Quadrature formulas with variable nodes and Jackson-Nikolskii inequalities for rational functions

TL;DR

This paper introduces new variable-node quadrature formulas for complex rational functions and derives sharp Jackson-Nikolskii inequalities, extending previous results and applicable to various classes of functions.

Contribution

The authors develop novel parametric quadrature formulas with variable nodes and establish sharp Jackson-Nikolskii inequalities for rational functions and their derivatives.

Findings

New quadrature formulas for integrals over circles and real segments.

Sharp Jackson-Nikolskii inequalities for rational functions.

Extensions and refinements of previous inequalities.

Abstract

We obtain new parametric quadrature formulas with variable nodes for integrals of complex rational functions over circles, segments of the real axis and the real axis itself. Basing on these formulas we derive $(q,p)$-inequalities of Jackson-Nikolskii type for various classes of rational functions, complex polynomials and their logarithmic derivatives (simple partial fractions). It is shown that our $(\infty,2)$- and $(\infty,4)$-inequalities are sharp in a number of main theorems. Our inequalities extend and refine several results obtained earlier by other authors.