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Non-2π-Periodic Trigonometric Approximation And Singular Quadrature Research

Posted on:2017-06-02Degree:MasterType:Thesis
Country:ChinaCandidate:J LiuFull Text:PDF
GTID:2310330512465211Subject:Computer Application Technology
Abstract/Summary:PDF Full Text Request
By use of the method of the change of variable,a set of the orthogonal polynomials is transformed to a set of the non-2π-periodic trigonometric polynomials(hereinafter referred to as the nonperiodic trigonometric polynomials).Using the nonperiodic trigonometric polynomials as the approximate tool,some fundamental research on numerical computation for the singular integral with Cauchy kernel is discussed in the present thesis which consists of six chapters.At first,some results by H.Tal-Ezer are introduced,which are the orthogonalization of the nonperiodic trigonometric polynomials and the nonperiodic trigonometric polynomials approximation.That offers the theory support for us to construct some quadrature formulae.The function is approximated by the orthogonal projection on the subspace,which is spanned by the nonperiodic trigonometric polynomials transformed from the Chebyshev polynomials by using the change of variable.For the nonperiodic functions that have large gradients,the result of the nonperiodic trigonometric polynomials approximation is better than the polynomials approximation.Next,the quadrature formulae of the proper integral with Legendre weight and Chebyshev weight are constructed,which the integrand of the proper integral is approximated by the function in the subspace,which is spanned by the nonperiodic trigonometric polynomials.The approximation is realized by interpolation polynomials.The third quadrature formula of the proper integral with Chebyshev weight is constructed,which the integrand is approximated by the orthogonal projection.The three quadrature formulae above are different from the traditional sense.At the same time,the precision of the three quadrature formulae is presented.And then,by using the change of the nonperiodic trigonometric variable,the singular integral with Cauchy kernel is transformed to a singular integral with a new kernel.The equivalence of the definition of the two singular integrals is demonstrated.Two kind of quadrature formulae of the singular integral with the new kernel are constructed,and then,making use of the inverse transform of the variable,the quadrature formulae aretransformed to the quadrature formulae based on the nonperiodic trigonometric polynomials of the singular integral with Cauchy kernel.At the same time,the third kind of quadrature formula is obtained,which the density function is approximated by the orthogonal projection on the subspace,which is spanned by the nonperiodic trigonometric polynomials.The precision of the three quadrature formulae is presented,and their convergence is discussed.At end,the six kinds of quadrature formulae above are emulated with Matlab.According to the error results and analytic images of the numerical examples in the experiments,we know that the density function is nonperiodic and non-uniform,the results of approximation of the quadrature formulae based on the nonperiodic trigonometric polynomials are better than the traditional sense,which accords with our theoretical analysis.
Keywords/Search Tags:Chebyshev polynomials, polynomial approximation, orthogonal polynomials, singular integrals, quadrature formulae
PDF Full Text Request
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