Uniform approximation on the sphere by least squares polynomials

Numerical Algorithms - The paper concerns the uniform polynomial approximation of a function f, continuous on the unit Euclidean sphere of ?3 and known only at a finite number of points that...     

Uniform approximation on [?1, 1] via discrete de la Vallée Poussin means

Starting from the function values on the roots of Jacobi polynomials, we construct a class of discrete de la Vallée Poussin means, by approximating the Fourier coefficients with a Gauss?CJacobi quadrature rule. Unlike the Lagrange interpolation polynomials, the resulting algebraic polynomials are uniformly convergent in suitable spaces of continuous functions, the order of convergence being comparable with the best polynomial approximation. Moreover, in the four Chebyshev cases the discrete de la Vallée Poussin means share the Lagrange interpolation property, which allows us to reduce the computational cost.     

Orthogonal Polynomial Wavelets

Recently Fischer and Prestin presented a unified approach for the construction of polynomial wavelets. In particular, they characterized those parameter sets which lead to orthogonal scaling functions. Here, we extend their results to the wavelets. We work out necessary and sufficient conditions for the wavelets to be orthogonal to each other. Furthermore, we show how these computable characterizations lead to attractive decomposition and reconstruction schemes. The paper concludes with a study of the special case of Bernstein–Szegö weight functions.     

Some Interpolating Operators of de la Vallée-Poussin Type

We consider discrete versions of the de la Vallée-Poussin algebraic operator. We give a simple sufficient condition in order that such discrete operators interpolate, and in particular we study the case of the Bernstein-Szeg weights. Furthermore we obtain good error estimates in the cases of the sup-norm and L¹-norm, which are critical cases for the classical Lagrange interpolation.