Relationships Between the Zeros, Weights, and Weight Functions of Orthogonal Polynomials: Derivative Rule Conjecture (the DRC) Applied to Stieltjes and Spectral Imaging

Abstract

Zeros, x[k,N], k = 1N, of orthogonal polynomials are useful for numerical applications, e.g. Gaussian quadrature where combined with quadrature weights w[k,N], integrals with weight function (x) may be performed, often to high precision. For broad classes of OPs the ratios w[k,N]/(x[k,N]) are independent of the OPs at hand if the x[k,N] lie within a smooth part of (x). This universality, as N, suggests that these ratios may be evaluated as x[k,N]= d(x[i,N])/di, i = k. This Derivative Rule Conjecture, or DRC, then gives, even for relatively small N, (x[k,N]) = w[k,N]/x[k,N], and Stieltjes imaging, or inversion, with exponential convergence. Similar methods apply to the numerics of Schrdinger resolvents. The example chosen to illustrate this latter indicates that the specific assumption of universality utilized, while suggesting the DRC, is not necessary.