Application of Mhaskar‐Prestin operators to the convergence of orthonormal expansions

Abstract

Let I be either R or (−1, 1), and let W : I → (0, ∞). Assume that W 2 is a weight. We study the quasi-interpolatory polynomial operators τ l,n,m introduced by Mhaskar and Prestin, for Freud weights, Erdős weights, and the exponential weights on (−1, 1). We investigate boundedness of τ l,n,m in weighted Lp spaces. We then use this result to show that lim n→∞ (f − sn (f )) W L∞(I) = 0, for even exponetial weights.