A free parameter depending family of polynomial wavelets on a compact interval

TL;DR

The paper addresses the challenge of constructing wavelet systems on a compact interval without relying on dilations and translations of a single mother function. It develops a non-standard multiresolution analysis on $[-1,1]$ using de la Vallée Poussin interpolation at Chebyshev zeros, with a free parameter $m$ that can be tuned to control Gibbs phenomena and approximation quality. It defines VP-based scaling functions $\Phi_{n,k}^m$ and interpolating wavelets $\psi_{n,k}^m$, establishing interpolation properties, Riesz stability in weighted $L^p$ spaces, and detailed decomposition/reconstruction algorithms that run in ${\cal O}(n\log n)$ via DCTs. The framework supports level-wise iteration with $m=\lfloor\theta n\rfloor$ and can extend to other odd scaling factors and Chebyshev weights, offering a scalable, boundary-friendly approach for polynomial wavelets with potential applications in imaging and numerical analysis.

Abstract

On a compact interval, we introduce and study a whole family of wavelets depending on a free parameter that can be suitably modulated to improve performance. Such wavelets arise from de la Vallée Poussin (VP) interpolation at Chebyshev nodes, generalizing previous work by Capobianco and Themistoclakis who considered a special parameter setting. In our construction, both scaling and wavelet functions are interpolating polynomials at some Chebyshev zeros of 1st kind. Contrarily to the classical approach, they are not generated by dilations and translations of a single mother function and are naturally defined on the interval $[-1,1]$ to which any other compact interval can be reduced. In the paper, we provide a non-standard multiresolution analysis with fast (DCT-based) decomposition and reconstruction algorithms. Moreover, we state several theoretical results, particularly on convergence, laying the foundation for future applications.

Figures (6)

• Figure 1: The oscillating behavior of the fundamental Lagrange polynomial compared to the fundamental VP polynomials with several $m$.
• Figure 2: The scaling function in function of $k$ with $m=\lfloor \theta n\rfloor$.
• Figure 3: Scaling functions for different values of $n = 3^j$ with $m=\lfloor \theta n\rfloor$.
• Figure 4: The interpolating wavelet function in function of $k$.
• Figure 5: Wavelet functions for different values of $n = 3^j$ .