Interpolating polynomial wavelets on [?1,1]

Abstract

The present paper gives a contribution of wavelet aspects to classical algebraic polynomial approximation theory. As is so often the case in classical approximation, the authors follow the pattern provided by the trigonometric polynomial case. Algebraic polynomial interpolating scaling functions and wavelets are constructed by using the interpolation properties of de la Vallée Poussin kernels with respect to the four kinds of Chebyshev weights. For the decomposition and reconstruction of a given function the structure of the involved matrices is studied in order to reduce the computational effort by means of fast discrete cosine and sine transforms.

Uniform weighted approximation on the square by polynomial interpolation at Chebyshev nodes

“…In literature (see e.g. [25,2,24,23,27,17]) different conditions, only sufficient to get this property, can be found. The main result of the present paper consists in stating the necessary and sufficient conditions in order to get the mentioned property (cf.…”

Filtered interpolation for solving Prandtl's integro-differential equations

“…[1,28] ), but it is based on non-standard projections obtained by applying de la Vallée Poussin (briefly VP) filters. They project the function into non-standard polynomial spaces, spanned by the so-called fundamental VP polynomials that share the interpolation property of fundamental Lagrange polynomial at Chebyshev nodes of second type [5,30,31,32,34]. The resulting filtered VP interpolation has been already applied to the field of Cauchy singular integral equation in order to solve the airfoil equation [20] and in other situations [7,18].…”

Generalized Gibbs phenomenon for Fourier partial sums and de la Vallée-Poussin sums

“…In general, the study of Gibbs behavior contributes to the problem of edge detection in computer vision as well as dealing with shocks in the numerical solution of certain partial differential equations. See [1] for asymptotics for de la Vallée Poussin means; [2,9] for application to wavelets; [3,10] for distributional approach to jumps; [4,5,7] for other orthogonal expansions.…”

“…ξ − cos 2ξ ξ2 dξ + o(1).Using the identity (6), we obtainτ n (h, x) − h(x) τ n (h, 0) − h(0) ξ − cos 2ξ) ξ − u ξ 2 dξ + o(1).The whole argument can be similarly repeated for negative u. Hence for u in a bounded set we haveτ n (h, x) − h(x) τ n (h, 0) − h(0) ξ − cos 2ξ) ξ − |u| ξ 2 dξ + o(1).…”

“…and study the generalized Gibbs phenomenon for F n (x) in the sense of Zygmund (see (2)). For any fixed x = x 0 , we show that (presumably, because of the difference of smoothness)…”