A Note on the Daubechies Approach in the Construction of Spline Type Orthogonal Scaling Functions

TL;DR

This paper presents an explicit solution method using Lorentz polynomials to efficiently verify and evaluate Daubechies' spline type orthogonal scaling functions, enhancing understanding of their construction.

Contribution

It introduces a novel explicit solution approach with Lorentz polynomials for Daubechies' equations, simplifying the proof and evaluation of spline type orthogonal scaling functions.

Findings

Explicit solutions for Daubechies' equations using Lorentz polynomials

Efficient verification of the existence of spline type orthogonal scaling functions

Simplified evaluation method for Daubechies scaling functions

Abstract

We use Lorentz polynomials to present the solutions explicitly of equations (6.1.7) of [I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, 61. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992] and (4.9) of [I. Daubechies, Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41 (1988), no. 7, 909--996] sot that we give an efficient way to prove Daubechies' results on the existence of spline type orthogonal scaling functions and to evaluate Daubechies scaling functions.