Polynomials of least deviation from zero in Sobolev $p$-norm

TL;DR

This paper extends the characterization of polynomials of least deviation from zero in Sobolev p-norms to the case p=1 and analyzes the zeros' distribution of such polynomials, including asymptotic behavior and location constraints.

Contribution

It provides new characterizations for p=1 and investigates the zeros' distribution and asymptotic properties of least deviation polynomials in Sobolev p-norms.

Findings

Characterization of least deviation polynomials for p=1.

Asymptotic distribution of zeros established.

At least n - d* zeros lie within the convex hull under certain conditions.

Abstract

The first part of this paper complements previous results on characterization of polynomials of least deviation from zero in Sobolev $p$-norm ($1<p<\infty$) for the case $p=1$. Some relevant examples are indicated. The second part deals with the location of zeros of polynomials of least deviation in discrete Sobolev $p$-norm. The asymptotic distribution of zeros is established on general conditions. Under some order restriction in the discrete part, we prove that, the $n$-th polynomial of least deviation has at least $n-\mathbf{d}^*$ zeros on the convex hull of the support of the measure, where $\mathbf{d}^*$ denotes the number of terms in the discrete part.