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}^* zeros on the convex hull of the support of the measure, where mathbf {d}^* denotes the number of terms in the discrete part.