ЧЕБЫШЕВСКИЙ СБОРНИК

Том 23. Выпуск 1.

УДК 517.5 DOI 10.22405/2226-8383-2022-23-1-21-32

О построении многомерных периодических фреймов всплесков1

П. А. Андрианов

Андрианов Павел Андреевич — Санкт-Петербургский государственный университет (г. Санкт-Петербург). e-mail: p.andrianov@spbu.ru

Аннотация

Изучаются многомерные периодические системы всплесков с матричным коэффициентом растяжения. В работе используется конструкция периодического кратномасштабпого анализа, наиболее общее определение которого дано И. Максименко и М. Скопиной в [25]. Описан алгоритмический метод построения двойственных фреймов всплесков по набору коэффициентов Фурье одной подходящей функции. Данная функция является первой функцией в масштабирующей последовательности, формирующей двойственные периодические кратномасштабные анализы, которые используется для конечного построения систем всплесков. Условия, накладываемые на исходную функцию, представляют собой ограничения на скорость убывания её коэффициентов Фурье, а также на взаимное расположение нулевых и ненулевых коэффициентов.

Ключевые слова: периодический кратномасштабный анализ, фреймы всплесков, Бессе-лева система, двойственные фреймы.

Библиография: 25 названий. Для цитирования:

П. А. Андрианов. О построении многомерных периодических фреймов всплесков, Чебышев-ский сборник, т. 23, вып. 1, с. 21-32.

1 Исследование выполнено за счет гранта Российского научного фонда (проект n0. 18-11-00055)

CHEBYSHEVSKII SBORNIK Vol. 23. No. 1.

UDC 517.5 DOI 10.22405/2226-8383-2022-23-1-21-32

On construction of multidimensional periodic wavelet frames

P. A. Andrianov

Andrianov Pavel Andreevich — Saint Petersburg State University (Saint Petersburg). e-mail: p.andrianov@sphu.ru

Abstract

Multidimensional periodic wavelet systems with matrix dilation in the framework of periodic multiresolution analyses are studied. In this work we use notion of a periodic multiresolution analysis, the most general definition of which was given by Maksimenko and M. Skopina in [25]. An algorithmic method of constructing multidimensional periodic dual wavelet frames from a suitable set of Fourier coefficients of one function is provided. This function is used as the first function in a scaling sequence that forms two periodic multiresolution analyses, which are used to construct wavelet systems. Conditions that the initial function has to satisfy are presented in terms of a certain rate of decay of its Fourier coefficients, and also mutual arrangement of zero and non-zero coefficients.

Keywords: wavelet function, periodic multiresolution analysis, wavelet frame, Bessel system, dual frames.

Bibliography: 25 titles. For citation:

P. A. Andrianov, "On construction of multidimensional periodic wavelet frames", Chebyshevskii sbornik, vol. 23, no. 1, pp. 21-32.

1. Introduction

A natural way to define periodic wavelet system is to periodize standard wavelet systems from L2(R), which is possible if wavelet functions have sufficient decay rate. Such systems are widely studied ([6, §9.3], [13], [19], [20], [22], [12]). But many periodic objects that can reasonably be classified as wavelet systems cannot be obtained that way, and thus there exist other approaches to defining periodic wavelets in a more general sense. Just as in nonperiodic case, wavelets can be obtained on the basis of multiresolution analyses. Specifically, orthogonal bases and tight frames are built using one periodic multiresolution analysis (for brevity, I'M ISA in the sequel), and biorthogonal bases and dual frames are built using two I 'M I! As (see [4], [14], [8], [23], [21]). In this paper we use the definition of I'M ISA given by I. Maksimenko and M. Skopina in [25] (also see [24, Chapter 9]). In [2] N. Atreas has shown that in order to establish that dual wavelet systems are frames, one should check that, along with a few technical conditions, these systems are Bessel. It is worth noting that similar constructions of tight frames do not require this check. Algorithmic methods for the construction of PMRA-based tight wavelet frames were suggested in [7], and in [2] for multidimensional case. However, the condition of systems being Bessel is critical for the construction of dual wavelet frames. Sufficient conditions, under which multidimensional periodic wavelet system is Bessel, were established in [1]. Basing on this result, we provide an algorithmic method of constructing multidimensional periodic dual wavelet frames, starting with any suitable set of Fourier coefficients. In the provided scheme these coefficients define a function that induces two scaling sequences, which generate dual frames.

