-spectrum of X in the framework

of the Schwartz distributions theory (see [29, 32]). For more discussion about PC processes

please see Example 1 in Section 6. A corresponding description of the SO-spectrum is available

for PC sequences (G = !). Let us remark here that a PC sequence is always harmonizable, but

there are continuous PC processes which are not, see e.g. [12, 13].

The above description of the SO-spectrum of a PC process, which originates from Glady-

shev’s papers [12, 13], can be easily extended to the case of coordinate-wise strongly periodi-

cally correlated fields over "n or !n (see e.g. [1, 6, 7, 11, 23]). The purpose of this work is to

describe the SO-spectrum of a K-periodically correlated field for any closed subgroup K of an

LCA group G and as a particular case when G = "m"!n. We also briefly address the question

of structure of K-PC fields.

3. Covariance Function of a PC Field

This section contains an extension of Gladyshev’s description of the covariance function

of one-parameter PC processes (see [12, 13]) to the case of K-PC fields. For any G/K-square

integrable K-PC field X , define the spectral covariance function of the field X (also called cyclic

covariance in signal theory, see e.g. [10]) by

a#(t) :=

(

G/K

&#, x'BX (t; x)#hG/K(d x), # $ !K . (6)

Let % be a fixed cross-section for G/K . For each # $ !K and t $ G let us define an ,X -

valued function Z#(t) on G/K by

Z#(t)(x) := &#, (ı(t) + x)'X
,
t + %(x)
-
, x $ G/K . (7)

Notice that Z#(t)(x) depends on the chosen cross-section %. From G/K-square integrability of

X it follows that for all # $ !K and t $ G, Z#(t) is an element of the Hilbert space L2(G/K;, ).

Theorem 1. Let X be an, -valued G/K-square integrable K-PC field, and let a#(t) and Z#(t)(x)

be as above. Then the cross-covariance function K
#,µ
Z (t, s) :=
,
Z#(t), Zµ(s)
-
7 of the family

{Z# :# $ !K} is given by

K
#,µ
Z (t, s) = &#, (t 2 s)' a#2µ(t 2 s) =: R#,µ(t 2 s). (8)

If additionally

[A] the function G 4 t .2% a0(t) is continuous at t = 0,

then {Z# :# $ !K} is a family of jointly stationary fields over G in L2(G/K;,X ).



D. Dehay, H. Hurd, A. Makagon / Eur. J. Pure Appl. Math, 7 (2014), 343-368 351

Proof. Let % be a fixed cross-section for G/K . Since X is G/K-square integrable and

ı ) %(x) = x for any x $ G/K , the function B(0; ·) is #hG/K -integrable, so Z#(t) $7 and

,
Z#(t), Zµ(s)
-
7 =

(

G/K

&#, (ı(t) + x)'
"
µ, (ı(s) + x)
#
bX

,
t, s; ı ) %(x)
-
#hG/K(d x)

=

(

G/K

&#, (ı(t) + x)'
"
µ, (ı(s) + x)
#
BX

,
t 2 s; ı(s) + x
-
#hG/K(d x)

= &#, (t 2 s)'
(

G/K

"
(#2µ), y
#
BX (t 2 s; y)#hG/K(d y)

= &#, (t 2 s)' a#2µ(t 2 s)

for all s, t $ G. In view of relation (8), in order to complete the proof it is enough to show

that for every # $ !K , the function G 4 t % Z#(t) $ 7 is continuous provided condition [A]

is satisfied, and this is obvious since by equality (8),
22Z#(t)2 Z#(s)
222
7 =K#,#

Z (t, t)2K#,#
Z (t, s)2K#,#

Z (s, t) +K#,#
Z (s, s)

=2a0(0)2 &#, (t 2 s)' a0(t 2 s)2 &#, (t 2 s)' a0(s2 t).

Proposition 1. The condition [A] in Theorem 1 is satisfied if either

(i) G is discrete, or

(ii) G/K is compact, or

(iii) X is bounded, and BX (0; ·)1/2 is #hG/K-integrable.

Proof. Property [A] is evident when the group G is discrete. When G/K is compact then X

is clearly bounded because -X (t)-, = -X (%(x))-, where x = ı(t) $ G/K , and

x .% -X (%(x))-, is continuous. Since #hG/K is finite, the continuity of the function

t .% a0(t) =
+

G/K
BX (t; x)#hG/K(d x) follows therefore from Lebesgue dominated convergence

theorem.

Suppose now that
+

G/K
BX (0; x)1/2#hG/K(d x)<3. In this case for all t, s, x

33BX (t; x)2 BX (s; x)
33=
33KX

,
t + %(x),%(x)
-
2KX

,
s+ %(x),%(x)
-33

8
22X
,
t + %(x)
-
2 X
,
s+ %(x)
-22
,
22X (%(x))
22
, 8 2 sup

t
-X-, BX (0, x)1/2,

and limu%t BX (u; x) = BX (t; x). Hence Lebesgue dominated convergence theorem applies and

we conclude that lims%t a0(s) = a0(t), so condition [A] is satisfied.

Relation (8) in Theorem 1 can be also obtained using Gladyshev’s technique, that is by

showing non-negative definiteness of
/
R#,µ(t)
0
#,µ$!K

, i.e. that

n4

j=1

n4

k=1

c j ck R# j ,#k(t j 2 tk)9 0, (9)



D. Dehay, H. Hurd, A. Makagon / Eur. J. Pure Appl. Math, 7 (2014), 343-368 352

for any finite set of complex numbers {c1, . . . , cn}. Our method, which is an adaptation of the

technique used in [28], has the advantage that it gives an explicit construction of an associated

stationary family of fields. We want to point out here that even in the case of PC processes on

" with period T not every matrix function [Rm,n(t)]m,n$! with continuous entries, which is

non-negative definite in the sense of (9) and such that Rm,n(t) ei2&mt/T depends only on m2n,

is associated with a continuous PC process of period T through the relation (8). To achieve

the one-to-one correspondence one has to consider not necessarily continuous PC processes

(see e.g. [28]).

To complete the analysis of the family of fields {Z# :# $ !K} defined by (7), consider the

space ,Z := span
5

Z#(t):# $ !K , t $ G
6
. Clearly ,Z is a subspace of L2(G/K;,X ). In the

case where G = !n ""m, these Hilbert spaces coincide. More precisely

Proposition 2. Let X be an , -valued G/K-square integrable K-PC field, and Z# be defined as

above by (7). Assume that the LCA group G admits a countable dense subset, which is true when

G = !n ""m. Then,Z = L2(G/K;,X ).

