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

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 $n-{\mathbf{d}}^{\ast }$ zeros on the convex hull of the support of the measure, where ${\mathbf{d}}^{\ast }$ denotes the number of terms in the discrete part.

arXiv comment21 pages