Semi-orthogonal Parseval frame wavelets and generalized multiresolution analyses

Abstract

We study Parseval frame wavelets inwhere A is an arbitrary expanding n × n matrix with integer coefficients, such that |det A| = 2. In our study we use generalized multiresolution analyses (GMRA) (V j ) in L 2 (R d ) with dilations D. We describe, in terms of the underlying multiresolution structure, all GMRA Parseval frame wavelets and, a posteriori, all semi-orthogonal Parseval frame wavelets in L 2 (R d ). As an application, we include an explicit construction of an orthonormal wavelet on the real line whose dimension function is essentially unbounded.

Scaling sets and orthonormal wavelets with dilations induced by expanding matrices

“…In contrast to the dyadic situation, here we obtain only an algorithm, rather than a formula, for the "high-pass filter" of the PF MRA in question. A similar result for GMRA's that admit orthonormal wavelets in the case d = 2 is proved in [2]. Theorem 4.2.…”

Matricial filters and crystallographic composite dilation wavelets

“…Many of the results below are part of the wavelet community folklore; however, we were not able to find a reference where they would be stated at this level of generality. A special case of the theory below can be found, for example, in [1]. The general case can be deduced, with some effort and additional (unnecessary) assumptions, from [18].…”

“…ξ. In this case we essentially use the standard [11] construction of the high pass filter modified for the multivariate case [1]. Let j(a) be an index such that β j(a) is not an element of the integer lattice and, hence, e 2πiβ j(a) = −1.…”

Some properties and a characterization of two-directional scaling function

“…Shouzhi Yang [2], introduces the concept of two-direction wavelet. As is known to all, refinement equation piays a very important role in wavelet theory and application [3][4][5][6][7][8], Two-directional refinement equation more general than fine equation also has a very important role in wavelet analysis.…”

“…Based on [8] this paper give two theorems in ) ( 2 R L . From two-directional refinement equation, gave the concept of two-directional multi-resolution analysis and two-directional scaling function, discussed the some properties of two-directional scaling function, and finally gave a sufficient and necessary condition for a function to be a two-directional scaling function.…”