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Asymptotic expansions of multiple integrals of rapidly oscillating functions

Authors: T. Iwaniec and A. Lutoborski
Journal: Math. Comp. 50 (1988), 215-228
MSC: Primary 41A60
MathSciNet review: 917829
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Abstract: Expansions of multiple integrals

$\displaystyle \int_{{a_1}}^{{b_1}} \cdots \int_{{a_n}}^{{b_n}} {w({\sigma _1}{x_1}, \ldots ,{\sigma _n}{x_n})g({x_1}, \ldots ,{x_n})\;d{x_1} \cdots d{x_n},} $

where w is a function on $ {{\mathbf{R}}^n}$ which is $ ({b_k} - {a_k})$-periodic in the kth variable, $ k = 1, \ldots ,n$, and g is smooth, are given in terms of negative powers of the integers $ {\sigma _1}, \ldots ,{\sigma _n}$. Estimates of the remainder term in the expansion are also given.

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

  • [1] M. Abramowitz & I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972.
  • [2] U. Banerjee, L. J. Lardy & A. Lutoborski, "Asymptotic expansions of integrals of certain rapidly oscillating functions," Math. Comp., v. 49, 1987, pp. 243-249. MR 890265 (88e:41063)
  • [3] A. Bensoussan, J. L. Lions & G. Papanicolaou, Asymptotic Analysis of Periodic Structures, North-Holland, Amsterdam, 1978. MR 503330 (82h:35001)
  • [4] P. J. Davis & P. Rabinowitz, Methods of Numerical Integration, 2nd ed., Academic Press, New York, 1984. MR 760629 (86d:65004)
  • [5] J. N. Lyness, "The calculation of Fourier coefficients by the Möbius inversion of the Poisson summation formula, Part I. Functions whose early derivatives are continuous," Math. Comp., v. 24, 1970, pp. 101-135. MR 0260230 (41:4858)
  • [6] H. J. Stetter, "Numerical approximation of Fourier transforms," Numer. Math., v. 8, 1966, pp. 235-249. MR 0198716 (33:6870)
  • [7] G. Talenti, "Best constant in Sobolev inequality," Ann. Mat. Pura Appl., v. 110, 1976, pp. 353-372. MR 0463908 (57:3846)

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Article copyright: © Copyright 1988 American Mathematical Society

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