In this paper we deal with sequences of polynomials orthogonal with respect to the discrete Sobolev inner productwhere ω is a weight function, ξ ≤ 0, and M, N ≥ 0. The location of the zeros of discrete Sobolev orthogonal polynomials is given in terms of the zeros of standard polynomials orthogonal with respect to the weight function ω. In particular, for ω(x) = x α e −x we obtain the asymptotics for discrete Laguerre-Sobolev orthogonal polynomials.
We have constructed a new sequence of positive linear operators with two variables by using Szasz-Kantorovich-Chlodowsky operators and Brenke polynomials. We give some inequalities for the operators by means of partial and full modulus of continuity and obtain a Lipschitz type theorem. Furthermore, we study the convergence of Szasz-Kantorovich-Chlodowsky-Brenke operators in weighted space of function with two variables and estimate the rate of approximation in terms of the weighted modulus of continuity.
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