2. Notation and auxiliary results

As usual, N is a set of positive integers, Rd is a ^dimensional euclidean space, x = (x1,..., Xd), y = (^i,... are its elements (vectors), (x,y) = x1y1 + ...Xdyd, 0 = (0,..., 0) € Rd,

= y/(x,x), Zd is integer lattice in Rd, Z = Z1, Z+ = {0,1,...} Td = (-i; i jd is a d-dimensional unit torus, 5n,k is Kronecker delta, f(k) = fJd f (t)e- is fc-th Fourier coefficient

of f € L2(Td), (f, g) is inner product in L2(Td).

If A is d x d matrix, then is its euclidean operator norm from Rd to Rd, A* is its Hermitian adjoint, A* = (A*y, Id is d x d identity matrix. If A is a d x d nonsingular integer matrix, we say that vectors k, n € Zd are congruent modulo A and write k = n (mod A) if k — n = Al, I € Zd. We denote bv Z0 A set of all I € Zd, such that I = 0 (mod A). The integer lattice Zd is partitioned into cosets with respect to this congruence. The number of these cosets equals to | det A\ (see, for instance, [11, Proposition 2.1.1]). Any set containing only one representative of each coset is called a set of digits of the matrix A. When it does not matter which set of digits is chosen, we assume that it is chosen arbitrarily and denote it by D(A). Let us also note that H(A) := Zd n ATd is a set of digits (see [11, Proposition 2.1.1]). Also, there is a following lemma that establishes connection between sets of digits of matrices A, A^ and A^+1.

Lemma 1 ([11], Lemma 2.1.3). Let, A be a nonsingular integer d x d matrix, | det Al > 1. Then the set {r + AJp} for all possible r € D(AJ) and p € D(A) is a set of digits of the matrix A>+1.

In this paper M denotes a square integer matrix with eigenvalues greater than one in modulus. We will also denote m := | det M^^^e that matrix M-1 has all eigenvalues less than one in modulus, and there is only finite number of them, and hence spectral radius of matrix M-1 is also less than one. This implies that

lim ||M~n|| =0. (1)

For any I € Zd, lj is & vwtor such that lj € H(M*), lj = ¿modM* (note that it is unique).

A matrix M is called isotropic if it is similar to a diagonal matrix such that numbers X1,... ,Xd are placed on the main diagonal and |A1| = ... = Thus, X1,..., Xd are eigenvalues of M and the spectral radius of M is equal to |A|, where A is one of the eigenvalues of M. Note that if matrix M is isotropic then M* is isotropic and M^ is isotropic for all j € Z. It is well known that for an isotropic matrices M and for any j € Z we have

Cf |A|J < HMj|| < Cif |A|J, (2)

where A is one of the eigenvalues of M.

For any sequence of functions {fj }jeZ+ C L2(Td) we will denote its sh ifts bv fjk := fj 0+Mk). Bv wavelet system we will mean a system of shifts {fjk}jeZ+,keD(Mj)> associated with a sequence

of functions {fj}j£Z+ C L2(Td), and denote it by {fjk}j,k■ If we have several sequences {f^}j£Z+, v = 1,... ,n, n € N, the system that represents a union of wavelet systems of each sequence we will also call a wavelet system and denote it by {f^k) }j,k,u ■ In the case if we will need to specify the sets

of indices, we will write {fjk}jeZ+,keD(Mi),u=i,...,n-

In this paper we rely on the following result that establishes sufficient conditions for wavelet systems to be Bessel.

