On the almost periodic at infinity functions from homogeneous spaces Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 7 (25), No. 2, 2018, pp. 3-19

3

DOI: 10.15393/j3.art.2018.4990

UDC 517.9

A. G. BASKAKOV, V. E. STRÜKÜV, I. I. STRÜKÜVA

ON THE ALMOST PERIODIC AT INFINITY FUNCTIONS FROM HOMOGENEOUS SPACES

Abstract. We consider homogeneous spaces of functions defined on the real axis (or semi-axis) with values in a complex Banach space. We study the new class of almost periodic at infinity functions from homogeneous spaces. The main results of the article are connected to harmonic analysis of those functions. We give four definitions of an almost periodic at infinity function from a homogeneous space and prove them to be equivalent. We also introduce the concept of a Fourier series with slowly varying at infinity coefficients (neither necessarily constant nor necessarily having a limit at infinity). It is proved that the Fourier coefficients of almost periodic at infinity function from a homogeneous space (not necessarily continuous) can be chosen continuous. Moreover, they can be extended on c to bounded entire functions of exponential type. Besides, we prove the summability of Fourier series by the method of Bochner-Fejer. The results were received with essential use of isometric representations and Banach modules theory. Key words: almost periodic at infinity function, homogeneous space, Banach module, almost periodic vector, Fourier series 2010 Mathematical Subject Classification: 33E30, 43A60,

46B04

1. Homogeneous function spaces. Let X be a complex Banach space, EndX be a Banach algebra of bounded linear operators on X. Let j be either r+ = [0, ro), or r = (-ro, ro).

By Ljoc(J,X) denote the space of Bochner measurable locally integrable on j (classes of) functions with values in X. By Sp(j, X), p E [1, ro), denote the Stepanov space of functions [15] x E L}oc(J,X) with the following norm:

i/p

|x||Sp = sup ( J ||x(s + t) IIXdt) , p E

sei

© Petrozavodsk State University, 2018

i

Stepanov spaces play an important role in studying differential equations in Banach spaces (see [4]).

Definition 1. A Banach space F(r, X) of functions defined on r with values in a Banach space X is called homogeneous, if the following conditions are satisfied:

(a) the space F(r, X) is injectively (which means injectivity of the inclusion operator) and continuously embedded in S:(r,X);

(b) a group of shift operators S(t), t E r, of the form

(S(t)x)(s) = x(s + t), s, t E r, x E F(r,X), (1)

is defined on F(r, X);

(c) for any functions f E L:(r), x E F(r,X) their convolution defined by

(f * x)(t) = y f (t)x(t - T)dr = J f (t)(S(-T)x)(t)dT, t E r, (2)

r r

belongs to F(r,X) and satisfies the condition ||f * x|| ^ C||f ||i|x| for some C ^ 1 (usually C =1);

(d) the inclusion px E F(r, X) holds for any x E F(r, X) and any infinitely differentiable function p E Cb(r) with compact support supp p; moreover, inequality ||px|| ^ ||p|1|x| holds and the mapping t M- pS(t)x : r ^ F(r, X) is continuous.

By Fo(r, X) we denote the least closed subspace of F(r,X) containing all functions px, x E F(r, X), where p E Cb(r,X) is infinitely differentiable and its support supp p is a compact set.

Definition 2. A Banach space F(r+,X) of functions from S 1(r+,X) is called homogeneous, if a homogeneous space F(r, X) associated with F(r+,X) exists, such that for any function x E F(r+,X) there is an extension y E F(r, X) with the following properties:

1) y(t) = x(t) for all t E r+;

2) ||y|| ^ C||x||, C > 0;

3) y EFo(r-,X);

4) S(t)x E F(r+,X) for all t ^ 0, x E F(r+,X);

5) for any other extension z E F(r, X) with the properties 1)-4) the condition y — z E F0 (r, X) holds.

In this article we denote a homogeneous space by F(j, X), while in case X = c the notation F(j) can be used. Denote a closed subspace of F(j, X) defined by {x E F(j,X) : function t ^ S(t)x : j ^ F(j,X) is continuous} by Fc(j,X).

Example 1. The following Banach spaces of functions defined on j with values in a Banach space X are homogeneous. All of them are linear subspaces of L/oc(j,X).

