Wavelet Basis Packets and Wavelet Frame Packets

This article obtains the nonseparable version of wavelet packets on ~a and generalizes the "'unstability" result of nonorthogonal wavelet packets in Cohen-Daubechies to higher dimensional cases. 1 . I n t r o d u c t i o n The wavelet packets introduced by R. Coifman, Y. Meyer, and M. V. Wickerhauser played an important role in the applications of wavelet analysis as shown, for example, in [CMW1, CMW2]. But the theory itself is worthy of further study. Some developments in the wavelet packets theory should be mentioned, such as the tensor product version (due to [CM]) and the non-tensor-product version (due to [S]) of wavelet packets on 11~ d, the nonorthogonal version of wavelet packets on ll~ ~ (due to [CL]), and the wavelet frame packets on R t (due to [C]). The higher dimensional version of wavelet packets obtained in [S] is very close to the expected one. But it seems that there is a shortcoming in Shen's result; specifically, the implied frequency index is denoted by the point ~ in Za+, which makes the correspondence between the index pair ($, j ) and the dyadic interval I~, j less natural than that in the one-dimensional cases. One task of this article is to set up a more natural framework for the wavelet packets in the higher dimensional case. Another task of this article is to study the lack of stability of nonorthogonal wavelet packets. As shown in [CD], starting from one-dimensional biorthogonal multiresolution analysis (MRA), a stable wavelet packet can hardly be constructed unless the matrix used in the splitting trick is unitary. We want to generalize the result to/~d. The notation and symbols used in this article are standard in wavelet theory. We list them as follows. For more detail see [LC]. An MRA is a nondecreasing family {Vj } ~ of closed subspaces of L2(~, d) satisfying: i. N Vj = {0}, U Vj = L2(~a); ii. f ( x ) ~ Vj ~ ,~ f ( 2 x ) ~ Vj+I,Vj; iii. 3~o(x) ~ Vo such that {~o(x k)}k is a Riesz basis of V0. ~o(x) is called the scaling function of MRA {Vj}~_~, and ~o(x) satisfies the refinement equation 3{dk} ~ l 2 such that qg(x) = 2 d ~ dk~o(2x -k) a.e. x ~ ~a. k *Professor Ruilin Long died on August 13, 1996. Math Subject Classifications. 41 A99, 42C99.