Wavelets with the Translation Invariance Property of Order N

TL;DR

This paper introduces a new property called translation invariance of order N for wavelets, demonstrating the existence of wavelets with this property for various dilation factors, enhancing multiresolution analysis understanding.

Contribution

It establishes the existence of wavelets with translation invariance of any order for dilation by 2 and arbitrary integers, expanding wavelet theory.

Findings

Wavelets with translation invariance of all levels exist for dilation by 2.

Such wavelets also exist for arbitrary integer dilation factors.

The concept generalizes multiresolution analysis by incorporating translation invariance of higher order.

Abstract

All wavelets can be associated to a multiresolution like structure, i.e. an incr easing sequence of subspaces of L^2(R). We consider the interaction of a wavel et and the translation operator in terms of which of the subspaces in this multi resolution like structure are invariant under the translation operator. This ac tion defines the notion of the translation invariance property of order n. In this paper we show that wavelets of all levels of translation invariance exist, first for the classic case of dilation by 2, and then for arbitrary integral di lation factors.