Proof. We know that ,Z / L2(G/K;,X ). To show the equality, let f $ L2(G/K;,X ) be

such that
,
Z#(t), f
-
7 = 0 for all # $ !K and t $ G, i.e.

(

G/K

&#, (ı(t) + x)'
,
X
,
t + %(x)
-
, f (x)
-
, #hG/K(d x) = 0, # $ !K , t $ G.

Since !K :'G/K and &#, ı(t)' ;= 0, the scalar product
,
X (t + %(x)), f (x)

-
, = 0, for #hG/K -

almost every x $ G/H and for every t $ G [14, Theorem 23.11]. This implies that for each t

there is a negligible Borel subset $t of G/K such that X
,
t + %(x)
-
<, f (x) for every x /$ $t .

Note that for every x , span
5

X
,
t + %(x)
-
: t $ G
6
= span {X (t): t $ G} = ,X . If G admits a

countable dense subset G!, which is true in the case when G = !m""n, then from continuity

of X , it follows that also span
5

X
,
t + %(x)
-
: t $ G!
6
=,X . Therefore f (x) <, ,X for all x

which are not in the negligible set
7

t$G! $t . Hence f (x) = 0 for #hG/K -almost every x $ G/K
and Proposition 2 is proved.

4. SO-spectrum of a PC Field

Let X be a G/K-square integrable K-PC over G and let KX (t, s) be its covariance function.

The objective is to describe the domain of the SO-spectrum of X , which by definition (see

Definition 1) is the domain of the spectrum of the function %(t, s) = KX (t,2s).

First we give a description of the domain of the domain of the SO-spectrum of the PC field

X in the simplest case where G/K is compact.

Lemma 1. Let X be a G/K-square integrable K-PC over G, and let !K = ı!('G/K) = !G. Then the

domain of the SO-spectrum of X is the subgroup L of !G " !G given by

L = {(*,*2#):# $ !K ,* $ !G}.



D. Dehay, H. Hurd, A. Makagon / Eur. J. Pure Appl. Math, 7 (2014), 343-368 353

Note that L can be viewed as the union of hyperplanes

L =
8

#$!K

L# where L# := {(*,*2#):* $ !G}. (10)

Proof. Since KX (t + u, s + u) is K-periodic in u, the function %(t, s) = KX (t,2s) is itself

a periodic function on G " G with the period D = {(k,2k) : k $ K} = G " G. The domain

of the SO-spectrum of X is therefore the dual group of (G " G)/D viewed as a subgroup of
!G" !G. Note that the subgroup D is the image of the subgroup {0}"K through the isomorphism

> : G " G 4 (t, s) .% (t + s,2s) $ G " G, and this induces an isomorphism from the quotient

group (G"G)/D onto (G"G)/({0}"K). Furthermore, since (G"G)/({0}"K) = G"G/K and

its dual is !G "!K , we deduce that the dual of (G " G)/D can be identified with the subgroup

L of !G " !G consisting of the elements of the form (! ,! 2#), ! $ !G, # $ !K .

We have not used the fact that KX is a covariance function of a process. It turns out that

this additional property of KX (i.e. the fact that it is nonnegative definite) implies that the

"part of the SO-spectrum" that sits on each L# is a measure. We want to point out that the set

!K may be uncountable.

Theorem 2. Suppose that X is a G/K-square integrable K-PC field that satisfies the condition

[A] of Theorem 1. Then for every # $ !K there is a unique Borel complex measure *# on !G such

that

a#(t) =

(

!G

"
! , t
#
*#(d!), t $ G. (11)

Furthermore sup# Var(*#) <3 and for each # $ !K, the measure *# is absolutely continuous

with respect to *0.

In this paper by a representation of G in a Hilbert space 7 we mean a weakly continuous

group ? := {U t : t $ G} of unitary operators in 7 (see [14, Section 22]). In this case there

exists a weakly countably additive orthogonally scattered (w.c.a.o.s) Borel operator-valued

measure E on !G such that for every Borel set # the operator E(#) is an orthogonal projection

in 7 , and for every u, v $7 ,

,
U tu, v
-
7 =

(

!G

"
! , t
# ,

E(d!)u, v
-
7 , t $ G.

“Orthogonally scattered” means that
,
E(#1)u, E(#2)v

-
7 = 0 for all disjoint #1, #2 and

u, v $ 7 . The measure E will be referred to as the spectral resolution of the unitary operator

group ? .

Proof. [Theorem 2] The joint stationarity of the fields {Z# :# $ !K} defined by (7), implies

that each Z#(t) = U t Z#(0) where ? := {U t : t $ G} is the common shift operators group.

Condition [A] guarantees the continuity of the representation ? of G in L2(G/K;,X ), and

hence

Z#(t) =

(

!G

"
! , t
#

E(d!)Z#(0),



D. Dehay, H. Hurd, A. Makagon / Eur. J. Pure Appl. Math, 7 (2014), 343-368 354

where E is the spectral resolution of? . Therefore for every #,µ $ !K there is a complex mea-

sure "#,µ on !G such that R
#,µ
Z (t) =
+
!G
"
! , t
#
"#,µ(d!), namely, "#,µ(#) =

,
E(#)Z#(0), Zµ(0)

-
7 ,

where 7 := L2(G/K;,X ). Consequently, from relations (6) and (8) we conclude that

a#(t) =

(

G/K

&#, x'BX (t; x)#hG/K(d x) = R0,2#
Z (t) =

(

!G

"
! , t
#
"0,2#(d!).

Then equality (11) is satisfied with *# := "0,2#. From Cauchy-Schwarz inequality we have
33"0,2#(#)
33=
333
,
E(#)Z0(0), E(#)Z2#(0)

-
7

333

8
9
"0,0(#)
9
"2#,2#(#) =
9
"0,0(#)
9
"0,0(#2#),

and we deduce the absolute continuity of *# with respect to *0 for any # $ !K .

Finally note that the total variations of the measures "#,#, # $ !K , are all equal to "0,0(!G).
Indeed, since the measures "#,#, # $ !K , are non-negative

Var
,
"#,#
-
= "#,#(!G) = R#,#

Z (0) =

(

G/K

BX (0; y)#hG/K(d y) = R0,0
Z (0) = "

0,0(!G).

