Multiresolution Analysis Based on Coalescence Hidden-variable FIF

TL;DR

This paper develops a multiresolution analysis framework using Coalescence Hidden-variable Fractal Interpolation Functions (CHFIFs), offering enhanced control and explicit construction of orthonormal wavelets in L2(R).

Contribution

It introduces the vector space of CHFIFs, determines its dimension, and constructs Riesz bases and orthogonal bases for multiresolution analysis, including explicit wavelet generation.

Findings

Constructed Riesz bases for CHFIF-based subspaces.

Explicitly generated orthonormal wavelets with compact support.

Enhanced control in function reconstruction over AFIF-based methods.

Abstract

In the present paper, multiresolution analysis arising from Coalescence Hidden-variable Fractal Interpolation Functions (CHFIFs) is accomplished. The availability of a larger set of free variables and constrained variables with CHFIF in multiresolution analysis based on CHFIFs provides more control in reconstruction of functions in L2(\mathbb{R})than that provided by multiresolution analysis based only on Affine Fractal Interpolation Functions (AFIFs). In our approach, the vector space of CHFIFs is introduced, its dimension is determined and Riesz bases of vector subspaces Vk, k \in \mathbb{Z}, consisting of certain CHFIFs in L2(\mathbb{R}) \cap C0(\mathbb{R}) are constructed. As a special case, for the vector space of CHFIFs of dimension 4, orthogonal bases for the vector subspaces Vk, k \in \mathbb{Z}, are explicitly constructed and, using these bases, compactly supported continuous orthonormal wavelets are generated.