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Zeros of orthogonal polynomials and some matrix inequalities

Carmen Escribano, Raquel Gonzalo

TL;DR

This work develops a matrix-analytic framework for Sobolev orthogonal polynomials by using infinite HPD matrices to define matrix Sobolev inner products. It characterizes when the multiplication operator is bounded in this setting, linking bounded zeros of Sobolev polynomials to bounded point evaluations and, crucially, to Wirtinger-type inequalities that relate a polynomial’s norm to that of its derivative. The results yield numerous examples where the zeros remain uniformly bounded without requiring sequential dominance of the measures, including cases with Lebesgue measures on circles and vectorial measures supported on circles and their interiors. The paper thus broadens the toolkit for zero localization in Sobolev orthogonality, showing new sufficient conditions beyond sequential dominance and connecting spectral properties of moment matrices to polynomial inequality bounds with concrete geometric interpretations on the support.

Abstract

The main aim of this work is to apply the matrix approach of ortho\-gonal polynomials associated with infinite Hermitian definite positive matrices in relation with an important question regarding the location of zeros of Sobolev orthogonal polynomials via the study of the boundedness of multiplication operator. We apply the notion of bounded point evaluations of a measure, and more generally to infinite HPD matrices, to the problem of boundedness of multiplication operator. Moreover, we introduce certain Wirtinger-type inequalities, relating the norm of the polynomials with the norm of their derivatives, in order to provide new examples of Sobolev polynomials for which we may ensure that the zeros of Sobolev polynomials are uniformly bounded. In particular, we consider the case of Lebesgue measures supported on circles. With these techniques, we may obtain many examples of vectorial measures such that the zeros of Sobolev orthogonal polynomials are bounded, and nevertheless they are not sequentially dominated, not even matrix sequentially dominated.

Zeros of orthogonal polynomials and some matrix inequalities

TL;DR

This work develops a matrix-analytic framework for Sobolev orthogonal polynomials by using infinite HPD matrices to define matrix Sobolev inner products. It characterizes when the multiplication operator is bounded in this setting, linking bounded zeros of Sobolev polynomials to bounded point evaluations and, crucially, to Wirtinger-type inequalities that relate a polynomial’s norm to that of its derivative. The results yield numerous examples where the zeros remain uniformly bounded without requiring sequential dominance of the measures, including cases with Lebesgue measures on circles and vectorial measures supported on circles and their interiors. The paper thus broadens the toolkit for zero localization in Sobolev orthogonality, showing new sufficient conditions beyond sequential dominance and connecting spectral properties of moment matrices to polynomial inequality bounds with concrete geometric interpretations on the support.

Abstract

The main aim of this work is to apply the matrix approach of ortho\-gonal polynomials associated with infinite Hermitian definite positive matrices in relation with an important question regarding the location of zeros of Sobolev orthogonal polynomials via the study of the boundedness of multiplication operator. We apply the notion of bounded point evaluations of a measure, and more generally to infinite HPD matrices, to the problem of boundedness of multiplication operator. Moreover, we introduce certain Wirtinger-type inequalities, relating the norm of the polynomials with the norm of their derivatives, in order to provide new examples of Sobolev polynomials for which we may ensure that the zeros of Sobolev polynomials are uniformly bounded. In particular, we consider the case of Lebesgue measures supported on circles. With these techniques, we may obtain many examples of vectorial measures such that the zeros of Sobolev orthogonal polynomials are bounded, and nevertheless they are not sequentially dominated, not even matrix sequentially dominated.

Paper Structure

This paper contains 5 sections, 25 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Characterization of boundedness of multiplication operator for matrix Sobolev inner products
  3. Bounded point evaluations and zeros of Sobolev orthogonal polynomials
  4. Polynomial Wirtinger-type inequalities.
  5. Applications to boundedness of zeros of Sobolev orthogonal polynomials

Key Result

Proposition 1

Let $\Vert \cdot \Vert_{\mathbf{M_S}}$ be a matrix Sobolev norm associated with a pair of HPSD matrices $\{\mathbf{M}_0,\mathbf{M}_1\}$ with $\mathbf{M}_0$ being HPD. Assume that $\mathcal{D}$ is bounded on $(\mathbb{P}[z],\Vert \cdot \Vert_{\mathbf{M}_0})$ and there exists $C_1>0$ such that Then, the following are equivalent:

Theorems & Definitions (51)

  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • Lemma 1
  • proof
  • Definition 3
  • ...and 41 more