1) The spaces Lp = Lp(j,X), p E [1, ro), of Lebesgue measurable and

integrable with power p E [1, ro) (classes of) functions with the norm

\ i/p

ixilp =(/ l|x(t)IIXdt) E [!> O

if1 \q/p\ 1/q

l|x(s + k)HpdsJ j < oo, p, q E [1, o).

Note that (Lp(j,X))c = Lp(j,X), (Lp(j,X))o = Lp(j,X).

2) The space L^ = L^(j,X) of essentially bounded (classes of) functions with the following norm: ||x||^ = vrai sup ||x(t)||X. Note that

tej

(L~(j,X ))c = C6)u(j,X).

3) Stepanov spaces Sp = Sp(j,X), p E [1, ro).

4) Wiener amalgam spaces (Lp,/q) = (Lp(j,X),/q(j,X)), p E [1, ro),

q E [1, ro], (see [4]) of functions x E L/oc(j,X) such that

q/p\ 1/q

Ix 11 p, q I

\fcez

The following equivalent norm can also be used:

if1 \q/p\1/q

y^l ||x(s + t + k)||pds <ro, p,q E [1,ro).

teI0,1] Vfce^o J J

5) The space Cb = Cb(j,X) of bounded continuous functions with

the norm ||x||^ = sup ||x(t)||X, x E Cb(j,X). Note that Cb(j,X) tej

is a closed subspace of L^(j,X) and (Cb(j,X))c = Cb,u(j,X), (Cb(j,X ))o = CO (j, X).

6) The subspace Cb,u = Cb,u(j,X) of Cb(j,X) of uniformly continuous functions from Cb. Note that (Cb,u(j,X))c = Cb,u(j,X) and (Cb,u(j,X ))o = Co (j, X).

7) The subspace C0 = C0(j,X) of Cb,u(j,X) of continuous functions, vanishing at infinity. These functions satisfy the condition

lim ||x(t)|| = 0, t e j.

8) The subspace Cs1,^ = Cs1,^(j,X) of Cb,u(j,X) of continuous functions slowly varying at infinity. These functions satisfy the condition

lim ||x(t + t) - x(t)|| = 0, t,T e j (see [12,17-19]).

9) The subspace Cw,^ = Cw,^(j,X) of Cb,u(j,X) of continuous u-periodic at infinity functions, u e r+. These functions satisfy the condition lim ||x(t + u) - x(t)|| = 0, t e j (see [12,17-19]).

10) The subspace AP^ = AP^(j,X) of Cb,u(j,X) of continuous almost periodic at infinity functions (see [3,4]).

11) Subspaces Ck = Ck(j, X), k e n, of k times continuously dif-ferentiable functions with bounded k-th derivative and the norm

||x||(k) ||x||to + ||x

12) Holder spaces Ck,a = Ck,a(j,X), k e n U {0}, a e (0,1],

nka i nk ,, (k)„ |x(k)(t) - x(k)(s)| Ck,a = < x e Ck : ||x(k)||Co,« = sup -V^-i—— < ^

||x|Cfc,a = ||x|Cfc + ||x(k)||C0,a .

13) The subspace v = v(j, X) of functions from L^(j, X) with bounded

variation ||x||V = sup (x) + sup ||x||X used as a norm. tej tej

Definition 1 implies that each of the mentioned homogeneous spaces F(r,X) is endowed with the structure of a Banach L1(r)-module on using the convolution (2), where S is a group of shifts defined by (1). Thus, there is an opportunity to use some notions and results of the theory of Banach L1(r)-modules given below. In particular, the spaces Fc(r,X) coincide with the spaces of S-continuous vectors (see Definition 3).

2. Almost periodic vectors from Banach L1(r)-modules. Let X be a complex Banach space and End X be a Banach algebra of linear operators on X. By L1(r) we denote the algebra of complex Lebesgue's measurable (classes of) functions summable on r with convolution as the multiplication: (f * g)(t) = f f (t - s)g(s)ds, t e r, f, g e L1'

We endow X with the structure of a non-degenerate Banach L1( module (see [5,10,14]) associated with some bounded isometric representation T : r ^ End X. It means that the following properties hold.

Assumption 1. A Banach L1(r)-module X fulfills the conditions below:

1) if fx = 0 for every function f e L1(r), then the vector x e X vanishes (a non-degeneracy property of X);

2) for every f e L1(r) and x e X the following equations hold:

T(t)(fx) = (T(t)f)x = f (T(t)x), t e r,

the module structure on X is associated with the representation T : r ^ End X.

