Sufficient Conditions for a Multidimensional System of Periodic Wavelets to be a Frame
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SUFFICIENT CONDITIONS FOR A MULTIDIMENSIONAL SYSTEM OF PERIODIC WAVELETS TO BE A FRAME P. A. Andrianov∗
UDC 517.5
We study multidimensional periodic wavelet systems with matrix dilations. We obtain conditions sufficient for such a system to be Bessel. The conditions are given in terms of Fourier coefficients. We propose a method for constructing a wavelet Riesz basis that starts with a suitable sequence of trigonometric polynomials. Bibliography: 19 titles.
1. Introduction A natural way to define periodic systems of wavelets is periodization of wavelets from L2 (Rd ); this is possible if the wavelet functions decay sufficiently fast at infinity. Such wavelet systems are widely studied in the literature (see [5–9, 11–14], [10, Sec. 2.6, Sec. 3.1]). However, many periodic objects that “deserve” to be called wavelet systems, cannot be obtained by periodization, so there are other approaches to define wavelets on a period, in a more general sense. As in the non-periodic case, the wavelets can be constructed on the basis of multiresolution analyses. Namely, orthogonal bases and tight frames are constructed on the basis of a periodic multiresolution analysis (PMRA, in brief); and biorthogonal bases and dual frames are constructed on the basis of two PMRA (see [1–4,17]). In the present work, we use the definition of multidimensional PMRA given by I. Maksimenko and M. Skopina [19] (see also [18, Chap. 9]). N. Atreas [16] showed that the Bessel property and certain technical conditions are sufficient for the dual wavelet systems to be frames. In the present work, we obtain conditions sufficient for the Bessel property of a multidimensional wavelet system. A one-dimensional analogue of this result was obtained in [15]. Also, based the result obtained, we provide a method for constructing biorthogonal dual wavelet bases for any suitable sequence of trigonometric polynomials. 2. Notation and auxiliary results We use the following standard notation: N is the set of natural numbers, x = (x1 , . . . , xd ), y = (y1 , . . . , yd ) are elements (vectors) ofthe d-dimensional Euclidean space Rd , (x, y) = x1 y1 + . . . xd yd , 0 = (0, . . . , 0) ∈ Rd , |x| = (x, x), Zd is the integer lattice in Rd , Z = Z1 , Z+ = {0, 1, . . .}, Td = (− 12 ; 12 ]d is the d-dimensional unit torus, δn,k is the Kronecker symbol, f(k) = Td f (t)e−2πi(k,t) dt is the Fourier coefficient of f ∈ L2 (Td ) with number k, f, g is the scalar product in L2 (Td ). If A is a d × d matrix, then A is its Euclidean operator norm from Rd to Rd , A∗ is its hermitian conjugate matrix, A∗j = (A∗ )j , Id is the unit d × d matrix. Given a nondegenerate integer d × d matrix A, we say that vectors k, n are congruent modulo A and we write k ≡ n (mod A) if k − n = Al, where l ∈ Zd . The integer lattice Zd splits into cosets with respect to this congruency relation. The number of these cosets is | det A| (see, e.g., [18, Proposition 2.2.1]). A set with exactly one element from each coset is called a set of digits of the matrix A. If the exact set of selected digits is not important, t
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