Hence all total variations Var
,
"#,µ
-
8
9
"#,#(!G)
9
"µ,µ(!G), #,µ $ !G, are bounded by the

same constant, and in consequence all measures *#, # $ !K , have uniformly bounded total

variations.

Remark that when the field X = P is K-periodic and G/K-square integrable, the field PK is

#hG/K -square integrable and thanks to Parseval equality, the spectral covariance function of the

field P can be expressed as

aP
#(t) =

(

G/K

&#, x'
,
PK(ı(t) + x), PK(x)

-
, #hG/K(d x)

=

(

!K

"
! , ı(t)
#,.PK(!),.PK(! 2#)

-
, #h!K

(d!)

where.PK is the Fourier Plancherel transform of the field PK . Then, we deduce that the function

! .%
,.PK(!),.PK(! 2#)

-
, is the density function of the SO-spectral measure *P

# of the field P

with respect to #h!K
,

*P
#(#) =

(

#+!K

,.PK(!),.PK(! 2#)
-
, #h!K

(d!) (12)

for any # $((!G). Particulary

*P
0 (#) =

(

#+!K

22.PK(!)
222
, #h!K

(d!).

Notice that the SO-spectral measure *P
# is concentrated on !K / !G : *P

#(#) = *
P
#(#+!K) for

any # $((!G). When in addition %Pk is #h!K
-integrable, then the fields PK and P are harmoniz-

able with



D. Dehay, H. Hurd, A. Makagon / Eur. J. Pure Appl. Math, 7 (2014), 343-368 355

KP(t, s) =KPK
(ı(t), ı(s))

=

( (

!K"!K

&#, ı(t)'
"
µ, ı(s)
# ,.PK(#),.PK(µ)
-
, #h!K

(d#)#h!K
(dµ)

=

( (

!G"!G

"
! , t
# "
) , s
#
%P(d! , d))

where %P is the measure on !G " !G concentrated on !K "!K defined by

%P(#) :=

( (

#+(!K"!K )

,.PK(!),.PK())
-
, #h!K

(d!)#h!K
(d)), # $((!G " !G).

From relation (12) we find out that the measure *P
# or more precisely its image %P

# := *P
# ) +21

#

through the mapping +# : !G % !G2 defined by +#(!) = (! ,! 2 #), ! $ !G, is the restriction of

the measure %P to the hyperplane L# := {(! ,! 2#):! $ !G}.
More generally, when X is a PC field, the family {*# :# $ !K} is commonly referred to as

the SO-spectral family of X . The measure *# or more precisely its image %# = *#)+21
#

represents

the part of the SO-spectrum of X that sits on the hyperplane L# = {(! ,! 2 #) : * $ !G}, see

relation (10).

Next, we give a sufficient condition for a PC field to be harmonizable.

Theorem 3. Let X be a G/K-square integrable K-PC field that satisfies the condition [A] of

Theorem 1, and let {*# : # $ !K} be the SO-spectral family of X . Suppose that there is an

#h!K
-integrable non-negative function , on !K such that for every # $ !K

|*#(#)|8,(#) for any Borel # $((!G). (13)

Then the field X is harmonizable and the SO-spectral measure of X is given by

%(#) =

(

!K

%#(#)#h!K
(d#), for any Borel # $((!G " !G), (14)

where %# := *# ) +21
#

and +#(!) := (! ,! 2#), ! $ !G.

Notice that condition (13) is satisfied by any G/K-square integrable K-periodic field P such

that.PK is #h!K
-integrable. Here we can take,(#) equal to the total variation of the SO-spectral

measure *P
# of the field P

,(#) =

(

!K

333
,.PK(!),.PK(! 2#)

-
,

333 #h!K
(d!).

The integrability condition on .PK : !K % , is always satisfied when !K is compact that is

when G/K is discrete, and in particular when G = !n.



D. Dehay, H. Hurd, A. Makagon / Eur. J. Pure Appl. Math, 7 (2014), 343-368 356

Proof. [Theorem 3] Let {Z# :# $ !K} be as in Theorem 1. From the proof of Theorem 2 it

follows that

*#(#) = "
0,2#(#) =
,
E(#)Z0(0), Z#(0)

-
7

Thanks to definition (7) and Lebesgue dominated convergence theorem it follows that the field

!K 4 # .% Z#(0) $ 7 is continuous, and hence by assumption (13), # .% *#(#) is integrable

over !K for every Borel # of !G. For all Borel D = !K and # = !G let us define

%̃(#" D) :=

(

D

*#(#)#h!K
(d#) =

(

D

,
E(#)Z0(0), Z#(0)

-
7 #h!K

(d#).

Condition (13) and again Lebesgue dominated convergence theorem entail that the func-

tion %̃(#" D) is countably additive in # and D separately. So to show that the bimeasure %̃

extends to a Borel measure on !G "!K , it is sufficient to show that its Vitali variation is finite

(see [9, 33]), that is

sup

:
n4

i=1

n4

j=1

33%̃(#i " Dj)
33:#i +# j = @ and Di + Dj = @ for i ;= j in {1, . . . , n}

;
<3.

Since
1n

i=1

33*#(# j)
338 Var(*#)8 4,(#) and the function , is #h#-integrable,

n4

i=1

n4

j=1

33%̃(#i " Dj)
338

n4

j=1

(

Dj

n4

i=1

33*#(#i)
33#h!K

(d#)

8
(

7
j Dj

4,(#)#h!K
(d#)8
(

!K

4,(#)#h!K
(d#)<3.

Hence %̃ is a measure and in particular Fubini and Lebesgue dominated convergence theorems

hold for %̃. Let # = !G be fixed and let -(#) :=
1n

j=1 bj1Dj
(#) be a simple function on !K .

Then (

!K

-(#)%̃(#, d#) =
n4

j=1

bj

(

Dj

*#(#)#h!K
(d#) =

(

!K

-(#)*#(#)#h!K
(d#).

From condition (13) we deduce that
+
!K
-(#) %̃(#, d#) =

+
!K
-(#)*#(#)#h!K

(d#) for any

bounded Borel function -. Consequently, for any simple function ' on !G and bounded - on

!K ((

!G"!K

'(!)-(#) %̃(d! , d#) =

(

!K

-(#)

<(

!G
'(!)*#(d!)

