Orthonormal polynomial wavelets associated with de la Vall\'ee Poussin-type interpolation on $[-1,1]$

TL;DR

This paper introduces a new family of orthonormal polynomial wavelet bases derived from de la Vallée Poussin interpolation, offering well-localized, orthogonal bases with proven convergence properties, suitable for applications requiring orthonormality.

Contribution

The study develops orthonormal polynomial wavelet bases associated with de la Vallée Poussin interpolation, enhancing previous interpolating bases with orthogonality and similar localization features.

Findings

Orthonormal bases are well-localized and comparable to interpolating bases.

Uniform boundedness of Lebesgue constants is established.

Convergence rates are comparable to best polynomial approximation.

Abstract

Starting from de la Vall\'ee Poussin type (VP) interpolation, the authors have recently introduced a family of interpolating polynomial scaling and wavelet bases generating the approximation and detail spaces of a non-standard multiresolution analysis. Motivated by the fact that, in many applications, orthonormal rather than interpolating bases are preferable, the present study develops a new family of scaling and wavelet polynomials that provide well-localized and orthonormal bases for the same approximation and detail spaces. We show that the proposed new bases have a behavior very similar to the interpolating bases already introduced, presenting similar features although they are not interpolating but orthonormal. In particular, we study the Fourier projection corresponding to the proposed orthonormal scaling basis, and introduce a discrete version of it by approximating the Fourier--like coefficients. For both continuous and discrete orthogonal projections, we prove the uniform boundedness of the Lebesgue constants and the uniform convergence with an asymptotic rate comparable with the best uniform polynomial approximation. Numerical experiments confirm the theoretical results and compare the new orthonormal VP scaling and wavelet bases with the interpolating case previously treated by the authors.