On Construction of Multivariate Wavelet Frames

Construction of wavelet frames with matrix dilation is studied. We found a necessary condition and a sufficient condition under which a given pair of refinable functions generates dual wavelet systems with a given number of vanishing moments. For image compression and some other applications, it is very desirable to have wavelets with vanishing moment property. In particular, vanishing moments are closely related to the approximation order of wavelet frames (see, e.g., [1]). If wavelet system is a basis, the number of its vanishing moments depends only on the dual generating refinable function. Situation is essentially different for frames. Two pairs of dual wavelet frames may be generated by the same refinable functions and have different number of vanishing moments. The goal of this paper is to describe refinable functions generating dual wavelet systems with vanishing moments and to present an explicit method for construction compactly supported wavelet frames with arbitrary number of vanishing moments. Very close problem were investigated by Ming-Jun Lai and A. Petukhov[2] for univariate wavelet frames. Their technique is not appropriate for multi-dimensional investigations because zero properties of multivariate masks can not be described by means of factorization in contrast to the one-dimensional case . Throughout the paper we will use the following notations. N is the set of positive integers, R denotes the d-dimensional Euclidean space, x = (x1, . . . , xd), y = (y1, . . . , yd) are its elements (vectors), (x, y) = x1y1+. . .+xdyd, |x| = √ (x, x), ej = (0, . . . , 1, . . . , 0) is the j-th unit vector in R, 0 = (0, . . . , 0) ∈ R, 1 = (1, . . . , 1) ∈ R; Z is the integer lattice in R. For x, y ∈ R, we write x > y if xj > yj, j = 1, . . . , d; Z+ = {x ∈ Z : x ≥ 0}. The paper is supported by RFBR, project N 03-01-00373