=
#h!K
(d#). (15)

If |'| is bounded by some finite c > 0 then by condition (13), the integral
+
!G |'(!)|*#(d!) is

bounded by 4c,(#)which is an #h#-integrable function of #. Therefore by Lebesgue dominated

convergence theorem, relation (15) holds for any two bounded measurable functions ' on !G



D. Dehay, H. Hurd, A. Makagon / Eur. J. Pure Appl. Math, 7 (2014), 343-368 357

and - on !K . In particular

((

!G"!K

"
! , t
#
&#, x' %̃(d! , d#) =

(

!K

&#, x'
<(

!G

"
! , t
#
*#(d!)

=
#h!K
(d#)

=

(

!K

&#, x'a#(t)#h!K
(d#) = BX (t; x) = KX (t + x , x)

(16)

for all t $ G and x $ G/K .

Let + : !G"!K % !G 2 be defined by +(! ,#) := +#(!) = (! ,! 2#), and let %= %̃)+21 be the

image of the measure %̃ through the mapping +, that is %(#) = %̃{(! ,#):(! ,! 2#) $#}. Then

% is a Borel measure on !G 2 and change of variables formula yields that

((

!G"!G
.(!1,!2)%(d!1, d!2) =

((

!G"!K

.(! ,! 2#) %̃(d! , d#) (17)

for any bounded Borel function . : !G " !G% $. In particular, in view of relation (16)

((

!G"!K

"
! , t
#
&#, s' %(d! , d#) =

((

!G"!K

"
! , t
# "
(! 2#), s
#
%̃(d! , d#)

=BX

,
t 2 s; ı(s)
-
= KX (t, s).

for all t, s $ G (recall that &#, s' = &#, ı(s)' for all # $ !K and s $ G). Thus the field X

is harmonizable and % is its SO-spectral measure. Note that relation (15) holds true if the

product '(!)-(#) is replaced by any bounded measurable function .(! ,#) of two variables.

Thanks to such upgraded relation (15) and to relation (17) with .= 1#, we get

%(#) =

((

!G"!K

1#(! ,! 2#) %̃(d! , d#) =

(

!K

<(

!G
1#(! ,! 2#)*#(d!)

=
#h!K
(d#)

So, by the definition of %̃, we deduce relation (14).

Note that if G = !n then condition [A] is satisfied, !K is compact, and condition (13) holds

true with ,(#) = Var(*#) <3. Therefore we generalize the property of harmonizability of

the PC sequences proved in [12].

Corollary 1. Any G/K-square integrable K-PC field over G = !n is harmonizable.

All the results above simplify significantly if G/K is compact, because then every K-PC field

over G is G/K-square integrable and condition [A] in Theorem 1 is always satisfied.

Theorem 4. Suppose that X is a K-PC field and that G/K is compact. Then for every # $ !K

there is a Borel complex measure *# on !G such that

a#(t) =

(

!G

"
! , t
#
*#(d!), t $ G.



D. Dehay, H. Hurd, A. Makagon / Eur. J. Pure Appl. Math, 7 (2014), 343-368 358

Moreover the set !K is countable,
1
#$!K
|a#(t)|2 <3 and for every t $ G

BX (t; x)
L2

=
4

#$!K

&#, x'a#(t) (18)

(series (18) converges in L2(G/K) with respect to x). Additionally:

(i) if
1
#$!K
|a#(t)| <3 for every t $ G, then the series (18) converges also pointwise and

uniformly with respect to x $ G/K, and for all t, s $ G we have

KX (t + s, s) =
4

#$!K

&#, s'a#(t);

(ii) if
1
#$!K

Var(*#)<3, then X is harmonizable, and for all t, s $ G we have

KX (t, s) =

((

!G"!G

"
! , t
# "
/, s
#
%(d! , d/),

where the SO-spectral measure % is given by %(#) =
1
#$!K
%#(#), %# := *# ) +21

#
and

+# : !G% !G " !G is defined as +#(!) := (! ,! 2#).

Proof. Existence of *# follows from Theorem 2. Since for each t $ G the function

x .% BX (t; x) is bounded, it is in L2(G/K) and hence its Fourier transform # .% a#(t) is

in L2(!K). Formula (18) is just the inverse formula for BX (t; ·). Item (i) follows from the

uniqueness of the Fourier transform, while item (ii) from Theorem 3.

5. Structure of PC fields

When X is a K-PC field then for every k $ K the mapping V k : X (t) .% X (t + k), t $ G,

is well defined and extends linearly to an isometry from ,X = span {X (t): t $ G} onto itself.

The group A := {V k :k $ K} is a unitary representation of K in,X and is called the K-shift of

X .

Theorem 5. A continuous field X over G is K-PC if and only if there are a unitary representation

? = {U t : t $ G} of G in,X , and a continuous K-periodic field P over G with values in,X such

that X (t) = U t P(t), t $ G.

Proof. The “if” part is obvious. Prove the other part. Let X be a K-PC field, A = {V k :k $ K}
be the K-shift of X , and E be the spectral resolution of A . Hence E is a w.c.a.o.s. Borel operator-

valued measure defined on !K . Since !K is isomorphic to !G/!K , the measure E can be seen as a

measure on !G/!K , see [37, Section 2.1.2]. Let 0 be a cross-section for !G/!K . For every Borel

subset # of !G let us define Ẽ(#) := E
,
021(#)
-
. Then Ẽ is a w.c.a.o.s. Borel operator-valued

measure on !G whose support is contained in a measurable set 0(!G/!K), and whose values are



D. Dehay, H. Hurd, A. Makagon / Eur. J. Pure Appl. Math, 7 (2014), 343-368 359

orthogonal projections in,X . Since for all ! $ !K and k $ K ,
"
! , k
#
=
"
0(!), k
#
, by change of

variable we obtain that

V k =

(

!G

"
! , k
#

Ẽ(d!), k $ K .

Following Gladyshev’s idea [12] for every t $ G define the operator on,X ,

U t :=

(

!G

"
! , t
#

Ẽ(d!), t $ G.

Clearly ? := {U t : t $ G} is a group of unitary operators indexed by G. Moreover for every

v $,X ,

-(U t 2 I)v-2, =
(

!G

333
"
! , t
#
2 1
333
2

µv(d x)

where µv(d x) = -E(d x)v-2, is a finite non-negative measure on !G. From Lebesgue dominated

convergence theorem we therefore conclude that the unitary operator group? is continuous,

and hence it is a unitary representation of G in,X . Note that for t = k $ K , we have Uk = V k.

