Sharp quadrature formulas and Nikol'skii type inequalities for rational functions

TL;DR

This paper develops precise quadrature formulas for integrating complex rational functions on various sets and uses them to establish sharp inequalities relating different function norms, advancing approximation theory.

Contribution

It introduces new sharp quadrature formulas for rational functions on circles, real axes, and segments, and derives Nikol'skii type inequalities based on these formulas.

Findings

Sharp quadrature formulas for complex rational functions

Exact calculation methods for L2 norms of rational functions

New inequalities relating different function metrics

Abstract

Sharp quadrature formulas for integrals of complex rational functions on circles, real axis and its segments are obtained. We also find sharp quadrature formulas for calculation of $L_2$-norms of rational functions on such sets. Basing on quadrature formulas for rational functions, in particular, for simple partial fractions and polynomials, we derive sharp inequalities for different metrics (Nikol'skii type inequalities).