Theorem 1 ([1]). Let Fourier coefficients of functions ^j € L2(Td), j € Z+, satisfy the following conditions

Vj € Z+,l € Zd |mj/%(l)| ^C min{ |-( l+£), ^ *~jl |a} (3)

for some C > 0 £> 0, a > 0. Then, the wavelet s ystem {tpjk }j,k ^ Bessel.

Let us now proceed to defining periodic multiresolution analysis.

Definition 1 ([24], Definition 9.1.1). A collections of sets {Vj}°=0, Vj c L2(Td), is called PMRA, if the following properties hold:

• MR1. Vj c Vj+i;

• MR2. UJc=o Vj = L2(Td);

• MR3. dimVj = mj;

. MRl dim{ f G V : /(• + M-n) = Xnf Vn G Zd} < 1 V{\n}n&d, An G C;

• MR5. f G Vj & f(- + M-jn) G Vj Vn G Zd;

. MR6. a) f G Vi ^ f(M■) G Vj+\; f G V+ ^ EseD(M) f (M-1 ■ +M-1 s) G Vj.

Definition 2 ([24], Definition 9.1.3). Let {Vj }c=0 be a PMRA in L2(Td). Sequence of functions {ipj }jçz+, Pj G Vj, is called a scaling sequence, if functions pjk, k G D(MJ ), form a basis for Vj.

Theorem 2 ([24], Theorem 9.1.4). Functions {pj}c=0 C L2(Td) form a scaling sequence for some PMRA if and only if:

• SI. Pp0(k) = 0, /or all k = 0;

• S2. for all j G Z+, and for all n G Zd exists m = n (mod M*J), such that p(k) = 0;

• S3, for all k G Zd exists j G Z+, such that p(k) = 0;

• S4- For all j G Z+, n G Zd, exists = 0, such that jhp (k) = PpJ+1(M*k) for all k = n (mod M * );

• S5. For all j G N n G Zd, exists fh, such that pp—1 (k) = fhp(k) for all k = n (mod M*J).

Let us note that in Theorem 2 the sequences of numbers {^f3k} keZd, { f3k } kezd are M ^-periodic with respect to k for everv j G Z+.

Now we define how scaling sequences generate wavelet systems. Let {pj^^ {pj }c=0 be two scaling sequences, s k - arbitrarily enumerated digits of the matrix M*, and matrices AW = {^toio, A« = o are such that

n(r) = ..3+1 = rrj+1 (A\

n0k = fr+M*isk , a0k = fr+M*isk ' W

and for any r G D(M * ) it is true that A(r)A(r)* = m Im. For v = 1,...,m - 1, let

v,j _ (r) ~v,j _ ~(r) /r\

ar+M*iSk = %k , ar+M*iSk = avk ■

By lemma 1, vectors r + M*JSk form a set of digits D(M*:>+1 ), i. e. we can M^^-periodically extend these sequences to Zd. Let us define functions \ by defining its Fourier coefficients

Pp (I) = a^P^d), = ^^1(1). (6)

Systems {p0}lJ {p^jp} } jez+, keD(Mj),v=i...,m-1 and {Pp0}^{'ll)t^jk}j,k,u we will call dual wavelet systems that are generated by scaling sequences {pj^^ {pj}'j=0- Now let us cite a theorem that establishes frame conditions for such systems.

Theorem 3 (f2j). Let {pj}j=0, {Pj}j=o scaling sequences that satisfy the condition

lim mjp(k)p(k) = 1 Vk e Zd, (7)

and let {p0} U {ipp}}j,k,v and {p0} U {ip(k }j,k,v be Bessel dual wavelet systems generated by them. Then these system,s are dual frames.

