On zeros of polynomials in best $L^p$-approximation and inserting mass points

TL;DR

This paper revisits classical ideas in $L^p$ spaces to analyze the zeros of orthogonal polynomials, providing simplified proofs and improvements on existing theorems related to the monotonicity of zeros and inserting mass points.

Contribution

It introduces a unified approach using $L^p$ space techniques to study zeros of orthogonal polynomials, improving previous results on the discrete Markov theorem.

Findings

Simplified proof of monotonicity of zeros in $L^p$ spaces.

Enhanced results on the effect of inserting mass points.

Unified approach connecting classical and discrete orthogonal polynomial theory.

Abstract

The purpose of this note is to revive in $L^p$ spaces the original A. Markov ideas to study monotonicity of zeros of orthogonal polynomials. This allows us to prove and improve in a simple and unified way our previous result [Electron. Trans. Numer. Anal., 44 (2015), pp. 271-280] concerning the discrete version of A. Markov's theorem on monotonicity of zeros.