Wendroff's theorem beyond consecutive degrees and related inverse spectral problems

TL;DR

This paper extends Wendroff's theorem to arbitrary polynomials, providing new inverse spectral problem solutions for orthogonal and paraorthogonal polynomials on the real line and the unit circle.

Contribution

It generalizes Wendroff's theorem beyond consecutive degrees and introduces explicit reconstruction methods for related inverse spectral problems.

Findings

Reconstruction via Vandermonde-type systems is possible for arbitrary polynomials.

Strict interlacing of zeros is necessary and sufficient for solvability.

Explicit algorithms and examples are provided for polynomial and matrix reconstruction.

Abstract

A classical theorem of Wendroff shows that one may reconstructs a sequence of orthogonal polynomials on the real line from two non-constant polynomials of consecutive degrees whose zeros strictly interlace on the real line. In this note we extend this result to arbitrary non-constant polynomials. The reconstruction may be formulated via a Vandermonde-type linear system and recast as an underdetermined inverse spectral problem, in which the spectra of a finite Jacobi matrix and of one of its leading principal submatrices are prescribed. In addition, the analogous result on the unit circle is established by reconstructing a sequence of paraorthogonal polynomials from two arbitrary non-constant polynomials whose zeros strictly interlace on the unit circle. In this setting, the Jacobi matrix is replaced by a finite unitary pentadiagonal matrix, and the spectral data consist of the spectrum of the full matrix together with that of a rank-one perturbation of a leading principal submatrix. Strict interlacing of zeros is shown to be a necessary and sufficient condition for solvability, and explicit constructions of the associated polynomial families and matrices are provided. Finally, an algorithm and several illustrative examples are presented.