Uniform Weighted Approximation by Multivariate Filtered Polynomials

Abstract

The paper concerns the weighted uniform approximation of a real function on the $d-$cube $[-1,1]^d$, with $d>1$, by means of some multivariate filtered polynomials. These polynomials have been deduced, via tensor product, from certain de la Vallée Poussin type means on $[-1,1]$, which generalize classical delayed arithmetic means of Fourier partial sums. They are based on arbitrary sequences of filter coefficients, not necessarily connected with a smooth filter function. Moreover, in the continuous case, they involve Jacobi–Fourier coefficients of the function, while in the discrete approximation, they use function’s values at a grid of Jacobi zeros. In both the cases we state simple sufficient conditions on the filter coefficients and the underlying Jacobi weights, in order to get near–best approximation polynomials, having uniformly bounded Lebesgue constants in suitable spaces of locally continuous functions equipped with weighted uniform norm. The results can be useful in the construction of projection methods for solving Fredholm integral equations, whose solutions present singularities on the boundary. Some numerical experiments on the behavior of the Lebesgue constants and some trials on the attenuation of the Gibbs phenomenon are also shown.