3. Main result

Theorem 4. Let M be an isotropic matrix such that Td c M*Td, and pi e L2(Td) with Fourier coefficients given by

{ao, if I = 0,

al(ili )a, if ie ZdM*, leQ,

0, otherwise,

where a > d/2, 0 < Ci ^ lall ^C2for I = 0 and al11 e Q, where Q c Zd is such that Qr\Z0^M* = 0, H(M*) c Q and satisfies the condition:

(Z) If I e Q and I e H(M*) for some j e N, then I + Mjk £ Q for every k e Zd. Then there exist scaling sequences {pj}j=0, {pj}°=0 that generate wavelet systems {p0} U {ipjk}j,k and

{p0} U {ipjk}j,k> which are dual frames.

For any vector I e Z0 M*, I eQ we set al = Cl5 and define {a**}, I e Zd, bv

'ai, if / = 0OTI£ Z0

a* = < ■ d'M* , ,

1 \ak, Hl = M*nk,n e N,k£ Zd

M

Next, we construct scaling sequences {pj}j=0, {pj}j=0 bv defining their Fourier coefficients. We start with setting

and, since Td С M*Td,

g (0:=K ^ ' iîleH (M

3 iq, me h (M *j ),

1Л/т, Hie H (M *i-1),

Tj3 =i „ (8)

^1 \Q, if i e h (M *j-1).

Thus, the functions pj are defined, and they are trigonometric polynomials.

Construction of [ipj}j is slightly more sophisticated. First of all we define the function on 0-th level,

po(0) := Vm -pï(0), po(0 := 0, 1 = 0. Note that the already have Fourier coefficients of p\. Next we define coefficients pj (I) for the rest

I. (I £ ZdM*) Define p(I) and find , for I £ Zd>Mk £ Zd, j > 1. 1) Let I £ H (M ; ). Two cases may occur:

a) ^(0=° * Pi (!):= , = M. (9)

b) (i)=0 ^ Pi (i):=m"1-1 ( M y a.;, rf = 0. (10)

Note that the case I = 0 is not described here, and hence | Ij-11 = 0.

2) Let I £ H(M*). Since numbers ^should be M*J-periodic with respect to I, we will periodically extend them from I £ H(M*), where we defined these numbers at previous step. Again, two cases may occur:

a) ^ =0 ^ P (l):= ^^; (11)

b) n =0 ^ p(I) := 0. (12)

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II. (I G Z0 M*) Now we define p(I) and find f\ for l G Z0 M*, j > 1.

1 - r*-1j\ ,J — ..3-1

P (l):= VmPJ-1(M ^ f =f1

(13)

Note that ^ = —!= to all I £ Zd due to this formula.

't ^Jm

Thus, we have defined all p(I). Obviously, the corresponding functions p are in L2. For I = 0 we, by definition (13), have a simple formula p(0) = —^p—l(0). Next, let us show that for the following inequality holds for all I = 0,

IP(1)1 < C*m-^ (M)°|a*|, (14)

where for I £ Z0 M* inequality turns into equality with C* = ^d C* = (C?f *)2a for I £ Z0 M For I £ Z0,M*, it'foUows directly from the formulas (9)-( 12). Now let I = M*nk, k £ Z0,M*, k £ Zd, and let p(I) = 0. Using definition (13) n times, we have

»5(') = (^)"n-n(M•-() = m-fmr-r1 ( ^^ pM-f

According to definition of a*, a*M= a*. Also, due to properties of matrix M and definition of I j, we know that (M*-n I )j-n = M *-nl + M *j-nr, where r £ Zd is such that M*-nl + M*3-nr £ M*3-nTd. This means that M*n(M*-nl)j-n = I + M£ M*Td, and hence I + M= Ij.Thas, (M*-nl)j-n = M*-nly Using these facts, we obtain

IPm=m-^ (W-iT™ ^

| M*

V IM*-nlim J 1 11

¥ ( Ui*-.B|M->M-H1 x-in*,

IM *-n 11 Ill) 1 11

^m-^ (\\M*-'-\\\\M-\\MyInU

It remains to recall that M is an isotropic matrix, which implies that \\M*-n\\\\M*ra\\ ^ (C2f *)2.

