Weighted polynomial approximation on the square by de la Vallée Poussin means

We consider the generalization of discrete de la Vallée Poussin means on the square, obtained via tensor product by the univariate case. Pros and cons of such a kind of filtered approximation are discussed. In particular, under simple, we get near-best discrete approximation polynomials in the space of all locally continuous functions on the square with possible algebraic singularities on the boundary, equipped with the weighted uniform norm. In the four Chebychev cases, these polynomials also interpolate the function. Moreover, for almost everywhere smooth functions, the Gibbs phenomenon appears reduced. Comparison with other interpolating polynomials are proposed.