The Theory of Cubature Formulas

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Springer Science & Business Media, Jun 30, 1997 - Mathematics - 418 pages
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This volume considers various methods for constructing cubature and quadrature formulas of arbitrary degree. These formulas are intended to approximate the calculation of multiple and conventional integrals over a bounded domain of integration. The latter is assumed to have a piecewise-smooth boundary and to be arbitrary in other aspects. Particular emphasis is placed on invariant cubature formulas and those for a cube, a simplex, and other polyhedra. Here, the techniques of functional analysis and partial differential equations are applied to the classical problem of numerical integration, to establish many important and deep analytical properties of cubature formulas. The prerequisites of the theory of many-dimensional discrete function spaces and the theory of finite differences are concisely presented. Special attention is paid to constructing and studying the optimal cubature formulas in Sobolev spaces. As an asymptotically optimal sequence of cubature formulas, a many-dimensional abstraction of the Gregory quadrature is indicated.
Audience: This book is intended for researchers having a basic knowledge of functional analysis who are interested in the applications of modern theoretical methods to numerical mathematics.
 

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Contents

IV
1
V
7
VI
20
VII
23
VIII
33
IX
37
X
43
XI
47
XXX
230
XXXI
235
XXXII
239
XXXIII
245
XXXIV
253
XXXV
257
XXXVI
270
XXXVII
275

XII
64
XIII
74
XIV
75
XV
82
XVI
86
XVII
93
XVIII
103
XIX
117
XX
122
XXI
131
XXII
142
XXIII
148
XXIV
154
XXV
164
XXVI
173
XXVII
200
XXVIII
211
XXIX
221
XXXVIII
280
XXXIX
288
XL
292
XLI
312
XLII
324
XLIII
331
XLIV
337
XLV
344
XLVI
351
XLVII
355
XLVIII
361
XLIX
368
L
375
LI
389
LII
411
LIII
413
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Page 408 - Cubature for the sphere and the discrete spherical harmonic transform', SIAM J.
Page 407 - Les coefficients optimaux des formules d'integration approximative," Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5, No. 3, 455-469 (1978). 260. Sobolev SL, "On the asymptotics of the roots of the Euler polynomials,
Page 404 - Lattice methods for multiple integration," J. Comput. Appl. Math., 12-13, 131-143 (1985). 221. Sloan IH, "Superconvergence," in: Mathematical Concepts and Methods in Science and Engineering, Plenum Press, New York, 1990, pp.
Page 404 - On the approximate integration of functions in Sobolev spaces with a weight,
Page 397 - The Euler-Maclaurin expansion for the simplex,
Page 397 - Best approximate integration formulas,

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