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

The paper deals with de la Vallée Poussin type interpolation on the square at tensor product Chebyshev zeros of the first kind. The approximation is studied in the space of locally continuous functions with possible algebraic singularities on the boundary, equipped with weighted uniform norms. In particular, simple necessary and sufficient conditions are proved for the uniform boundedness of the related Lebesgue constants. Error estimates in some Sobolev-type spaces are also given. Pros and cons of such a kind of filtered interpolation are analyzed in comparison with the Lagrange polynomials interpolating at the same Chebyshev grid or at the equal number of Padua nodes. The advantages in reducing the Gibbs phenomenon are shown by means of some numerical experiments.