Indefinite integration of oscillatory functions

Paweł Keller

Applicationes Mathematicae (1998)

  • Volume: 25, Issue: 3, page 301-311
  • ISSN: 1233-7234

Abstract

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A simple and fast algorithm is presented for evaluating the indefinite integral of an oscillatory function ∫ x y i f ( t ) e i ω t d t , -1 ≤ x < y ≤ 1, ω ≠ 0, where the Chebyshev series expansion of the function f is known. The final solution, expressed as a finite Chebyshev series, is obtained by solving a second-order linear difference equation. Because of the nature of the equation special algorithms have to be used to find a satisfactory approximation to the integral.

How to cite

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Keller, Paweł. "Indefinite integration of oscillatory functions." Applicationes Mathematicae 25.3 (1998): 301-311. <http://eudml.org/doc/219205>.

@article{Keller1998,
abstract = {A simple and fast algorithm is presented for evaluating the indefinite integral of an oscillatory function $\int _x^yi f(t) e^\{iωt\} dt$, -1 ≤ x < y ≤ 1, ω ≠ 0, where the Chebyshev series expansion of the function f is known. The final solution, expressed as a finite Chebyshev series, is obtained by solving a second-order linear difference equation. Because of the nature of the equation special algorithms have to be used to find a satisfactory approximation to the integral.},
author = {Keller, Paweł},
journal = {Applicationes Mathematicae},
keywords = {indefinite integration; second-order linear difference equation; oscillatory function; trigonometric approximation; algorithm; Chebyshev series},
language = {eng},
number = {3},
pages = {301-311},
title = {Indefinite integration of oscillatory functions},
url = {http://eudml.org/doc/219205},
volume = {25},
year = {1998},
}

TY - JOUR
AU - Keller, Paweł
TI - Indefinite integration of oscillatory functions
JO - Applicationes Mathematicae
PY - 1998
VL - 25
IS - 3
SP - 301
EP - 311
AB - A simple and fast algorithm is presented for evaluating the indefinite integral of an oscillatory function $\int _x^yi f(t) e^{iωt} dt$, -1 ≤ x < y ≤ 1, ω ≠ 0, where the Chebyshev series expansion of the function f is known. The final solution, expressed as a finite Chebyshev series, is obtained by solving a second-order linear difference equation. Because of the nature of the equation special algorithms have to be used to find a satisfactory approximation to the integral.
LA - eng
KW - indefinite integration; second-order linear difference equation; oscillatory function; trigonometric approximation; algorithm; Chebyshev series
UR - http://eudml.org/doc/219205
ER -

References

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  1. [1] W. Gautschi, Computational aspects of three-term recurrence relations, SIAM Rev. 9 (1967), 24-82. Zbl0168.15004
  2. [2] W. M. Gentleman, Implementing Clenshaw-Curtis quadrature. II. Computing the cosine transformation, Comm. ACM 15 (1972), 343-346. Zbl0234.65024
  3. [3] T. Hasegawa and T. Torii, Indefinite integration of oscillatory functions by the Chebyshev series expansion, J. Comput. Appl. Math. 17 (1987), 21-29. Zbl0613.65145
  4. [4] T. Hasegawa and T. Torii, Application of a modified FFT to product type integration, ibid. 38 (1991), 157-168. Zbl0754.65023
  5. [5] T. Hasegawa and T. Torii, An algorithm for nondominant solutions of linear second-order inhomogeneous difference equations, Math. Comp. 64 (1995), 1199-1204. Zbl0831.65140
  6. [6] T. Hasegawa, T. Torii and H. Sugiura, An algorithm based on the FFT for a generalized Chebyshev interpolation, ibid. 54 (1990), 195-210. Zbl0685.65003
  7. [7] F. W. J. Olver, Numerical solution of second-order linear difference equations, J. Res. Nat. Bur. Standards 71 (B) (1967), 111-129. Zbl0171.36601
  8. [8] S. Paszkowski, Numerical Applications of Chebyshev Polynomials and Chebyshev Series, PWN, Warszawa, 1975 (in Polish). Zbl0423.65012

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