Towards the best constant in front of the Ditzian-Totik modulus of smoothness

José A Adell,Alberto Lekuona

doi:10.1186/s13660-016-1078-0

Abstract

We give accurate estimates for the constants $$ K\bigl(\mathcal{A}(I), n, x\bigr)=\sup_{f\in\mathcal{A}(I)}\frac{|L_{n} f(x)-f(x)|}{\omega_{\sigma}^{2} ( f; 1/\sqrt{n} )}, \quad x\in I, n=1,2,\ldots, $$ where $I=\mathbb {R}$ or $I=[0,\infty)$ , $L_{n}$ is a positive linear operator acting on real functions f defined on the interval I, $\mathcal{A}(I)$ is a certain subset of such function, and $\omega _{\sigma}^{2} (f;\cdot)$ is the Ditzian-Totik modulus of smoothness of f with weight function σ. This is done under the assumption that σ is concave and satisfies some simple boundary conditions at the endpoint of I, if any. Two illustrative examples closely connected are discussed, namely, Weierstrass and Szasz-Mirakyan operators. In the first case, which involves the usual second modulus, we obtain the exact constants when $\mathcal{A}(\mathbb {R})$ is the set of convex functions or a suitable set of continuous piecewise linear functions.

Highlights

Let I be a closed real interval with nonempty interior set I
If σ ≡, we denote by ω (f ; ·) = ωσ (f ; ·) the usual second modulus of smoothness of f
We give different upper estimates of the aforementioned constants, heavily depending on the set of functions under consideration and on the kind of convergence we are interested in, namely, pointwise convergence or uniform convergence. Both examples are connected in the sense that, roughly speaking, the upper estimates for Szàsz-Mirakyan operators are, asymptotically, the same as those for the Weierstrass operator

Summary

We give accurate estimates for the constants

F ∈A(I) ωσ[2] (f ; 1/ n) where I = R or I = [0, ∞), Ln is a positive linear operator acting on real functions f defined on the interval I, A(I) is a certain subset of such function, and ωσ[2] (f ; ·) is the Ditzian-Totik modulus of smoothness of f with weight function σ. This is done under the assumption that σ is concave and satisfies some simple boundary conditions at the endpoint of I, if any.

Introduction

This readily implies that