Abstract
We study some cubature formulae for integrals on I 2 = [-1, 1]2 that use two types of information for the integrand: line integrals over either the boundary of I 2 or the coordinate axes, and evaluations at the points of a uniform grid. The error of these cubature formulae is analyzed, in particular the exact Peano constants are found for some classes of functions of low smoothness.
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References
B. D. Bojanov, D. P. Dryanov, W. Haußmann, G. P. Nikolov: Best one-sided L 1 -approximation by blending functions, in: Advances in Multivariate Approximation, W. Haußmann, K. Jet-ter, M. Reimer (eds.), Wiley-VCH, Berlin-Weinheim-New York-Chichester-Brisbane-Singapore-Toronto 1999, pp. 86–106.
H. Braß: Quadraturverfahren, Vandenhoeck & Ruprecht, Göttingen 1977.
H. Braß, K.-J. Förster: On the application of the Peano representation of linear functionals in numerical analysis,in: Recent Progress in Inequalities, G. Milovanovic (ed.), Kluwer, Dordrecht 1998.
P. J. Davis, P. Rabinovitz: Methods of Numerical Integration, Academic Press, New York-San Francisco-London 1975.
K.-J. Förster: Exit criteria and monotonicity in compound quadratures, Numer. Math. 66 (1993), 321–327.
K.-J. Förster: A survey of stopping rules in quadrature based on Peano kernel methods,Suppl. Rend. Circ. Mat. Palermo, Ser. II, 33 (1993), 311–330.
K.-J. Förster, P. Köhler, G. P. Nikolov: Monotonicity and stopping rules for compound Gauss-type quadrature formulae,East J. Approx. 4 (1998), 55–74.
V. I. Krylov: Approximate Calculation of Integrals, Nauka, Moscow 1967 (in Russian). English translation of the first edition: MacMillan, New York-London 1962.
I. Mysovskikh: Interpolatory Cubature Formulae, Nauka, Moscow 1982.
A. Sard: Linear Approximation, Math. Surveys and Monographs 9, Amer. Math. Soc., Providence, R. I., 1982 (Second printing).
G. Schmeißer: Optimale Quadraturformeln mit semidefiniten Kernen, Numer. Math. 20 (1972), 32–53.
A. H. Stroud: Approximate Calculation of Multiple Integrals, Prentice-Hall, Englewood Cliffs, N. J., 1971.
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Gushev, V., Nikolov, G. (2001). Some Cubature Formulae Using Mixed Type Data. In: Haussmann, W., Jetter, K., Reimer, M. (eds) Recent Progress in Multivariate Approximation. ISNM International Series of Numerical Mathematics, vol 137. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8272-9_13
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DOI: https://doi.org/10.1007/978-3-0348-8272-9_13
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9498-2
Online ISBN: 978-3-0348-8272-9
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