Define P(t) := U2t X (t), t $ G. Then P is continuous and

P(t + k) = U2t U2kX (t + k) = U2t V2kX (t + k) = U2t X (t) = P(t), t $ G, k $ K .

So P is a continuous K-periodic field with values,X and X (t) = U t P(t), for every t $ G.

Theorem 5 gives a good insight on the origin of the measures *#, # $ !K . Indeed let X be

a G/K-square integrable K-PC field, ? and P be as defined in Theorem 5. Then

-X (t)-, = -P(t)-, and
,
X (t + u), X (u)
-
, =
,
U t P(t + u), P(u)

-
,

for all t,u $ G. The PC field X being G/K-square integrable, the field PK : G/K %,X defined

by P = PK ) ı is square integrable as well as the field x .% U t PK(ı(t) + x) defined on G/K , for

any t $ G. Denoting by .PK : !K % ,X the Fourier Plancherel transform of PK , the Fourier

Plancherel transform of U t PK(ı(t) + ·) coincides with the function

µ .%
(

!G

"
! +µ, t
#

Ẽ(d!).PK(µ).

where Ẽ is the Borel operator-valued measure on !G defined in the proof of Theorem 5. Then

thanks to Parseval equality, the spectral covariance function of the PC field X verifies

a#(t) =

(

G/K

&#, x'
,
U t PK(ı(t) + x), PK(x)

-
, #hG/K(d x)

=

(

!K

)(

!G

"
! +µ, t
#

Ẽ(d!).PK(µ),.PK(µ2#)
*

,
#h!K
(dµ)

=

(

!G

"
1, t
#
>(

!K

,
Ẽ(d1 2µ).PK(µ),.PK(µ2#)

-
, #h!K

(dµ)

?



D. Dehay, H. Hurd, A. Makagon / Eur. J. Pure Appl. Math, 7 (2014), 343-368 360

for all # $ !K and t $ G. The SO-spectral measure of the field X is

*#(#) =

(

!K

,
Ẽ(#2µ).PK(µ),.PK(µ2#)

-
, #h!K

(dµ), # $((!G),# $ !K .

In comparison with expression (12), we see that the spectral resolution Ẽ of the unitary oper-

ators group ? , in some sense, “spreads” the SO-spectral measure *P
# over !G to form *#.

Theorem 5 also suggests a possibility to decompose a PC field into stationary components.

If the K-PC field X is G/K-square integrable, we can therefore formally write

X (t)B
(

!K

&#, t'X#(t)#h!K
(d#) (19)

where {X#(t) := U t.PK(#) : t $ G}, # $ !K , is a family of jointly stationary fields over G.

If in addition .PK is integrable, then integral (19) exists, and we have equality for any t. The

integrability condition on.PK being satisfied if !K is compact, that is in particular when G = !n,

we deduce the following result.

Corollary 2. Let X be a G/K-square integrable K-PC field over G = !n. Then there exists a family

{X# :# $ !K} of jointly stationary fields over G in,X such that

X (t) =

(

!K

ei#t 0X#(t)#h!K
(d#), t $ G,

Note that the pair (? , P) in Theorem 5 is highly non-unique since there are many ways to

extend A = {V k :k $ K} into ? = {U t : t $ G}. Consequently, the family {X# :# $ !K} above

is likewise not unique.

If G/K is compact, then every K-PC field is G/K-square integrable, !K is countable, and

the integrals above become series. If G = !n and G/K is compact (and hence finite), then !K

is finite and Corollary 2 yields the following !n version of Gladyshev’s representation of PC

sequences included in [12].

Corollary 3. Suppose that X is a K-PC field over !n and that !n/K is compact. Then !K is finite

and there is a finite family {X# :# $ !K} of jointly stationary fields over G in ,X such that for

every t $ G

X (t) =
4

#$!K

ei#t 0X#(t).

If !K is not compact then even in the case of a periodic function, its Fourier transform does

not have to converge everywhere.

6. Examples

In this section G = !n ""m, the Haar measure on !n is the counting measure, the Haar

measure on "m is d t/(
*

2&)m where d t is the Lebesgue measure on "m, .!n will be identified



D. Dehay, H. Hurd, A. Makagon / Eur. J. Pure Appl. Math, 7 (2014), 343-368 361

with [0,2&)n with addition mod 2&, @"m will be identified with "m, the Haar measures on

[0,2&)n is d t/(2&)n, !G = [0,2&)n ""m, elements of G and !G are row vectors, and the value

of a character ! $ !G at t $ G is
"
! , t
#
= e2i#t 0 , where t 0 is transpose of t. Remembering

previous sections, in order to describe the domain of the SO-spectrum of a K-PC field X over

G the only task is to identify G/K and'G/K as concrete subsets Q and !K of G = !n""m and
!G = [0,2&)n ""m, respectively, in the way that the value of character # $ !K at t $ Q is still

&#, t' = e2i#t 0 . This identification, which is obvious when X is coordinate-wise PC, may be

less trivial in the case of more complex K . It may be helpful, and is worth, to note that any

closed nontrivial subgroup K of G = !n ""m is isomorphic to !k ""l for some k, l $ & such

that l 8 m and 1 8 k + l 8 n+m. This isomorphism, which at least in the case of G = !n or

G = "m can be found by selecting a proper basis for G (see [14, Theorem 9.11 and A.26]),

provides a description and a parametrization of the sets Q and !K . As before
/
a
0

b
will denote

the remained in integer division of a by b, b > 0.

First we briefly revisit a one-parameter case and its slight extension.

Example 1. Suppose that X is a PC process with period T > 0. Then K =
5

kT : k $ !
6
,

"/K = [0, T ) with addition modulo T and !K =
5

2&k
T :k $ !
6
. Moreover for every

# = 2&k
T $ !K, a#(t) := ak(t) =

1
T

+ T
0

e2is 2&k
T KX (t + s, s) ds and there is a measure *k on " such

that ak(t) =
+
"

eitu *k(du) (Theorem 2). The domain of the SO-spectrum of X is L =
7

k$! Lk,

where Lk :=
5,

u,u2 2&k
T

-
:u $ "
6
. The part of the SO-spectrum that sits on Lk is a measure %k

defined as %k = *k ) +21
k

where +k : "% "2, +k(u) =
,
u,u2 2&k

T

-
. If
1

k Var(*k) <3 then the

process X is harmonizable and %:=
1

k %k is a measure on "2 which satisfies relation (5).