Let us show that {pj^^ {pj^o are scaling sequences. Condition SI is obviously fulfilled. Since

p(i) = 0, p(I) = 0

whenever l G H (M * ), conditions S2 and S3 are also granted. Condition S4 (periodicity of is also fulfilled, because all ^3k are equal to each others. The last, condition S5 (periodicity of fk) is granted by the fact that for every j G Z+ we defined fk on H (M *J ), and then extended it to Zd. It

is also worth noting that the fulfillment of condition (Z) grants us absence of collisions during the process of defining

Noting that lj = I for sufficiently large j , we can see that the equality

lim mjp (l)pj (I) = 1 Vie Zd

follows from inequality (14) for I £ Z0 Mwhich, as it was mentioned above, turns into equality with C * = 1; and from equality (15) for le Z0 M *.

Now, we introduce and analvze wavelet svstems generated bv the scaling sequences {pj}j=Lo) }f=o-

Let us define Fourier coefficients of ^j, tpj. It will be suitable for us to represent a set of digits of the matrix M*J as given in Lemma 1, i. e.

D(M * ) = U [r + M *j-l'p}-

(16)

reD(M *j-1) PeD(M *)

But we should note that this set is not necessarily the same as H(M*j). However, when speaking about due to its M*j-periodicitv we can safely regard it as defined on any set of digits (particularly on H(M*j)), whenever they are defined on at least one set of digits. It follows from (8) that

/4+1 _ 0 for к eH(M*j),

= 0 for к e H (M *j+1) \ H (M * ).

(17)

Using lemma 1, with D(M *j ) = H (M * ), D(M *) = H (M *), we can rewrite it as

We H (M - ) *, _0, forp = 0,

v > ^r+м*рЛ _ о, for p _ 0,P eH(M*).

(18)

Let us now build matrices A^ fa every r e H(M*j). First, enumerate digits

p e H(M*) such that p0 = 0. Then we define the first row as

(r) _ 3+1

a0k _ lr+M*jpk ,

~(r) _ ~j+1 a0k _ lr+M*j

Pk1

к _ 0,1,... ,m — 1.

(19)

It is easy to see that, due to (18), a^ = 0 for k = 1,... ,m — 1. Extend these matrices to square matrices in the following fashion

A(r) _

' 3 + 1 3 + 1 1 lr+M*ipi

0 —Jii+1

A(r) _

l

ùj+1 3+1

'r+M *i pi

3 + 1

3 + 1 lr+M *i pm-i

+1

¡4+

0 0

l

3 + 1

'r+M *i Pm-i

3 + 1 1+

0

0

0

0

Due to (9), = Vm(p^j) = Vm since r £ H(M*). Using this equality and (8), it is easy to check that A(r)A(r)* = mIm. Now we let

(r) —v,j Mr)

rv = /7 rv — it '

ar+M*jpk avk , ar+M*jpk avk '

Vectors r + M *jpk, k = l,...,m —1 we a set of digits D(M *+1 ), since r £ H (M *j), pk £ H (M *). Thus, we can M*J+l-periodically extend the coefficients a"'3, av'3 to Zd. Now, for u = 1,... ,m — 1, we let

vf (I) = a^P+id), ^(l) = a^p+id).

We can see that

{—Vmp]^(I), for 1 = r + pv (mod M*+1),

r £ H(M*'); (20)

0, ;

__ (—VmpJ+i (I), ioil£H (M *i+1)\H (M *);

tf\l) = \ n+M(I), for I £ H(M*), (21)

0, ;

To estimate them we consider two cases:

1) Let I £ H (M *). In Ms case, IM *-II < -f, and hence,

IM*-jlIa < Cda\M*-jlI-a, (22)

-2a

where Cdta = From (20), |tp^(Z)\ = 0. Next,

I^t ](l)I = in+M *j pv Im-^ laf11,

\u+ \ = m I(l+M*3p-)*I r

Wi+M*jp„I = ^m{ I(i + M*!pv)+iI) .

