Sobolev spaces with respect to measures in curves and zeros of Sobolev orthogonal polynomials

TL;DR

This paper establishes criteria for bounded multiplication operators in Sobolev spaces with measures on curves, linking these properties to the zeros and asymptotics of Sobolev orthogonal polynomials, and proving these spaces are Banach.

Contribution

It provides practical criteria for bounded multiplication operators in Sobolev spaces with measures on curves, characterizes such spaces for various weights, and proves their Banach space structure.

Findings

Bounded multiplication operators imply uniform zero bounds for Sobolev orthogonal polynomials.

Characterization of weighted Sobolev spaces with bounded multiplication operators.

Sobolev spaces with measures on curves are Banach spaces.

Abstract

In this paper we obtain some practical criteria to bound the multiplication operator in Sobolev spaces with respect to measures in curves. As a consequence of these results, we characterize the weighted Sobolev spaces with bounded multiplication operator, for a large class of weights. To have bounded multiplication operator has important consequences in Approximation Theory: it implies the uniform bound of the zeros of the corresponding Sobolev orthogonal polynomials, and this fact allows to obtain the asymptotic behavior of Sobolev orthogonal polynomials. We also obtain some non-trivial results about these Sobolev spaces with respect to measures; in particular, we prove a main result in the theory: they are Banach spaces.