Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Evaluation of Fourier integrals using $ B$-splines


Authors: M. Lax and G. P. Agrawal
Journal: Math. Comp. 39 (1982), 535-548
MSC: Primary 65D30; Secondary 42A15, 65R10
DOI: https://doi.org/10.1090/S0025-5718-1982-0669645-5
Corrigendum: Math. Comp. 43 (1984), 347.
Corrigendum: Math. Comp. 43 (1984), 347.
MathSciNet review: 669645
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Finite Fourier integrals of functions possessing jumps in value, in the first or in the second derivative, are shown to be evaluated more efficiently, and more accurately, using a continuous Fourier transform (CFT) method than the discrete transform method used by the fast Fourier transform (FFT) algorithm. A B-spline fit is made to the input function, and the Fourier transform of the set of B-splines is performed analytically for a possibly nonuniform mesh. Several applications of the CFT method are made to compare its performance with the FFT method. The use of a 256-point FFT yields errors of order $ {10^{ - 2}}$, whereas the same information used by the CFT algorithm yields errors of order $ {10^{ - 7}}$--the machine accuracy available in single precision. Comparable accuracy is obtainable from the FFT over the limited original domain if more than 20,000 points are used.


References [Enhancements On Off] (What's this?)

  • [1] F. Abramovici, "The accurate calculation of Fourier integrals by the fast Fourier transform technique," J. Comput. Phys., v. 11, 1973, pp. 28-37. MR 0413436 (54:1550)
  • [2] M. Abramowitz & I. A. Stegun (Editors), Handbook of Mathematical Functions, Dover, New York, 1965, p. 300.
  • [3] G. P. Agrawal & M. Lax, "Fraunhofer diffraction in the beam approximation from two longitudinally separated slits," J. Opt. Soc. Amer., v. 72, 1982, pp. 164-166.
  • [4] P. J. Davis, Interpolation and Approximation, Blaisdell, New York, 1963, Section 3.7. MR 0157156 (28:393)
  • [5] C. de Boor, "On calculating with B-splines," J. Approx. Theory, v. 6, 1972, pp. 50-62. MR 0338617 (49:3381)
  • [6] C. de Boor, "Package for calculating with B-splines," SIAM J. Numer. Anal., v. 14, 1977, pp. 441-472; C. de Boor, A Practical Guide to Splines, Springer-Verlag, New York, 1978. MR 0428691 (55:1711)
  • [7] B. Einarsson, "Numerical calculation of Fourier integrals with cubic splines," BIT, v. 8, 1968, pp. 279-286; BIT, v. 9, 1969, pp. 183-184 (errata). MR 0239757 (39:1114)
  • [8] B. Einarsson, "Use of Richardson extrapolation for the numerical calculation of Fourier transforms," J. Comput. Phys., v. 21, 1976, pp. 365-370. MR 0416080 (54:4156)
  • [9] L. N. G. Filon, "On a quadrature formula for trigonometric integrals," Proc. Roy. Soc. Edinburgh, v. 49, 1928, pp. 38-47.
  • [10] S. M. Flatte & F. D. Tappert, "Calculation of the effect of internal waves on oceanic sound transmission," J. Acoust. Soc. Amer., v. 58, 1975, pp. 1151-1159.
  • [11] J. Gazdag, "Numerical convective schemes based on accurate computation of space derivatives," J. Comput. Phys., v. 13, 1973, pp. 100-113.
  • [12] R. W. Hockney, "A fast direct solution of Poisson's equation using Fourier analysis," J. Assoc. Comput. Mach., v. 17, 1965, pp. 95-113. MR 0213048 (35:3913)
  • [13] H. Jeffreys & B. S. Jeffreys, Methods of Mathematical Physics, Cambridge Univ. Press, Oxford, 1978, p. 262.
  • [14] M. Lax, G. P. Agrawal & W. H. Louisell, "Continuous Fourier transform spline solution of unstable resonator field distribution," Opt. Lett., v. 9, 1979, pp. 303-305.
  • [15] R. C. le Bail, "Use of Fast Fourier Transforms for solving partial differential equations in physics," J. Comput. Phys., v. 9, 1972, pp. 440-465. MR 0309342 (46:8452)
  • [16] M. J. Marsden & G. D. Taylor, "Numerical evaluation of Fourier integrals," in Numerische Methoden der Approximations Theorie, Band I (Vortragsauszuge einer Tagung, Oberwolfach, 1971), pp. 61-76; Internat. Schriftenreibe zur Numer. Math. Band 16, Birkhauser, Basel, 1972. MR 0386234 (52:7092)
  • [17] J. Marti, "An algorithm recursively computing the exact Fourier coefficients of B-splines with non-equidistant knots," J. Appl. Math. and Phys., v. 29, 1978, pp. 301-305. MR 498770 (81f:65010)
  • [18] R. Piessens & M. Branders, "Computation of oscillating integrals," J. Comput. Appl. Math., v. 1, 1975, pp. 153-164.
  • [19] R. Piessens & A. Haegemans, "Algorithm for the automatic integration of highly oscillatory functions," Computing, v. 13, 1974, pp. 183-193. MR 0455302 (56:13541)
  • [20] I. J. Schoenberg, "Contributions to the problem of approximation of equidistant data by analytic functions," Quart Appl. Math., v. 4, 1946, pp. 45-49, 121-141. MR 0015914 (7:487b)
  • [21] I. J. Schoenberg, Cardinal Spline Interpolation, SIAM, Philadelphia, Pa., 1973, p. 2. MR 0420078 (54:8095)
  • [22] A. E. Siegman & E. A. Sziklas, "Mode calculations in unstable resonators with flowing saturable gain, 2: Fast Fourier transform method," Appl. Optics, v. 14, 1975, pp. 1874-1889.
  • [23] S. D. Silliman, "The numerical evaluation by splines of Fourier transforms," J. Approx. Theory, v. 12, 1974, pp. 32-51. MR 0356556 (50:9026)
  • [24] R. C. Singleton, "An algorithm for computing the mixed radix fast Fourier transform," IEEE Trans. Audio Electroacoust., v. AU-17, 1969, pp. 93-103.
  • [25] F. D. Tappert, Numerical Solutions of the Korteweg-de Vries Equation and its Generalizations by the Split-Step Fourier Method, Lectures in Appl. Math., vol. 15, Amer. Math. Soc., Providence, R. I., 1974, pp. 215-216.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65D30, 42A15, 65R10

Retrieve articles in all journals with MSC: 65D30, 42A15, 65R10


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1982-0669645-5
Keywords: Fourier integral, B-splines, fast Fourier transform, continuous Fourier transform
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society