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 < ∞) 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 − d * zeros on the convex hull of the support of the measure, where d * denotes the number of terms in the discrete part.