It is not hard to see that (l + M*pv)j = I, and since pv = 0, (l + M*jpv)+ £ H(M*i+1) \H(M*), which means that I(l + M*pu)j+^^I ^ 2\\M*-jy • Using this and the fact that M* is isotropic, we have

^.pjI=H (^MX v w )a < *-J""IM *-3lIY

< vm2a ( cMM * )2a[ iM *-j i\y,

and thus, according to (21), we have

mj/2ItPj (0I < m 2 2a(CMM * IM *-j l^Ia*- \ < C^m 3 2a(CM * )2a( IM *-j '"laf11-2) Let I £ H(M*). In Ms case IM*-jII ^ ±> and hence

IM*-jlI-a < [aIM*-jlIy.

By (21), Itpj(0| = for I e H(M*j+i ) \ H(M*j), i. e. where IM*-jl| < \\M*\\Vd, and

0 otherwise. Thus, we have the following estimate

Imj/2ijj (0| = I-m la*-11 < I-m 2a**-1 I\\M *\\ad f IM *-jl |-a < I-m 2a**-1 I\\M *\\ad f ^4|M *-jl ^

Next, from (14), for non-zero coefficients we have |55>j+i(OI < C*m-2^^j+p) Ia*I, where

1 Ij+iI < since lj+i e H (M *j+1). Using to and the fact that M * is isotropic,

155— mi < C *m-2 ( IM *-(j+1)l II lj+iI Y\a*\ < C *m-2 ( ^d\\M *j+1\\\\M I^j+i(l)I <Cm ^ IM*-(j+in / ) Ia*I <Cm 2{-2|M*-(j+Dl /-)

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< C*m-2(C2M *Y\\M*-i\\-aIM*-jl|-a

< C*m-2(C¥*y4a\\M*-i\\-aIM*-jl|a.

Bv definition,

mj/2\^j(0| = mj/2\ - < С*)2a(^)a\\M*-1\\-а1М*-jl\-a

a

< C*Vm(Cf *)2a(^)V||M*-l\\-alM*-jl|a.

As for coefficients that are equal to zero, the same estimates are obviously held.

Thus, we have shown that all conditions of theorems 1 and 3 are satisfied, and hence, wavelet systems {(p0} U {ipjk}j,k and {(p0} U {ipjk}j,k are dual frames.

Corollary 1. Let M be an isotropic matrix such that Td c M*Td, and ( e L2(Td) with Fourier coefficients given by

{ao, if 1 = 0,

ai(Tryr, ie K,m*, 0, if le Z0,m* , 1 = 0,

where a > d/2, 0 <C\ ^ lai| ^ C2 for I = 0 and all I £ Z0 m*■ Then there exist scaling sequences

{(j }j=o> {(i }(jL0 that generate wavelet systems {(p0}U{ipjk }jkk and {¡p0}U{ipjk }j,k> which are dual frames.

It suffices to check that, in this case, Q = {I : I £ Z0 m* }• This set obviously satisfies condition (Z) from Theorem 4.

4. Conclusion

We have presented a method of constructing periodic dual wavelet frames with an isotropic matrix dilation, starting with only one suitable function. Its Fourier coefficients have to have a sufficient rate of decay, and also satisfy the condition (Z) on mutual arrangement of zero and non-zero coefficients. The resulting wavelet systems can be built layer by layer, with the provided recurrent formulas for its Fourier coefficients.

СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ

1. P. A. Andrianov, "Sufficient conditions for a multidimensional system of periodic wavelets to be a frame", Zap. Nauchn. Sem. POMI (Russian), 2019, Volume 480, 48-61.

2. N. D. Atreas, 2002, "Characterization of dual multiwavelet frames of periodic functions", International Journal of Wavelets, Multiresolution and Information Processing, Vol. 14, No. 03, 1650012.

