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 ($$1<p<\infty $$ 1 < p < ∞ ) for the case $$p=1$$ 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 $$n-\mathbf {d}^*$$ n - d ∗ zeros on the convex hull of the support of the measure, where $$\mathbf {d}^*$$ d ∗ denotes the number of terms in the discrete part.

Differential Properties of Jacobi-Sobolev Polynomials and Electrostatic Interpretation

“…Let us begin by recalling the definition of a sequentially ordered Sobolev inner product, which was stated in [20] (Definition 1) or [21] (Definition 1). Definition 2.…”