On conjectures of Stenger in the theory of orthogonal polynomials

Walter Gautschi,Ernst Hairer

doi:10.1186/s13660-019-2107-6

Abstract

The conjectures in the title deal with the zeros x_{j}, j=1,2, ldots ,n, of an orthogonal polynomial of degree n>1 relative to a nonnegative weight function w on an interval [a,b] and with the respective elementary Lagrange interpolation polynomials ell _{k} ^{(n)} of degree n-1 taking on the value 1 at the zero x_{k} and the value 0 at all the other zeros x_{j}. They involve matrices of order n whose elements are integrals of ell _{k}^{(n)}, either over the interval [a,x_{j}] or the interval [x_{j},b], possibly containing w as a weight function. The claim is that all eigenvalues of these matrices lie in the open right half of the complex plane. This is proven to be true for Legendre polynomials and a special Jacobi polynomial. Ample evidence for the validity of the claim is provided for a variety of other classical, and nonclassical, weight functions when the integrals are weighted, but not necessarily otherwise. Even in the case of weighted integrals, however, the conjecture is found by computation to be false for a piecewise constant positive weight function. Connections are mentioned with the theory of collocation Runge–Kutta methods in ordinary differential equations.

Highlights

Let w be a nonnegative weight function on [a, b], –∞ ≤ a < b ≤ ∞, and pn be the orthonormal polynomial of degree n relative to the weight function w
The Stenger conjectures relate to the eigenvalues of matrices of order n whose elements are certain integrals involving the elementary Lagrange polynomials (1), the claim being that the real part of all eigenvalues is positive
Proof We present the proof for the extended conjecture, the one for the restricted conjecture being the same (just drop the factor w(t) in all integrals)

Summary

Introduction

Let w be a nonnegative weight function on [a, b], –∞ ≤ a < b ≤ ∞, and pn be the orthonormal polynomial of degree n relative to the weight function w. Vn in the restricted as well as in the extended Stenger conjecture are the same if w is a symmetric weight function. 3 we show that, both in the restricted and extended conjecture, the matrix Un(α,β) belonging to the Jacobi weight function w(x) = (1 – x)α(1 + x)β on [–1, 1] with parameters α, β is the same as the matrix Vn(β,α) with the Jacobi parameters interchanged. 8 the extended Stenger conjecture is challenged in the case of a piecewise constant positive weight function. Theorem 1 If w is symmetric, the eigenvalues of Vn are the same as those of Un, both in the case of the restricted (where b < ∞) and the extended Stenger conjecture.

Proof of the Stenger conjecture for Legendre polynomials

Block-discrete and ε-block-discrete weight functions