If T : r ^ End X is a strongly continuous isometric representation, then the formula

T(f )x = fx = J f (t)T(-t)xdt, f e L1 (r), x e X,

r

endows X with the structure of a Banach L1(r)-module satisfying the conditions of Assumption 1; this structure is associated with the representation T.

Remark 1. Associate a unique representation T : r ^ End X (see [10]) with every non-degenerate Banach L1(R)-module X. In order to emphasise this, sometimes the notation (X, T) is used.

The theory of Banach L1(r)-modules was constructed in [7] and studied in [1,2,5,6,9,10,14].

Definition 3. A vector from a Banach L1(R)-module (X, T) is called continuous (with respect to the representation T) or T-continuous if a function x : r ^ X defined by <^x(t) = T(t)x, t e r, is continuous for t = 0 (hence, it is continuous on r).

A set of all continuous vectors from a Banach L1(r)-module X denoted by Xc or (X,T)c is a closed submodule in X, i.e., Xc is a closed linear subspace in X invariant under shift operators T(f), T(t) for f e L1(r) and t e r.

Every homogeneous space F(r, X) is endowed with the structure of a Banach L1(r)-module using the convolution (2), where S: r ^ End F(r, X) is the group of shifts defined in (1). However, formula (2) does not define the structure of L1(r)-module for F(r+,X). Nevertheless, the quotient space F(j,X)/F0(j,X) is endowed with this structure.

Given a function f from L1 (r), the Fourier transform f : r ^ c is defined as

/(A) = / f (t)e-iAt dt, A E r.

r

Definition 4. The Beurling spectrum of a vector x E X is the set of numbers A(x) in r defined by

A(x) = {Ao E r : fx = 0 for every function f E L1(r) with /(Ao) = 0}.

The definition implies that A(x) = r\{^o E r : there exists a function f E L1(r) such that /(^o) = 0 and fx = 0}.

The Beurling spectrum of vectors in a Banach L1(r)-module X has the following properties (see [5,10]):

Lemma 1. For every x E X and f E L1(r) the following properties hold:

1) the set A(x) is closed and A(x) = 0 if and only if x = 0 ;

2) A(fx) C (supp f) U A(x);

3) fx = 0 when (supp /) P|A(x) = 0 and fx = x if the set A(x) is compact and f =1 in its neighbourhood;

4) the set A(x) is a singleton ((A(x) = {Ao})) if and only if x = 0 and T(t)x = eiAotx for t E r.

Remark 2. As we indicated above, every homogeneous space F(r, X) is a Banach L1 (R)-module. If a function x E F(r, X) has the property A(x) = {Ao}, then it can be represented as x(t) = xoeiAot for t E r, where xo E X.

Now let us introduce Ao-nets, bounded approximate identities (BAIs), and invariant integrals (see [2,5,6,10]), which are essential for our study.

Definition 5. Let U be a directed set and Ao E r. A bounded net (fa,a E U) of functions from L1(r) is called a Ao-net if the conditions below hold:

1) fa (Ao) = 1 for all a E U;

2) lim fa*f = 0 for any function f E L1(r) with the property /(Ao) = 0.

a

As an example of a Ao-net from L1(r) one can consider the functions ga(t) = fa(t)eiAot, ^a(t) = Pa(t)eiAoi, a > 0, where 0-nets (f«,a > 0) and (pa, a > 0) are defined by

fa(t) = { Va(t) = { 02'

\ 0, t < 0,

(2a)-1, t G [-a, a], 0, t G [-a, a],

ae-at, t ^ 0,

a > 0,

a > 0.

The set U = (0, ro) for (fa,a > 0) is directed in the descending order, while for (pa, a > 0) it is in the ascending order.

Definition 6. Given a directed set U, a bounded net (ea,a e U) of functions from L1(r) is called the bounded approximate identity (BAI) of L1(r) if the conditions below are met:

Definition 7. Given a directed set U, a bounded net (fa,a e U) of functions from L1(r) is called an invariant integral if the conditions below are true:

Below we use the following concept of an almost periodic vector in a Banach space X (see [2,5,6]), carrying a strongly continuous isometric representation T : r ^ End X.