3. С. K. Chui, H. N. Mhaskar, 1993, "On trigonometric wavelets", C'onstr. Approx., 9, 2-3, 167-190.

4. С. K. Chui, J. Z. Wang, 1992, "A general framework of compact supported splines and wavelets", J. Approx. Theory 71, 263-304.

5. I. Daubechies, B. Han, A. Ron, Z. Shen, 2003, "Framelets: MRA-based constructions of wavelet frames", ACHA, Volume 14, Issue 1 Pages 1-46.

6. I. Daubechies, 1992, "Ten lectures on Wavelets", CBMS-NSR Series in Appl. Math., SIAM.

7. S. S. Goh, B. Han, Z. Shen, 2011, "Tight periodic wavelet frames and approximation orders", ACHA, Volume 31, Issue 2, Pages 228-248.

8. S. S. Gon, S. Z. Lee, Z. Shen, W. S. Tang, 1998, "Construction of Schauder decomposition on banach spaces of periodic functions", Proceedings of the Edinburgh Mathematical Society, Volume 41, Issue 1, pp. 61-91.

9. B. Han, 1997, "On dual wavelet tight frames", ACHA, V. 4. P. 380-413.

10. B. Han, 2003, "Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix", J. Comput. Appl. Math., Vol. 155. P. 4367.

11. A. Krivoshein, V. Protasov, M. Skopina, 2016, "Multivariate Wavelet Frames", Springer Singapore, P. 182.

12. E. Lebedeva, 2017, "On a connection between nonstationarv and periodic wavelets", J. Math. Anal. Appl, 451:1, 434-447

13. Y. Meyer, 1990, "Ondelettes", Herman, Paris.

14. A. P. Petukhov, 1997, "Periodic wavelets", Mat. Sb., 188:10, 69-94.

15. A. Ron, Z. Shen, 1995, "Gramian analysis of affine bases and affine frames", Approximation Theory VIII, V. 2: Wavelets (C.K. Chui and L. Schumaker, eds), World Scientific Publishing Co. Inc (Singapore). P. 375-382.

16. A. Ron, Z. Shen, 1995, "Frame and stable bases for shift-invariant subspaces of L2(Rd)", Canad. J. Math., V. 47. N. 5. P. 1051-1094.

17. A. Ron, Z. Shen, 1997, "Affine systems in L2(Rd): the analysis of the analysis operator", J. Func. Anal, V. 148. P. 408-447. *

18. A. Ron, Z. Shen, 1997, "Affine systems in L2(Rd): dual systems", J. Fourier. Anal Appl, V. 3. P. 617-637.

19. M. Skopina, 1998, "Local convergence of Fourier series with respect to periodized wavelets", J. Approx. Theory., V. 94. P. 191-202.

20. M. Skopina, 2000, "Wavelet approximation of periodic functions", J. Approx. Theory., V. 104. P.302-329.

21. M. Skopina, 1997, "Multiresolution analysis of periodic functions", East Journal On Approximations, Volume 3, Number 2, 203-224.

22. G. G. Walter, L. Cai., 1999 "Periodic Wavelets from Scratch", Journal of Computational Analysis and Applications, Volume 1, Issue 1, pp 25-41.

23. V. A. Zheludev, 1994, "Periodic splines and wavelets", Proc. of the Conference "Math. Analysis and Signal Processing", Cairo, Jan. 2-9, 1994.

24. Novikov, I. Y., Protasov, V. Y., Skopina, M. A., 2011, 'Wavelet Theory", American Mathematical Society.

25. I. Maksimenko, M. Skopina, 2004, " Multivariate periodic wavelets", St. Petersburg Math. J., 15, 165-190.

REFERENCES

1. P. A. Andrianov, "Sufficient conditions for a multidimensional system of periodic wavelets to be a frame", Zap. Nauchn. Sem. POMI (Russian), 2019, Volume 480, 48-61.

