Approximation properties of Poisson integrals for orthogonal expansions

Abstract

In the present paper we introduce Poisson type integrals for orthogonal expansions. We first give some direct computations for the moments and compute the rates of convergence by means of the modulus of continuity and the Lipschitz functionals; and also we prove that our results are stronger and more general than the results obtained by Toczek and Wachnicki [J. Approx. Theory 116 (2002), 113-125]. We obtain a statistical approximation theorem by using the concept of T-statistical convergence which is a (non-matrix) summability transformation. Furthermore, we give a general Voronovskaya type theorem for these operators. Finally, introducing a higher order generalization of Poisson integrals we discuss their approximation properties.