Orthonormal polynomial wavelets associated with de la Vallée Poussin-type interpolation on $[-1,1]$

Abstract

Starting from de la Vallée 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.

Figures (13)

• Figure 1: The interpolating VP scaling functions $\Phi_{n,k}^m$ for $n=13$, $m=6$, and $k=3, 7, 11$. The open dots denote the set of nodes $X_n$.
• Figure 2: The polynomials $q_{13,6}^6$ (left) and $q_{13,12}^6$ (right) .
• Figure 3: The orthonormal VP scaling functions $\tilde{\varphi}_{n,k}^m$ for $n=13$, $m=6$, and $k=3,7,11$. The open dots denote the set of nodes $X_n$.
• Figure 4: Lebesgue functions $\lambda_{10}^{5}(x)$ (left) and $\lambda_{100}^{50}(x)$ (right).
• Figure 5: Lebesgue functions $\tilde{\lambda}_{10}^{5}(x)$ (left) and $\tilde{\lambda}_{100}^{50}(x)$ (right) .