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

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 Vall\'ee 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.