In this paper we bridge local and global approximation theorems for positive linear operators via Ditzian Totik moduli | 2 , ( f, $) of second order whereby the step-weights , are functions whose squares are concave. Both direct and converse theorems are derived. In particular we investigate the situation for exponential-type and Bernstein-type operators.1998 Academic Press
Lubinsky and Totik's decomposition [11] of the Cesàro operators σ (α,β) n of Jacobi expansions is modied to prove uniform boundedness in weighted sup norms, i.e., w (a,b) 2 and a, b are within the square around α 2 + 1 4 , α 2 + 1 4 having a side length of 1. This approach uses only classical results from the theory of orthogonal polynomials and various estimates for the Jacobi weights. The present paper is concerned with the main theorems and ideas, while a second paper [7] provides some necessary estimations.
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