2. N. D. Atreas, 2002, "Characterization of dual multiwavelet frames of periodic functions", International Journal of Wavelets, Multiresolution and Information Processing, Vol. 14, No. 03, 1650012.

3. C. K. Chui, H. N. Mhaskar, 1993, "On trigonometric wavelets", C'onstr. Approx., 9, 2-3, 167-190.

4. C. K. Chui, J. Z. Wang, 1992, "A general framework of compact supported splines and wavelets", J. Approx. Theory 71, 263-304.

5. I. Daubechies, B. Han, A. Ron, Z. Shen, 2003, "Framelets: MRA-based constructions of wavelet frames", ACHA, Volume 14, Issue 1 Pages 1-46.

6. I. Daubechies, 1992, "Ten lectures on Wavelets", CBMS-NSR Series in Appl. Math., SIAM.

7. S. S. Goh, B. Han, Z. Shen, 2011, "Tight periodic wavelet frames and approximation orders", ACHA, Volume 31, Issue 2, Pages 228-248.

8. S. S. Gon, S. Z. Lee, Z. Shen, W. S. Tang, 1998, "Construction of Schauder decomposition on banach spaces of periodic functions", Proceedings of the Edinburgh Mathematical Society, Volume 41, Issue 1, pp. 61-91.

9. B. Han, 1997, "On dual wavelet tight frames", ACHA, V. 4. P. 380-413.

10. B. Han, 2003, "Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix", J. Comput. Appl. Math., Vol. 155. P. 4367.

11. A. Krivoshein, V. Protasov, M. Skopina, 2016, "Multivariate Wavelet Frames", Springer Singapore, P. 182.

12. E. Lebedeva, 2017, "On a connection between nonstationarv and periodic wavelets", J. Math. Anal. Appl, 451:1, 434-447

13. Y. Meyer, 1990, "Ondelettes", Herman, Paris.

14. A. P. Petukhov, 1997, "Periodic wavelets", Mat. Sb., 188:10, 69-94.

15. A. Ron, Z. Shen, 1995, "Gramian analysis of affine bases and affine frames", Approximation Theory VIII, V. 2: Wavelets (C.K. Chui and L. Schumaker, eds), World Scientific Publishing Co. Inc (Singapore). P. 375-382.

16. A. Ron, Z. Shen, 1995, "Frame and stable bases for shift-invariant subspaces of L2(Rd)", Canad. J. Math., V. 47. N. 5. P. 1051-1094.

17. A. Ron, Z. Shen, 1997, "Affine systems in L2(Rd): the analysis of the analysis operator", J. Func. Anal, V. 148. P. 408-447. *

18. A. Ron, Z. Shen, 1997, "Affine systems in L2(Rd): dual systems", J. Fourier. Anal. Appl, V. 3. P. 617-637.

19. M. Skopina, 1998, "Local convergence of Fourier series with respect to periodized wavelets", J. Approx. Theory., V. 94. P. 191-202.

20. M. Skopina, 2000, 'Wavelet approximation of periodic functions", J. Approx. Theory., V. 104. P.302-329.

21. M. Skopina, 1997, "Multiresolution analysis of periodic functions", East Journal On Approximations, Volume 3, Number 2, 203-224.

22. G. G. Walter, L. Cai., 1999 "Periodic Wavelets from Scratch", Journal of Computational Analysis and Applications, Volume 1, Issue 1, pp 25-41.

23. V. A. Zheludev, 1994, "Periodic splines and wavelets", Proc. of the Conference "Math. Analysis and Signal Processing", Cairo, Jan. 2-9, 1994.

24. Novikov, I. Y., Protasov, V. Y., Skopina, M. A., 2011, 'Wavelet Theory", American Mathematical Society.

25. I. Maksimenko, M. Skopina, 2004, " Multivariate periodic wavelets", St. Petersburg Math. J., 15, 165-190.

Получено 22.10.2021 г. Принято в печать 27.02.2022 г.