On the evaluation of double integrals

Author:
Moshe Levin

Journal:
Math. Comp. **39** (1982), 173-177

MSC:
Primary 65D32

DOI:
https://doi.org/10.1090/S0025-5718-1982-0658221-6

MathSciNet review:
658221

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Abstract: A cubature formula consisting of line integrals which is optimal on a set of functions satisfying given boundary conditions is obtained. The line integrals of this formula may be evaluated by optimal quadrature formulas. The advantage of this formula over the optimal cubature formula with a rectangular lattice of knots is shown. This approach to optimal cubatures was stimulated by the idea of blending [1], [2].

**[1]**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, New York, 1977, pp. 53-64. MR**0487203 (58:6863)****[2]**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, London, 1969, pp. 223-277. MR**0275021 (43:779)****[3]**M. Levin, "On approximation in and optimal cubature formulae," in*Fourier Analysis and Approximation Theory*, Mathematica Societatis János Bolyai, Budapest, 1976, pp. 495-501. MR**540326 (81k:41019)****[4]**M. Levin & J. Girshovich,*Optimal Quadrature Formulas*, Teubner-Verlag, Leipzig, 1979. MR**572264 (81j:65051)****[5]**M. Levin & J. Girshovich, "Extremal problems for cubature formulas,"*Soviet Math. Dokl.*, v. 18, 1977, pp. 1355-1358. MR**505720 (80a:41029)**

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DOI:
https://doi.org/10.1090/S0025-5718-1982-0658221-6

Article copyright:
© Copyright 1982
American Mathematical Society