Interpolating polynomial wavelets on \([-1,1]\) (Q1776160)

The authors use a system of orthogonal polynomials with respect to the four Chebyshev weights, \(1/\sqrt(1-x^2)\), \(\sqrt(1-x^2)\), \(\sqrt{[(1-x)/(1+x)]}\) and \(\sqrt{[(1+x)/(1-x)]}\), with positive leading coefficients and Darboux kernels to construct four interpolating scaling functions and interpolating wavelets with a multiresolution structure. These wavelets, based on de la Vallée Poussin interpolation are more localized and give better approximation in the uniform weighted norm than polynomial wavelets based on Lagrange interpolation. However, the matrices involved in the two-scale relations are not orthogonal. The structure of these matrices is studied in detail: the elements of the inverse matrices are explicitly known and the computation of matrix-vector products can be performed by means of fast discrete cosine and sine transforms.