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

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.

Figures (2)

• Figure 1: The sets $Z_n=\{1,2,3,4\}$ (disks) and $Z_m=\{\tfrac{3}{2},\tfrac{7}{2}\}$ (squares), with bands $I_0=\{1\}$, $I_1=\{2,3\}$, $I_2=\{4\}$.
• Figure 2: The sets $Z_n=\{i,e^{4\pi i/3},e^{5\pi i/3}\}$ (disks) and $Z_m=\{1,-1\}$ (squares) on the unit circle, with bands $I_1=(0,\pi)$ and $I_2=(\pi,2\pi)$.

Theorems & Definitions (12)

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