Point values and normalization of two-direction multiwavelets and their derivatives

TL;DR

This paper explores methods for determining point values and normalizing derivatives of two-direction multiscaling functions and multiwavelets, enhancing their application flexibility in signal processing.

Contribution

It introduces eigenvalue-based techniques for point value computation and normalization conditions for two-direction multiscaling functions and multiwavelets.

Findings

Eigenvalue approach effectively finds point values.

Normalization based on zeroth continuous moment.

Illustrative examples demonstrate the theory.

Abstract

Two-direction multiscaling functions $\boldsymbol{\phi}$ and two-direction multiwavelets $\boldsymbol{\psi}$ associated with $\boldsymbol{\phi}$ are more general and more flexible setting than one-direction multiscaling functions and multiwavelets. In this paper, we investigate how to find and normalize point values and those of derivatives of the two-direction multiscaling functions $\boldsymbol{\phi}$ and multiwavelets $\boldsymbol{\psi}$. %associated with $\boldsymbol{\phi}$. For finding point values, we investigate the eigenvalue approach. For normalization, we investigate the normalizing conditions for them by normalizing the zeroth continuous moment of $\boldsymbol{\phi}$. Examples for illustrating the general theory are given.