## The Theory of Cubature FormulasThis 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 |

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analogous arbitrary Assume asymptotically optimal Banach space belongs bounded calculate coincides compactly-supported compactly-supported functions condition Consequently Consider constant independent construct convolution coordinate origin corresponding cubature formula cube defined degree less Denote derivatives elementary errors English transl equality holds equation error l(x error with regular estimate Euler polynomial expansion extremal function finite follows formula with regular Fourier coefficients Fourier series Fourier transform function p(x fundamental parallelepiped given Hilbert space inequality holds inner product integral interpolation invariant L2 norm lattice of nodes lp space Math matrix H Moreover multi-index Nauk norm Novosibirsk number of nodes obtain operator orthogonal parallelepiped period matrix periodic functions points polyharmonic polyhedron polynomial of degree problem proof of Lemma proof of Theorem properties quadrature formulas regular boundary layer right side satisfies sequence solution space spherical harmonics subspace summand Theory of Cubature valid values vanishes vector weights

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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,