Sets of Fourier coefficients using numerical quadrature

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One approach to the calculation of Fourier trigonometric coefficients f(r) of a given function f(x) is to apply the trapezoidal quadrature rule to the integral representation f(r) = {line_integral}{sub 0}{sup 1} f(x)e{sup -2{pi}irx}dx. Some of the difficulties in this approach are discussed. A possible way of overcoming many of these is by means of a subtraction function. Thus, one sets f(x) = h{sub p-1}(x) + g{sub p}(x), where h{sub -1}(x) is an algebraic polynomial of degree p-1, specified in such a way that the Fourier series of g{sub p}(x) converges more rapidly than that of f(x). To obtain the Fourier ... continued below

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24 pages

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Lyness, J. N. October 10, 2001.

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One approach to the calculation of Fourier trigonometric coefficients f(r) of a given function f(x) is to apply the trapezoidal quadrature rule to the integral representation f(r) = {line_integral}{sub 0}{sup 1} f(x)e{sup -2{pi}irx}dx. Some of the difficulties in this approach are discussed. A possible way of overcoming many of these is by means of a subtraction function. Thus, one sets f(x) = h{sub p-1}(x) + g{sub p}(x), where h{sub -1}(x) is an algebraic polynomial of degree p-1, specified in such a way that the Fourier series of g{sub p}(x) converges more rapidly than that of f(x). To obtain the Fourier coefficients of f(x), one uses an analytic expression for those of h{sub p-1}(x) and numerical quadrature to approximately those of g{sub p}(x).

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24 pages

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  • International Conference on Numerical Algorithms, Marrakesh (MA), 10/01/2001--10/05/2001

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  • Report No.: ANL/MCS/CP-105987
  • Grant Number: W-31-109-ENG-38
  • Office of Scientific & Technical Information Report Number: 792107
  • Archival Resource Key: ark:/67531/metadc741052

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  • October 10, 2001

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  • Oct. 19, 2015, 7:39 p.m.

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  • March 30, 2016, 4:34 p.m.

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Lyness, J. N. Sets of Fourier coefficients using numerical quadrature, article, October 10, 2001; Illinois. (digital.library.unt.edu/ark:/67531/metadc741052/: accessed January 19, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.