Definition 8. A vector xo e X is called almost periodic (with respect to representation T) if one of the following conditions is met:

1) for every e > 0 the set Q(e,x0) = (w e r : ||T(w)x0 — x0|| < e} of e-periods of the vector x0 is relatively dense in r;

2) the orbit (T(t)x0, t e r} of x0 is precompact in X;

3) t M- p(t) = T(t)x0, t e r, is a continuous almost periodic function, i.e., p e AP(r,X) (see [5,15]);

4) for any e > 0 there are eigenvalues A1,..., AN and associated eigenvectors x1,... ,xN of the representation T, i. e., T (t)xk = eiAfcixk for t e r and k = 1,..., N such that

1) ea(0) = 1 for all a G U;

2) lim ea * f = f for all f from L1(R).

a

1) fa (0) = 1 and fa > 0 for all a G U;

2) lim/ |fa(t + u) - fa(t)|dt = 0 for all u G r.

a r

N

fc=1

The set AP(X) = AP(X, T) of almost periodic vectors (with respect to a representation T) is a closed submodule of X. Observe that AP(Cm(r,X),S) = AP(r,X) and AP(X) c Xc (see [3,4,7]). A unique linear operator J e EndAP(X) with the properties

1) IIJII = 1;

2) J (T (t)x) = J x for t e j and x eX;

exists on a Banach space AP(X) of almost periodic vectors. Consider a function xb : r ^ X, cb(A) = J(TAx), where TA(t) = T(t)e-iAt for t e r, for every vector x from AP(X). This function is called the Bohr transform of the vector x. Its support supp Xcb is at most a countable set, i. e., suppcb = {Ai, A2,... }, and

T(t)xk = eiAfeixfc, t e r, k ^ 1,

where xk, k ^ 1, are eigenvectors of the representation T; they are also eigenvectors of the generator iA of the operator group T, i.e., iAxk = = ¿AfcXfc, k ^ 1. Moreover, A(xk) = {Ak} for k ^ 1. The set AB(x) = = {A1, A2,... } is called the Bohr spectrum of the vector x e AP(X). Note that

a e

XCB (A) = lim 1 [ T(r)xe-iATdr = lim £ [ T(r)xe-(e+iA)rdr =

a—y^o a J 0<e—0 J

0 0

= lim £R(£ + iA,iA)x = lim fax, A e r,

0<e—0 a

where (fa,a e U) is an arbitrary A-net from L1(r). The series

X ~ xfc (3)

fc^i

is called the Fourier series of the vector x e AP(X) and xk, k ^ 1,

are called the Fourier coefficients of x. Note that if the series absolutely

converges, then x = xk.

fc^i

Note the uniqueness property of the Bohr transform: if the Bohr transform cb of a vector x e AP(X) is equal to zero, then x = 0.

Lemma 2. [13] For any function f from L1(r) and any almost periodic vector x e AP(X) with the Fourier series (3), the vector fx is almost periodic with the Fourier series of the form

fx ~ •C(Afc )xfc.

fc>i

The following lemma uses the (BAI) (/n,n ^ 1) in L1(r) (see Definition 6) constructed below (see [2,5,6]).

Let us consider a function /0 from L1(r) of Fourier transforms of functions from L1(r) (with pointwise multiplication) with compact support supp /0 on the interval [— 1,1], such that /o(0) = 1. Then the sequence (/n), n ^ 1, of functions defined by /n(t) = n/0(nt), t G r, is a (BAI) in L1(r). Note that ||/n|| = ||/0|| for n ^ 1. Lemma 2 implies

Lemma 3. [13] Let x be an almost periodic vector from AP(X) with the Bohr spectrum AB (x) = |A1, A2,... } satisfying the condition lim |An| =

In this case

lim ||x — /nx|| = lim

n^-œ n^œ

x — S fo V 'Xfc

| Afc |<n V

where xk, k ^ 1, are the Fourier coefficients of x.

Lemma 4. The Bohr transform : r — X of a vector x from AP(X) can be estimated as follows

||xB(A)|| ^ ||x||, A g r.

