Equations For Parseval's Frame Wavelets In $L^2(\R^d)$ With Compact Supports

TL;DR

This paper constructs Parseval's wavelet functions with compact support in multi-dimensional space using matrices with specific properties, providing a systematic approach for wavelet design in $L^2( ^d)$.

Contribution

It introduces a method to derive Parseval's wavelets with compact support from matrices similar to a given expansive integral matrix with determinant ±2.

Findings

Convergence of iterated sequences to a scaling function in $L^2( ^d)$.

Explicit construction of wavelet functions with compact support.

Framework applicable to matrices with determinant ±2.

Abstract

Let $d\geq 1$ be a natural number and $A_0$ be a $d\times d$ expansive integral matrix with determinant $\pm 2.$ Then $A_0$ is integrally similar to an integral matrix $A$ with certain additional properties. A finite solution to the system of equations associated with the matrix $A$ will result in an iterated sequence $\{\Psi^k \chi_{[0,1)^d}\}$ that converges to a function $\varphi_A$ in $L^2(\R^d)$-norm. With this (scaling) function $\varphi_A,$ we will construct the Parseval's wavelet function $\psi$ with compact support associated with matrix $A_0.$