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