If X is stationary then K = ", "/K = {0}, !K = {0},

a0(t) =

(

{0}
KX (t + s, s)(0(ds) = KX (t, 0).

By Theorem 2 there is a measure *0 on " such that a0(t) =
+
"

eitu *0(du). Consequently, the

SO-spectrum of X is the measure %= %0 = *0)+21
0 , which sits on the diagonal L0 =

5
(u,u):u $ "
6
.

To see the need for the square integrability assumption, let us add one parameter to the above

process; that is, let us consider a field X = {X (s, t):(s, t) $ "2} such that

KX

,
(s, t), (u, v)
-
= KX

,
(s+ T, t), (u+ T, v)

-
, s, t,u, v $ "

(T > 0 is fixed). Then K =
5
(kT, 0) :k $ !
6
, "2/K = [0, T )"" with addition modulo T on the

first coordinate, !K =
5,

2&k
T , x
-
:k $ !, x $ "
6

and

a#(s, t) := ak,x(s, t) =
1

T
*

2&

( T

0

(

"

e2i
,

u 2&k
T +vx
-
KX

,
(s+ u, t + v), (u, v)

-
dudv

for # =
,

2&k
T , x
-
$ !K. The square integrability assumption

+ T
0

A+
"
-X (s, t)-2, d t
B

ds < 3
assures that the above integral exists. If it does then, in view of Theorem 2, for every k $ !



D. Dehay, H. Hurd, A. Makagon / Eur. J. Pure Appl. Math, 7 (2014), 343-368 362

and x $ " there exists a measure *k,x on "2 such that ak,x(s, t) =
+
"

ei(su+t v) *k,x(du, dv).

The domain of the SO-spectrum of X is L =
7

k$!
7

x$" Lk,x , where Lk,x is a two-dimensional

plane in "4, Lk,x :=
5,

u, v,u 2 2&k
T , v 2 x
-

: u, v $ "
6
. The “part” of the SO-spectrum that

sits on Lk,x is a measure %k,x defined as %k,x := *k,x ) +21
k,x

, where +k,x : "2 % "4 is defined by

+k,x(u, v) :=
,
u, v,u2 2&k

T , v 2 x
-
. If Var(%k,x)8,(k, x) and

1
k

+
"
,(k, x) d x <3, then X is

harmonizable and the SO-spectral measure of X is %= 1*
2&

1
k

+
"
%k,x d x (see Theorem 3). Note

that L above is, in fact, the union of countably many three-dimensional hyperplanes Dk in "4,

Dk :=
7

x$" Lk,x =
5
(u, v,u2 2&

T , v 2 x):u, v, x $ "
6
, which are parallel to the “diagonal” D0.

If the field X = {X (s, t) : (s, t) $ "2} is stationary in s, then !K =
5
(0, x) : x $ "
6
, the

condition of the square integrability of X means that
+
"
-X (0, t)-2, d t <3 and, if the latter is

satisfied, the SO-spectrum of X sits on the three-dimensional hyperplane in "4,

L := D0 =
5
(u, v + x ,u, x):u, v, x $ "

6
.

Next example contains a complete analysis of the SO-spectrum of a weakly PC field.

Example 2. Let T and S be two non-zero integers. Suppose that the field X on !2 is weakly PC

with period (T,S) [23], that is

KX

,
(m, n), (u, v)
-
= KX

,
(m+ T, n+ S), (u+ T, v + S)

-
, for all n, m,u, v $ !

Here K =
5

k(T,S) :k $ !
6
. We assume that at least one of T or S is positive. Let d := gcd(T,S)

be the greatest common positive integer divisor of T and S, so that (T,S) = d " (T1,S1) and

gcd(T1,S1) = 1. From Bezout’s lemma there are integers q, p such that T1q2 S1p = 1. Let ' be

a mapping of !2 onto itself, given by '(m, n) = (m, n)%0, where % =

)
T1 p

S1 q

*
, and %0 stands

for the transpose matrix of the matrix %. Because det% = 1, the mapping ' is an isomorphism.

Since '(dk, 0) = (kT, kS) for k $ !, we have K = '(d!" {0}) and we identify G/K to

Q := '
,
{0, . . . , d 2 1}"!

-
=
5
(kT1 + l p, kS1 + lq):k = 0, . . . , d 2 1, l $ !

6
.

The dual mapping.(s, t) =
/
(s, t)%21
0

2&
=
,/

qs2S1 t
0

2&
,
/
2 ps+ T1 t
0

2&

-
, s, t $ [0,2&), maps5

2&k
d : k = 0, . . . , d 2 1

6
" [0,2&), which is the dual of {0, . . . , d 2 1}" !, onto the dual !K of

Q. The construction that we use produces a convenient parametrization of !K as the union of d

lines: !K =
7d21

k=0!k where

!k :=

CDE
2&kq

d
2 S1 t

F

2&
,

E22&kp

d
+ T1 t

F

2&

G
: t $ [0,2&)

H
.

Note that the value of a character (u, v) = .(s, t) $ !K at (m, n) = '(k, l) $ Q is equal to

e2i(mu+nv) = e2i(s,t)%21%(k,l)0 = e2i(ks+l t), as required. Assume that X is G/K-square integrable,

for example that
13

n=23 -X (m, n)-2, <3, for any m = 1, . . . , T 2 1. Then from the previous

discussion and the results of Section 4 we deduce the following properties.



D. Dehay, H. Hurd, A. Makagon / Eur. J. Pure Appl. Math, 7 (2014), 343-368 363

(i) If (u, v) $ !K, then u =
/2&kq

d 2 S1 t
0

2&
, v =
/22&kp

d + T1 t
0

2&
for some unique t $ [0,2&)

and unique k = 0, . . . , d 2 1. Hence the spectral covariance a(u,v)(m, n) =: ak,t(m, n) of X

at (m, n) $ !2 is equal to

ak,t(m, n) =
1

d

d214

j=0

34

l=23
e2i
,

j
2&kq

d +l t
-
KX

,
(m+ jT1+ l p, n+ jS1+ lq), ( jT1+ l p, jS1+ lq)

-
.

(ii) For each k = 0, . . . , d 2 1 and t $ [0,2&) there exists a measure *k,t on [0,&)2 such that

ak,t(m, n) =

( 2&

0

( 2&

0

ei(mx+ny) *k,t(d x , d y).

