Hermite-Gauss Quadrature with Generalized Hermite Weight Functions and Small Sample Sets for Sparse Polynomials
Vu, Brian-Tinh D.
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This thesis derives a Gaussian quadrature rule from a complete set of orthogonal lacunary polynomials. The resulting quadrature formula is exact for polynomials whose even part skips powers, with a set of sample values that is much smaller than the degree. The weight for these quadratures is a generalized Gaussian, whose negative logarithm is an even monomial; the powers of this monomial make up the even part of the polynomial to be integrated. We first present Rodrigues formulas for generalized Hermite polynomials (GHPs) that are complete and orthogonal with respect to the generalized Gaussian. From the Rodrigues formula for even GHPs we establish a three-term recursion relation and find the normalization constants. We present a slight modification to the Christoffel-Darboux identity and the Lagrange interpolation polynomials, and proceed to derive the roots, weights, and estimate of the error for the generalized Hermite-Gauss quadrature rule applied to sufficiently smooth functions. We illustrate the quadrature rule by applying it to two examples. Finally, we apply a major result from compressive sensing relating a matrix's coherence and sparse recovery guarantees to the quadrature setting.