A new form of trigonometric orthogonality and Gaussian-type quadrature

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摘要

Gauss's (2n+1)-point trigonometric interpolation formula, based upon f(xi), i = 1(1)2n+1, gives a trigonometric sum of the nth order, S2n+1(x = a0 + ∑jn = 1(ajcos jx + bjsin jx), which may be integrated to provide formulas for either direct quadrature or stepwise integration of differential equations having periodic (or near-periodic) solutions. An “orthogonal” trigonometric sum S2r+1(x) is one that satisfies ∫abS2r+1(x)S2r′+1(x)dx=0, r′ 0 and that 2r-1 of the nodes must lie within the interval [a,b], and the remaining node (which may or may not be in [a,b]) must be real. Unlike Legendre polynomials, any [a′,b′] other than a translation of [a,b], requires different and unrelated sets of nodes and weights. Gaussian-type quadrature formulas are applicable to the numerical integration of the Gauss (2n+1)-point interpolation formulas, with extra efficiency when the latter are expressed in barycentric form. S2r+1(x), xj and Aj, j = 1(1)2r, were calculated for [a,b] = [0, π/4], 2r = 2 and 4, to single-precision accuracy.

论文关键词:42A52,65D30,42A12,65D05,Trigonometric orthogonality,Gaussian-type quadrature,trigonometric interpolation,numerical integration,differential equations,periodic solutinos

论文评审过程:Available online 20 April 2006.

论文官网地址:https://doi.org/10.1016/0771-050X(76)90045-0