(iii) The SO-spectrum of X sits on the set L =
7d21

k=1

7
t$[0,2&) Lk,t , where Lk,t is a two-dimensional

plane in [0,2&)4

Lk,t :=

CD
x , y,

E
x 2 2&kq

d
+ S1 t

F

2&
,

E
y +

2&kp

d
2 T1 t

F

2&

G
: x , y $ [0,2&)

H
.

Note that Dk =
7

t$[0,2&) Lk,t is a three-dimensional hyperplane in [0,2&)4, so L is, in fact,

the union of d three-dimensional hyperplanes. If d = gcd(T,S) = 1, then

L = D0 =
IJ

x , y,
/
x + St
0

2&
,
/

y 2 T t
0

2&

K
: x , y, t $ [0,2&)

L
. In this case the field X is a

rotation of the field Y defined by Y (m, n) := X
,
(m, n)%0
-
, m, n $ !, which is stationary in

m. Indeed X (m, n) = Y
,
(m, n)(%0)21
-
.

(iv) If there is an integrable function , : [0,2&)% [0,3) such that Var(*k,t) 8 ,(t) for all

k and t, then X is harmonizable and

KX

,
(m, n), ( j, r)
-
=

( 2&

0

( 2&

0

( 2&

0

( 2&

0

ei(mu+nv2 j x2r y) %(du, dv, d x , d y).

The SO-spectral measure % of X is given by %(#) =
1d21

k=0

+ 2&
0
%k,t(#) d t, where %k,t is the

complex measure on [0,&)4 whose support is contained in the plane Lk,t and defined by

%k,t(#):= *k,t

C
(x , y) $ [0,2&)2 :

D
x , y,

E
x 2 2&kq

d
+ S1 t

F

2&
,

E
y +

2&kp

d
2 T1 t

F

2&

G
$#
H

.

Figure 1 is the graph of the set !K defined previously in the case when T = 12 and S = 9 (d = 3,

p = q = 1). Then !K = !0C!1C!2 consists of three lines, which are shown with different width

pattern. If now from each point on the graph we draw the rectangle [0,2&) " [0,2&) then the

resulting three-dimensional body in [0,2&)4 is the domain of the SO-spectrum of the field X .



D. Dehay, H. Hurd, A. Makagon / Eur. J. Pure Appl. Math, 7 (2014), 343-368 364

0 1 2 3 4 5 6
0

1

2

3

4

5

6

 

 

 

Figure 1: Graph of !K

The last example is a particular example of strongly PC fields over "" !2. It combines a

mixture of continuous and discrete structures.

Example 3. Suppose that X is a field over G := ""!2 such that

X (t, m, n) = X (t + 4, m, n) = X (t, m+ 1, n+ 3) = X (t, m+ 2, n)

for all t $ ", m, n $ !. Then K = {k(4,0,0) + j(0,1,3) + l(0,2,0) : k, l, j $ !}. In order to

describe G/K and !K we consider a change of basis of G = ""!2 defined by the mapping

'(t, m, n) := (t, m, n)%0, where %=

M

N
1 0 0

0 1 0

0 3 1

O

P .

Then the mapping ' is an isomorphism of G onto itself and K = '(P), where

P = {(4k, j, 6l):k, l, j $ !} = 4!"!" 6!.

To see this note that 2(0,1,3)2(0,2,0) = (0,0,6), so that K is generated by the 3-tuples (4,0,0),

(0,1,3) and (0,0,6), which are respectively equal to '(4,0,0), '(0,1,0), and '(0,0,6). The

quotient G/P = [0,4)"{0}"{0, . . . , 5}, so we take Q := '(G/P) = {(s, 0, l):s $ [0,4), l = 0, . . . 5}.
The dual of G/P is !P =

2&
T !" {0}"
5
&r
3 : r = 0, . . . , 5
6

and hence the dual of G/K can be rep-

resented as !K = .(!P), where . is the isomorphism of !G = "" [0,2&)2 onto itself defined by

.(t,u, v) := (t,u, v)%21 =
,
t,
/
u2 3v
0

2&
, v
-
. Therefore

!K =

CD
2&k

T
,
/
2&r
0

2&
,
&r

3

G
:k $ !, r = 0, . . . , 5

H
,

is countable. Note that
/
2&r
0

2&
is either & (if r is odd) or 0. For each

#k,r =

D
2&k

T
,
/
2&r
0

2&
,
&r

3

G
$ !K ,



REFERENCES 365

the corresponding spectral covariance is given by

ak,r(t, m, n) =
1

24

54

l=0

( 4

0

e2i( 2&ks
T +

&rl
3 )KX

,
(t + s, m, n+ l), (s, 0, l)

-
ds,

and for each k, r there exists a measure *k,r on "" [0,2&)2 such that

ak,r(t, m, n) =

(

"

( 2&

0

( 2&

0

ei(ts+mu+nv)
, *k,r(ds, du, dv).

The SO-spectrum of the field X sits on the union of countably many hyperplanes

Lk,r :=

CD
s,u, v, s2 2&k

T
, [u+&r]2& ,

E
v 2 &r

3

F

2&

G
:s $ ",u, v $ [0,2&)

H
,

k $ !, r = 0, . . . , 5, of " " [0,2&)2 " " " [0,2&)2. If the sum of total variations of measures

*k,r is finite, then X is harmonizable and its SO-spectral measure %=
13

k=23
15

r=0 %k,r , where

%k,r = *k,r ) +21
k,r

and +k,r(s,u, v) :=
,
s,u, v, s2 2&k

T ,
/
u+&r
0

2&
,
/
v 2 &r

3

0
2&

-
.

Note that if we define Y (t, n, m) := X
,
(t, n, m)%0
-
, then Y is PC in t with period T = 4,

stationary in n, and PC in m with period M = 6. Since X (t, n, m) = Y
,
(t, n, m)(%0)21
-
, one can

therefore say that the field X is periodically correlated in direction (1,0,0) with period T = 4,

stationary in direction of (0,1,3) and periodically correlated in direction (0,0,1) with period

M = 6.

ACKNOWLEDGEMENTS The paper was partially written during the author’s stay at Univer-

sité Rennes 2, Rennes, France, in June 2011.

References

[1] V.G. Alekseev. On spectral density estimates of Gaussian periodically correlated random

fields. Probability and Mathematical Statistics 11(2), 157–167. 1991.