Proof. Given a directed set U, in the equality (A) =lim /ax, x G AP(X), we should use a A-net (/a,a G U) from L1(r), which is an invariant integral (see Definition 7). In this case ||/a|| = /a (0) = 1 for any a G U and, consequently, ||xB(A)|| ^ sup ||/a||||x|| = ||x||, A G r. □

a

Theorem 1. For any vector x from AP(X) with the Bohr spectrum AB(x) = {A1, A2,... } the equality lim ||xb(An)|| = 0 holds true.

n—y^o

Proof. Condition 4) of Definition 8 implies that there exists a sequence (xn, n G n) of vectors from AP(X) with finite Bohr transform supports

supp (xn)B, n ^ 1. Lemma 4 implies that sup ||xb(A) — (xn)B(A)|| — 0,

agr

n —y to. Hence, lim ||xb(An)|| = 0. □ Given m > 0 and x G AP(X), consider

|Ak |

Tm(x) = Y^ 1 —

m

|Afc|<m, AfceAB (x)

Xfc,

where AB(x) = (Ai, A2,... } and xk with k ^ 1 are the Fourier coefficients of x (see (3)).

Theorem 2. [15] The Fourier series (3) of a vector x from AP(X) is summable by the Bochner-Feier method, i. e., the equality lim ||x — rm(x)|| = 0 holds.

m—x

If X0 is a closed submodule of X invariant under operators T(t), t e r, then the quotient space X/X0 is also a Banach L1(r)-module, whose structure for all f e L1(r) and equivalence classes X = x + X0, x eX, is defined as fx = fx + X0 = fx. This structure is associated with the representation

T : r ^ End X/X0, T(t)X = t(i)X = T(t)x + X0, x e X.

Denote the quotient space F(j,X)/F0(j,X) by X(j), j e {r+,r}; this is a Banach space under the norm ||X|| = inf ||y||, where X = x +

+ F0(j,X) is the equivalence class that contains the function x e F(j,X). A Banach space X(j) is a Banach algebra with the multiplication rule defined by XX = Xy, X, X e X(j).

Define the strongly continuous group of isometries X : r ^ End X (r) on X (r) by the formula

X(t)X = X(t)X, x e F(r,X), t e r.

The quotient space X(r) is endowed with the structure of a Banach L1(r)-module using the formula fX = / * x, f e L1(r), x e F(r, X).

Remark 3. Assume that j = r+. Each function x e F(r+,X) can be extended to the function y on r, so that y satisfied all five conditions of Definition 2. Note that the equivalence class X e X(r) does not depend on the certain extension and, consequently, the Banach space X(r+) is isometrically embedded in X(r) as a closed submodule. In this case, the group X is well-defined on X(r+).

3. Almost periodic at infinity functions. Consider a homogeneous function space F(j,X) satisfying the conditions (a)-(d) from Definition 1 for j = r (the conditions 1) - 5) from Definition 2 for j = r+). Consider the (semi-)group X : j ^ EndF(j,X) of operators of the form

(X(t)x)(r) = x(t + r), t,r e j

in a Banach space F(j, X).

Definition 9. A function x e Fc(j,X) is called slowly varying at infinity if and only if the condition X(a)x — x e F0(j,X) is fulfilled for every a e j.

Denote the set of slowly varying at infinity functions from F(j,X) by Fs1,^(j,X). Definition 9 directly implies that Fs1,^(j,X) is a closed subspace of F(j,X) that is invariant under the shift operators S(t), t G j. Slowly varying and periodic at infinity functions from homogeneous spaces were studied in [16].

In the case F(j,X) = Cb,u(j,X) the above definition is equivalent to the classical definition of continuous slowly varying at infinity function (see [3,4,8]). The set of these functions is denoted by Cs1,^(j,X). Particularly, the solutions of the heat equation were established to belong to Csi, ^(j,X) in [8].

Lemma 5. For every x G Fs1,^(r, X) there exists a function xo G Csi,^(r,X) such that x — xo G Fo(r,X). Moreover, for every x G Fs1, ^(r, X) there is a function y : r — X that can be extended on c to a bounded entire function of the exponential type and such that y — x G Fo(r,X).

For j = r this result was proved in [16]. The result for j = r+ follows from Remark 3.

For example, a function x = c + xo G F(j, X), where c is a vector from X and xo is a function from Fo(j,X), belongs to Fs1,^(j,X).

Let us give four definitions of almost periodic at infinity functions from the homogeneous space F(j, X). After that, we are going to prove them to be equivalent and study their Fourier series.

First, let us introduce a definition of a continuous almost periodic at infinity function (see [3, 4]) that is based on the notion of e-period at infinity.

Definition 10. Assume e > 0. A number u > 0 is called an e-period at infinity of x G Cb(j, X) if there exists a number a(e) > 0 such that

sup ||x(t + u) — x(t)|| < e.

