Summary
The purpose of the paper is the study of formulas and methods for numerical integration based on Euler summation formulas. Cubature formulas are developed from multidimensional generalizations. Estimates of the truncation error are given in adaptation to specific properties of the integrand.
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References
Brass, H.: Quadraturverfahren. Göttingen: Vandenhoeck & Ruprecht 1977
Brosowski, B., Kreß, R.: Einführung in die numerische Mathematik I und II. Mannheim, Wien, Zürich: B.I. Wissenschaftsverlag 1975, 1976
Cassels, J.W.S.: An Introduction to the Geometry of Numbers. Berlin, Heidelberg, New York: Springer 1981
Courant, R., Hilbert, D.: Methoden der Mathematischen Physik I, II. Berlin, Heidelberg, New York: Springer 1968
Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration, 2nd ed. New York, San Francisco, London: Academic Press 1984
Engels, H.: Numerical Quadrature and Cubature. London, New York, Toronto, Sydney, San Francisco: Academic Press 1980
Ewald, P.P.: Die Berechnung optischer und elektrostatischer Gitterpotentiale. Ann. Phys.64, 253–287 (1921)
Freeden, W.: Über eine Verallgemeinerung der Hardyschen Identität. Dissertation, Aachen 1975
Freeden, W.: Eine Verallgemeinerung der Hardy-Landauschen Identität. Manuscripta Math.24, 205–216 (1978)
Freeden, W., Reuter, R.: A Class of Multidimensional Periodic Splines. Manuscripta Math.35, 371–386 (1981)
Freeden, W.: Multidimensional Euler Summation Formulas and Numerical Cubature. ISNM57, 77–88 (1982)
Fricker, F.: Einführung in die Gitterpunktlehre. Basel: Birkhäuser 1982
Gauss, C.F.: De nexu inter multitudinem classicum, in quas formae binariae secundi gradus distribuuntur, earumque determinantem. Werke2, 269–291 (1863)
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals Series and Products. New York, London: Academic Press 1965
Hilbert, D.: Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. Leipzig, Berlin: Teubner 1912
Hlawka, E.: Über Integrale auf konvexen Körpern I. Monatshefte für Mathematik54, 1–37 (1950)
Henrici, P.K.: Elements of Numerical Analysis. New York: John Wiley 1964
Kellogg, O.D.: Foundations of Potential Theory. New York: Dover 1954
Lekkerkerker, C.G.: Geometry of Numbers. Amsterdam, London: North Holland 1969
Lyness, J.N.: The Calculation of Trigonometric Fourier Coefficients. J. Comput. Phys.54, 57–73 (1984)
Magnus, W., Oberhettinger, F.: Formeln und Sätze für die speziellen Funktionen, der Mathematischen Physik. Berlin, Göttingen, Heidelberg: Springer 1948
Mikhlin, S.G.: Mathematical Physics, and Advanced Course. Amsterdam, London: North Holland 1970
Mordell, L.J.: Poissons Summation Formula in Several Variables. Cambr. Phil. Soc.25, 412–420 (1928/29)
Müller, C.: Foundations of the Mathematical Theory of Electromagnetic Waves. Berlin, Heidelberg, New York: Springer 1969
Müller, C., Freeden, W.: Multidimensional Euler and Poisson Summation Formulas. Result. Math.3, 33–63 (1980)
Nijboer, B.R.A., de Wette, F.W.: On the Calculation of Lattice Sums. PhysicaXXIII, 309–321 (1957)
Oberhettinger, F.: Fourier Expansions: A Collection of Formulas. New York, London: Academic Press 1973
Stetter, H.J.: Numerical Approximation of Fourier transforms. Numer. Math.8, 235–249 (1966)
Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. New York, Heidelberg, Berlin: Springer 1980
Törnig, W.: Numerische Mathematik für Ingenieure und Physiker. Bd. 2: Eigenwertprobleme und numerische Methoden der Analysis. Berlin, Heidelberg, New York: Springer 1979
Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge: Cambridge University Press 1944
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Freeden, W., Fleck, J. Numerical integration by means of adapted Euler summation formulas. Numer. Math. 51, 37–64 (1987). https://doi.org/10.1007/BF01399694
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DOI: https://doi.org/10.1007/BF01399694