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 and H. Posdorf,*𝑛-th order blending*, Constructive theory of functions of several variables (Proc. Conf., Math. Res. Inst., Oberwolfach, 1976) Springer, Berlin, 1977, pp. 53–64. Lecture Notes in Math., Vol. 571. MR**0487203****[2]**William J. Gordon,*Distributive lattices and the approximation of multivariate functions.*, Approximations with Special Emphasis on Spline Functions (Proc. Sympos. Univ. of Wisconsin, Madison, Wis., 1969) Academic Press, New York, 1969, pp. 223–277. MR**0275021****[3]**M. I. Levin,*On approximation in 𝐿₂ and optimal cubature formulae*, Fourier analysis and approximation theory (Proc. Colloq., Budapest, 1976), Vol. II, Colloq. Math. Soc. János Bolyai, vol. 19, North-Holland, Amsterdam-New York, 1978, pp. 495–501. MR**540326****[4]**Meishe Levin and Jury Girshovich,*Optimal quadrature formulas*, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1979. With German, French and Russian summaries; Teubner-Texte zur Mathematik. [Teubner Texts on Mathematics]. MR**572264****[5]**J. Girshovich and M. Levin,*Extremal problems for cubature formulas*, Eesti NSV Tead. Akad. Toimetised Füüs.-Mat.**27**(1978), no. 2, 151–158 (English, with Russian and Estonian summaries). MR**505720**

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

Article copyright:
© Copyright 1982
American Mathematical Society