Relationships among Interpolation Bases of Wavelet Spaces and Approximation Spaces

TL;DR

This paper explores the relationships between interpolation bases in wavelet and approximation spaces within multiresolution analysis, providing conditions for the existence of interpolation wavelets and a new construction algorithm.

Contribution

It establishes necessary and sufficient conditions for interpolation wavelet existence and introduces a novel algorithm for their construction based on interpolation scaling functions.

Findings

Theorems confirm existence conditions for interpolation wavelets.

New algorithm effectively constructs interpolation wavelets.

Simulations validate theoretical results across typical wavelet spaces.

Abstract

A multiresolution analysis is a nested chain of related approximation spaces.This nesting in turn implies relationships among interpolation bases in the approximation spaces and their derived wavelet spaces. Using these relationships, a necessary and sufficient condition is given for existence of interpolation wavelets, via analysis of the corresponding scaling functions. It is also shown that any interpolation function for an approximation space plays the role of a special type of scaling function (an interpolation scaling function) when the corresponding family of approximation spaces forms a multiresolution analysis. Based on these interpolation scaling functions, a new algorithm is proposed for constructing corresponding interpolation wavelets (when they exist in a multiresolution analysis). In simulations, our theorems are tested for several typical wavelet spaces, demonstrating our theorems for existence of interpolation wavelets and for constructing them in a general multiresolution analysis.