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On the approximate calculation of double integrals

Author: Moshe Levin
Journal: Math. Comp. 40 (1983), 273-282
MSC: Primary 65D32; Secondary 41A55, 65D30
MathSciNet review: 679445
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Abstract: Cubature formulas are obtained which are optimal or asymptotically optimal on given sets of functions. These formulas consist of line integrals which may be evaluated by optimal or asymptotically optimal quadrature formulas. The advantage of these formulas over the optimal and asymptotically optimal cubature formulas with rectangular-lattices of knots is shown.

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

  • [1] A. H. Stroud, Approximate Calculation of Multiple Integrals, Prentice-Hall, Englewood Cliffs, N. J., 1971. MR 0327006 (48:5348)
  • [2] S. Sobolev, Introduction to the Theory of Cubature Formulas, "Nauka", Moscow, 1974. (Russian) MR 0478560 (57:18037)
  • [3] M. Levin & J. Girshovich, Optimal Quadrature Formulas, Teubner Verlag, Leipzig, 1979. MR 572264 (81j:65051)
  • [4] S. M. Nikolsky, Quadrature Formulas, "Nauka", Moscow, 1980. (Russian)
  • [5] W. J. Gordon, "Distributive lattices and the approximation of multivariate functions," in Approximations with Special Emphasis of Spline Functions (I. J. Schoenberg, ed.), Academic Press, New York and London, 1969, pp. 223-277. MR 0275021 (43:779)
  • [6] F. J. Delvos & H. Posdorf, "N-th order blending," in Constructive Theory of Functions of Several Variables (W. Schempp and K. Zeller, eds.), Springer-Verlag, Berlin, Heidelberg, and New York, 1977, pp. 53-64. MR 0487203 (58:6863)
  • [7] M. Levin & J. Girshovich, "Extremal problems for cubature formulas," Soviet Math. Dokl., v. 18, 1977, pp. 1355-1358. MR 505720 (80a:41029)
  • [8] A. Sard, "Best approximate integration formulas, best approximation formulas," Amer. J. Math., v. 71, 1949, pp. 80-91. MR 0029283 (10:576a)
  • [9] S. M. Nikolsky, "To the question of estimations of approximation with quadrature formulas," Uspekhy Mat. Nauk. v. 2 (36), 1950, pp. 165-177. (Russian)
  • [10] I. J. Schoenberg, "Spline interpolation and best quadrature formulas," Amer. J. Math., v. 70, 1964, pp. 143-148. MR 0157157 (28:394)
  • [11] M. Levin, "Extremal problem for one class of functions," Izv. Akad. Nauk ESSR, Fiz., Mat. i Tekhn. Nauk, v. 12, 1963, pp. 141-145. (Russian) MR 0152151 (27:2131)
  • [12] A. Zensykbaev, "On one property of best quadrature formulas," Mat. Zametki v. 23, 1978, pp. 551-562. (Russian) MR 0493104 (58:12140)
  • [13] V. I. Krylov, Approximate Calculation of Integrals, Macmillan, New York, 1962. MR 0144464 (26:2008)

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Article copyright: © Copyright 1983 American Mathematical Society

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