Denote the set of e-periods at infinity of a function x G Cb(j,X) by (e,x).

Definition 11. A subset ^ of j is called relatively dense on j if there exists an l > 0 with [t, t + l] fl ^ = 0 for every t G j.

Definition 12. A function x G Cb,u(j,X) is called almost periodic at infinity if for every e > 0 the set Q^(e,x) of its e-periods at infinity is relatively dense on j.

The set of almost periodic at infinity functions from Cb,u(j,X) is denoted by APx(j,X) and studied in [3,4,11,13]. Definitions 10 and 12 imply that every function x e Cb,u(r,X) almost periodic in the Bohr sense (x e AP(r,X), see [5,15]) is almost periodic at infinity. By AP(r+,X) denote the set of almost periodic Bohr functions that are restrictions to r+ of functions from AP(r, X). Therefore, AP(j,X) C APx(j,X).

Definition 13. Assume that £ > 0. A number u > 0 is called an £-period at infinity of x e F(j,X) if there exists a function x0 e F0(j,X) such that

||X(u)x — x — X0|| < £.

For the set of £-periods at infinity of x e F(j,X) we use the same denotation Qx(£,x).

Definition 14. A function x from Fc(j,X) is called almost periodic at infinity if for any £ > 0 the set Qx(£,x) of its £-periods at infinity is relatively dense on j.

Denote the set of functions from F(j, X) almost periodic at infinity by APxF(j,X). Note that APxF(j,X) is a closed subspace of F(j,X) invariant under the shift operators X(t), t e j. Definition 9 directly implies that Qx(£,x) = j for any x e Fs1, x(j,X) and £ > 0, hence, x e APxF(j,X). Consequently, Fsi, x(j,X) C AP^ F(j,X).

In the case F(j,X) = Cb,u(j,X) the above definition is equivalent to Definition 12.

Definition 15. A set of functions M C F (j, X) is called precompact at infinity if for any £ > 0 there exist finitely many functions b1,..., e M called an £-net at infinity, such that for every x e M there exists a function , 1 ^ k ^ N, and a function e F0(j,X) such that

||x — — a£|| < £.

Definition 16. A function x e Fc(j,X) is called almost periodic at infinity if the set M = {X(k)x, k e j} is precompact at infinity.

For F(j,X) = Cb,u(j,X) Definition 16 corresponds to the Bochner's criterion (see [15]) of almost periodicity. Observe that the functions of the form

N

x(t) = ^ Xk (t)eiAfc *, X1,... ,Xn e Fsi, x(j,X), A1,..., An e r, t e j, k=1

called generalized trigonometric polynomials, are almost periodic at infinity in the sense of Definition 16.

Definition 17. A function x G Fc(j,X) is called almost periodic at infinity if, given e > 0, we can indicate a finite collection A1,..., AN of real numbers and functions x1,..., xN from Fs1,^(j, X) such that

N

x ^ ^ xfc6fc

< e,

k=1

where ek, 1 ^ k ^ N, are functions defined by ek(t) = eiAfc 1, t G r.

Definition 18. A function x from Fc(j,X) is called almost periodic at infinity if the equivalence class X = x + Fo(j,X) is an almost periodic vector from X(j) = F(j,X)/Fo(j,X) with respect to the isometric representation S : r — End X.

The almost periodic at infinity functions from Cb,u(j, X) appeared, for the first time, in [3,4]. In these articles the definition corresponding to Definition 18 was used. The main results of those articles deal with the asymptotic behaviour of bounded operator semigroups.

Theorem 3. All definitions of almost periodic at infinity functions from F(j, X) (Definitions 14, 16, 17, 18) are equivalent.

Proof. We assume that j = r (the result for j = r+ follows from Remark 3). Let us consider the quotient space X = F(r,X)/Fo(r,X) and the group of isometries T = S : r — End X defined above. For this representation, Definition 17 corresponds to property 4) of Definition 8. Since all of the properties of Definition 8 are equivalent, it suffices to show that the first three properties are equivalent to Definitions 14, 16, and 17, respectively.

Given x G F(r,X), take the equivalence class X in X, constructed from x. Then for each e > 0 the set Q^(e, x) U (—Q^(e, x)) coincides with the set Q(e,x) of e-periods of X. Hence, the corresponding definitions are equivalent.

