Exact error bounds for the reconstruction processes using interpolating wavelets

Abstract

It is known that a multiresolution scheme without control on the infinity norm can produce numerical artifacts. This work is intended to provide explicit error bounds for the reconstruction process associated with the interpolating wavelets introduced by Donoho [D. Donoho, Interpolating Wavelet Transforms, Preprint, Department of Statistics, Stanford University, 1992]. The stability constants related to the interpolatory wavelets defined by the use of the Daubechies filters [I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992], also called the Deslaurier–Dubuc wavelets, are tabulated. Our study present two important facts: The first is that the obtained stability constants are much better approximated than those given by other approaches motivated by [S. Amat, J. Liandrat, On the stability of the PPH multiresolution algorithm, Appl. Comp. Harmon. Anal. 18 (2005) 198–206] and the second is that our analysis uses basic rules easily understood by a wide part of the scientific community interested in this kind of explicit numerical results.