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Stieltjes, Perron, and Markov in analysis of the moment problem, forabsolutely continuous measures, constructed the underlying measure as thediscontinuity across the cut of a Cauchy representation of an otherwisereal-analytic function. Introduction of Gaussian quadratures suggests moredirect approaches. The real, positive, and continuous weight function, rho(x),is constructed directly from interpolation of the zeros, x[k], and weights,w[k], of the polynomials orthogonal for that measure. This novel numericalapproach rests on a simple theorem: namely, rho(x[k]), evaluated at thequadrature point x[k], is approximated by w[k]/(dx[k]/dk), with k now beingassumed to be a continuous real variable. This derivative rule is shown to beclosely related to a simple type of universal behavior, relating the clock-rulespacings of the x[k] to the derivatives dx[k]/dk which, once established,allows an immediate derivation of the derivative rule, via known and wellunderstood clock-rule results for Nevai-Blumenthal class polynomials. Thisframework also leads to a statement of a universality for quadrature weightsand weight functions. This is implicit in statements of clock-rule universalityfor the N-B class, but not made explicit. This new universality is illustratedfor several classical orthogonal polynomials, and the non-classical Pollaczekpolynomials. The derivative rule is then applied to the problem of numericalStieltjes inversion, for both finite and unbounded intervals giving exponentialconvergence. Exponential convergence stands in stark contrast to the power lawconvergence of historically well-known histogram techniques A final sectiontraces the origin of the derivative rule in developing novel approaches to theuse of discrete spectral resolutions of Schr\"odinger operators.

Universality Properties of Gaussian Quadrature, The Derivative Rule, and a Novel Approach to Stieltjes Inversion