The equivalence of Definition 16 and property 2) of Definition 8 follows directly from the definition of the quotient module X.

In order to verify the equivalence of Definition 17 and property 4) of Definition 8, it suffices to establish that the Beurling spectrum A(y) of the equivalence class y G X with y = y + Fo (r,X) is the singleton

(A(y) = {A0}) if and only if y e F(r, X) can be represented in the form y(t) = y0(t)eiAoi for t e r,jvhere y0 e Fsi,oo(r,X).

If A(y) = {A0}, then y(t)y = eiAoty for every t e r (see property 4) of Lemma 1). Hence, A(y0) = {0}, where y0(s) = y(s)e-iAos for s e r. Therefore, y(t)y0 = y0 for every t e j. Thus, X(t)y0 — y0 e F0(r,X) for t e r, i.e., y0 eFsi,o(r,X).

Conversely, if y(t) = y0(t)eiAot for t e r, where y0 e F0(j,X), then y(t)y = eiAoty for t e r, and so property 4) of Lemma 1 implies that A(y = {A0} (see Remark 2). □

Given x e AP^,F(j,X), let us consider the series

x

J^Xn, Ab(x) = (Al, A2,... }, A(xn) = An,

n>1

which is the Fourier series of the equivalence class y = x + F0(j,X) e AP (X).

Definition 19. The series

;(t) - Xn(t)eiAni, t G j,

n^l

where functions zn, n ^ 1, of the form zn(t) = xn(t)eiAnt, t G j, xn G Fc(j,X), belong to the corresponding equivalence classes xn with n ^ 1, is called the Fourier series of x. The functions xn with n ^ 1 are called the Fourier coefficients of x.

Note that the Fourier series defined this way is ambiguous, i.e., the functions xn with n ^ 1 can be chosen differently. In [3,4] the analogous definition was given for functions from Cb,u(j,X).

Theorem 4. Coefficients of any Fourier series (4) of a function x G apqoF(j,X) belong to Fsl, oo(j,X) and satisfy the condition lim ||xn||f = 0.

n—^cXD

The condition xn G Fsl,oo(j,X) follows from Definition 17 and the equality lim ||xn= 0 follows from Theorem 1.

n—

Theorem 4 and Lemma 5 directly imply

Theorem 5. Given a function x G AP^,F(j,X), one can construct the Fourier series (4) such that xn G Csl, oc(j,X), n ^ 1. Moreover, the

functions xn with n ^ 1 can be extended on c to bounded entire functions of the exponential type.

Given m > 0 and x G APxF(j, X), consider a function Tm(x, ■) : j ^ X defined by

Tm(x,t)= ^ (l -1M) xfc(t)eiAni, t G j, |An|<m,AneAB(c) ^ '

where xk, k ^ 1, are the Fourier coefficients of x.

Definition 20. The Fourier series (4) of a function x G APxF(j,X) is summable at infinity by the Bochner-Feier method, if a sequence (y^, m G n) of functions from F0(j,X) such that

lim ||x - Tm(x,-) - ymIf = 0 (5)

exists.

For F(j,X) = Cb,u(j,X) condition (5) is equivalent to

lim sup 11x(t) - Tm(x,t) - ym(t)||x = 0. tej

Theorem 6. The Fourier series (4) of a function x G APxF(j, X) is summable at infinity by the Bochner-Feier method.

Proof. Assume that j = r (the result for j = r+ follows from Remark 3). Let us consider the quotient space X = F(r,X)/F0(r,X) and the group of isometries T = S : r ^ End X defined above. For a function x G APxF(r,X) an equivalence class x belongs to AP(X). Therefore, it satisfies the conditions of Theorem 2. The statement of Theorem 6 follows directly from Theorem 2. □

Note that the choice of Fourier coefficients in the last theorem is not essential.

Acknowledgment. The first author of this work was supported by RFBR according to the research project 16-01-00197, the second author was supported by RFBR according to the research project 18-31-00097 and the third author was supported by RFBR according to the research project 18-31-00097.

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Received July 8, 2018.

In revised form, December 3, 2018.

Accepted December 4, 2018.

Published online December 19, 2018.

Voronezh State University

1 Universitetskaya sq., Voronezh 394006, Russia

E-mails:

Baskakov A. G. [email protected], Strukov V. E. [email protected], Strukova I. I. [email protected]