[2] T. Bose, M.Q. Chen, K.S. Joo, and G.F. Xu. Stability of two-dimensional discrete systems

with periodic coefficients. IEEE Transactions on Circuits and Systems II: Analog and Digital

Signal Processing 45(7), 839–847. 1998.

[3] T. Bose, R. Thamvichai, and T. Radenkovic. Stability of the 2D Fornasini-Marchesini

model with periodic coefficients. IEEE International Conference on Acoustics, Speech, and

Signal Processing, ICASSP’01, Salt Lake City, UT, vol.3, 1925–1928. 2001.

[4] W. Chen, G.B. Giannakis, and N. Nandhakumar. Spatio-temporal approach for time-

varying image motion estimation. IEEE Transactions on Image Processing 10, 1448–1461.

1996.



REFERENCES 366

[5] D. Dehay and H.L. Hurd. Representation and estimation for PC and almost PC random

processes. In: Cyclostationarity in Communications and Signal Processing (Ed. W.A. Gard-

ner), IEEE Press, New York, 295–329. 1994.

[6] D. Dehay and H.L. Hurd. Spectral estimation for strongly periodically correlated random

fields defined on "2. Mathematical Methods of Statistics 11, 135–151. 2002.

[7] Y.P. Dragan and I.N. Yavorskii. The periodic correlation random field as a model for bidi-

mensional ocean waves. Otbor i Peredacha informasii 51, 15–21 (in Russian). 1982.

[8] N. Dinculeanu. Vector Integration and Stochastic Integration in Banach Spaces, J. Wiley &

Sons, New York, 2000.

[9] N. Dunford and J. Schwartz. Linear Operators I and II (re-edition), J. Wiley & Sons, New

York, 1988.

[10] W.A. Gardner, A. Napolitano, and L. Paura. Cyclostationarity: Half a century of research.

Signal Processing 86(4), 639–697. 2006.

[11] P. Gaşpar. On Operator periodically correlated random fields. Operator Theory: Advances

and Applications 153, 143–156. 2004.

[12] E.G. Gladyshev. Periodically correlated random sequences. Soviet Mathematics 2, 385–

388. 1961.

[13] E.G. Gladyshev. Periodically and almost periodically correlated random processes with

continuous time parameter. Theory of Probability and its Applications 8, 173–177. 1963.

[14] E. Hewitt and K.A. Ross. Abstract Harmonic Analysis I and II, 2nd. ed., Springer-Verlag,

Berlin, 1979.

[15] E. Hille. Functional Analysis and Semi-groups, A.M.S. Colloquium Pub. 31, American

Mathematical Society, New York, 1948.

[16] I. Honda. On the spectral representation and related properties of periodically correlated

stochastic processes. IEICE Transactions (1976–1990), E65, 12, 723–729. 1982.

[17] C. Houdré. Linear Fourier and stochastic analysis. Probability Theory and Related Fields

87, 167–188. 1990.

[18] H.L. Hurd. An Investigation of Periodically Correlated Stochastic Processes, Ph.D. Disserta-

tion, Duke University, Durham, North Carolina, 1969.

[19] H.L. Hurd. Periodically correlated processes with discontinuous correlation functions.

Theory of Probability and its Applications 19, 804–808. 1974.

[20] H.L. Hurd. Stationarizing properties of random shifts. SIAM Journal on Applied Mathe-

matics 26, 203–211. 1974.



REFERENCES 367

[21] H.L. Hurd. Representation of strongly harmonizable periodically correlated processes

and their covariance. Journal of Multivariate Analysis 29, 53–67. 1989.

[22] H.L. Hurd and G. Kallianpur. Periodically correlated processes and their relationship to

L1(0, T )-valued stationary processes. In: Nonstationary Stochastic Processes and Their

Applications (Ed. A. G. Miamee), World Scientific, Singapore, 256–284. 1991.

[23] H.L. Hurd, G. Kallianpur, and J. Farshidi. Correlation and spectral theory for periodically

correlated random fields indexed on !2. Journal of Multivariate Analysis 90, 359–383.

2004.

[24] H.L. Hurd and A.G. Miamee. Periodically Correlated Random Sequences; Spectral Theory

and Practice, J. Wiley & Sons, Hoboken, New Jersey, 2007.

[25] E.T. Kehlet. Cross-sections for quotient maps of locally compact groups. Mathematica

Scandinavica 55, 152–160. 1984.

[26] L.H. Loomis. An Introduction to Abstract Harmonic Analysis, Van Nostrand Company, New

York, 1953.

[27] A. Makagon, A.G. Miamee, and H. Salehi. Continuous time periodically correlated pro-

cesses : spectrum and prediction. Stochastic Processes and their Applications 49, 277–295.

1994.

[28] A. Makagon. Induced stationary process and structure of locally square integrable peri-

odically correlated processes. Studia Mathematica 136(1), 71–86. 1999.

[29] A. Makagon. Characterization of the spectra of periodically correlated processes. Journal

of Multivariate Analysis 78(1), 1–10. 2001.

[30] A.G. Miamee. PC processes and their stationary dilation. SIAM Journal on Applied Math-

ematics 50, 1194–1199. 1990.

[31] A.G. Miamee and H. Salehi. On the prediction of periodically correlated stochastic pro-

cess. In: Multivariate Analysis V (Ed. P. R. Krishnaiah), North Holland, Amsterdam, 167–

179. 1980.

[32] H. Niemi. Stochastic processes as Fourier transforms of stochastic measures. Annales

Academiae Scientiarum Fennicae. Series A 1, Mathematica 591, 1–47. 1975.

[33] M.M. Rao and D. K. Chang. Bimeasures and nonstationary processes. In: Real and

Stochastic Analysis (Ed. M.M.Rao.), J. Wiley & Sons, 7–118. 1986.

[34] M.M. Rao. Characterizations of harmonizable fields. Nonlinear Analysis 63, 935-947.

2005.

[35] H. Reiter. Classical Harmonic Analysis and Locally Compact Groups, Oxford University

Press, Oxford, 1968.



REFERENCES 368

[36] W. Rudin. Functional Analysis, McGraw-Hill, New-York, 1973.

[37] W. Rudin. Fourier Analysis on Groups, J. Wiley & Sons, New York, 1990.

[38] E. Serpedin, F. Panduru, I. Sari, and G.B. Giannakis. Bibliography on cyclostationarity.

Signal Processing 85, 2233–2303. 2005.

[39] V.S. Varadarajan. Geometry of Quantum Theory, vol.2, Springer-Verlag, New York, 1985.