An Equivalence Relation between Multiresolution Analysis ofL2(R)

Abstract

Abstract-This paper is concerned with the development of an equivalence relation between two multiresolution analysis of L 2 (R). The relation called unitary equivalence is created by the action of a unitary operator in such a way that the multiresolution structure and the decomposition and reconstruction algorithms remain invariant. A characterization in terms of the scaling functions of the multiresolution analysis is given. Distinct equivalence classes of multiresolution analysis are derived. Finally, we prove that B-splines give rise to nonequivalent examples.

Unitary mappings between multiresolution analyses of L2 (R) and a parametrization of low-pass filters

“…Scaling functions determine uniquely their corresponding low-pass filter by the two-scale relation (1.1). As it was shown in [16] this is not an one-to-one correspondence. Scaling functions having the same low-pass filters produce MRAs, which were called equivalent and a characterization of all such scaling functions was given in [ 16].…”

“…As it was shown in [16] this is not an one-to-one correspondence. Scaling functions having the same low-pass filters produce MRAs, which were called equivalent and a characterization of all such scaling functions was given in [ 16]. Set/z = ~22 m. converges almost everywhere in R, then Mallat [15] proved that this infinite product converges to a square-integrable function defined on R. The a.e.…”

“…I). Based on the characterization of equivalent MRAs given in [ 16] this question is equivalent to the following: Given an arbitrary scaling function 4~ is it possible to find unimodular functions defined on R m I 2zr-periodic and h satisfying h (2t) = h (t) a.e. such that .T'-l (m l hq~) is a scaling function which can be given by the infinite product (3.1)?…”

“…then co is scaling function for an MRA different from the Haar MRA and the corresponding lowpass filter for co is again m [16,Cor. 2.12].…”

“…Since the zero-indexed subspace of every MRA is the only subspace that has an orthonormal basis consisted of the integer translations of a single function [16], it is natural to search within a class of unitary operators that preserve this property. This leads us to the unitary operators in the commutant of the shift operator.…”