Approximation of a~Class of Singular Integrals by Algebraic Polynomials with Regard to the Location of a~Point on an Interval

V. P. Motornyi

Informatics and Automation, Function spaces, harmonic analysis, and differential equations, Tome 232 (2001), pp. 268-285

Résumé

A pointwise approximation of singular integrals $S(f)(x)=\frac 1\pi \int _{-1}^1\frac {f(t)}{t-x}\frac 1{\sqrt {1-t^2}}\,dt$, $x\in (-1,1)$, of functions from the class $W^rH^{\omega }$ by algebraic polynomials is analyzed ($\omega(t)$ is a convex upward modulus of continuity such that $t\omega '(t)$ is a nondecreasing function). The estimates obtained cannot be improved simultaneously for all moduli of continuity.