Hermite-Gauss Quadrature for Generalized Hermite Weight Functions and Polynomials



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Traditionally, the derivation of Gaussian quadrature rules from orthogonal polynomials hinged on the reality of the polynomials’ roots. As a result, the abscissas and weights were confined to the real line, with all the roots of the polynomial serving as the abscissas. We introduce a class of weight functions that are a generalization of that for the Hermite-Gauss case, as well as a set of generalized Hermite polynomials that are complete and orthogonal with respect to the weight function. We develop an analog to the Christoffel-Darboux identity, which suggests a natural measure for the development of our quadrature rule. We show that although our generalized Hermite polynomials lack the property of having all real roots like the regular case, we are still able to develop a quadrature rule where sampling of the function is performed only at the roots corresponding to the natural measure. Finally, we derive an expression for the error in the quadrature approximation.