On conjectures of Stenger in the theory of orthogonal polynomials
Walter Gautschi and Ernst Hairer,
Abstract. The conjectures in the title deal with the zeros x_j, j = 1,2,...,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 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 the interpolation polynomial,
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.
Key Words. Regularization, discontinuous dynamical system, Filippov solution.