[PDF] Approximation By Multivariate Singular Integrals eBook

Author : George A. Anastassiou
Publisher : Springer Science & Business Media
Page : 88 pages
File Size : 47,23 MB
Release : 2011-07-25
Category : Mathematics
ISBN : 1461405890

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Approximation by Multivariate Singular Integrals is the first monograph to illustrate the approximation of multivariate singular integrals to the identity-unit operator. The basic approximation properties of the general multivariate singular integral operators is presented quantitatively, particularly special cases such as the multivariate Picard, Gauss-Weierstrass, Poisson-Cauchy and trigonometric singular integral operators are examined thoroughly. This book studies the rate of convergence of these operators to the unit operator as well as the related simultaneous approximation. The last chapter, which includes many examples, presents a related Korovkin type approximation theorem for functions of two variables. Relevant background information and motivation is included in this exposition, and as a result this book can be used as supplementary text for several advanced courses. The results presented apply to many areas of pure and applied mathematics, such a mathematical analysis, probability, statistics and partial differential equations. This book is appropriate for researchers and selected seminars at the graduate level.

Author : George A. Anastassiou
Publisher :
Page : 410 pages
File Size : 13,9 MB
Release : 2012
Category : Approximation theory
ISBN :

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"This monograph is the first one to deal exclusively with the study of the approximation of singular integrals to the identity-unit operator. The authors study quantitatively the basic approximation properties of the general Picard, Gauss-Weierstrass and Poisson-Cauchy singular integral operators over the real line, which are not positive linear operators. In particular the authors study the rate of convergence of these operators to the unit operator, as well as the related simultaneous approximation and the global smoothness preservation property of these operators. The corresponding general approximation theory of general singular integral operators is presented with many applications to the trigonometric singular integral. For the convenience of the reader, the chapters of this book are written in a self-contained style. This monograph is intended for researchers, graduate students working in many areas of pure and applied mathematics, including mathematical analysis, probability, statistics, ordinary and partial differential equations." -- Publisher website.

Author : George A. Anastassiou
Publisher : Springer Science & Business Media
Page : 554 pages
File Size : 37,9 MB
Release : 1999-12-22
Category : Mathematics
ISBN : 9780817641511

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We study in Part I of this monograph the computational aspect of almost all moduli of continuity over wide classes of functions exploiting some of their convexity properties. To our knowledge it is the first time the entire calculus of moduli of smoothness has been included in a book. We then present numerous applications of Approximation Theory, giving exact val ues of errors in explicit forms. The K-functional method is systematically avoided since it produces nonexplicit constants. All other related books so far have allocated very little space to the computational aspect of moduli of smoothness. In Part II, we study/examine the Global Smoothness Preservation Prop erty (GSPP) for almost all known linear approximation operators of ap proximation theory including: trigonometric operators and algebraic in terpolation operators of Lagrange, Hermite-Fejer and Shepard type, also operators of stochastic type, convolution type, wavelet type integral opera tors and singular integral operators, etc. We present also a sufficient general theory for GSPP to hold true. We provide a great variety of applications of GSPP to Approximation Theory and many other fields of mathemat ics such as Functional analysis, and outside of mathematics, fields such as computer-aided geometric design (CAGD). Most of the time GSPP meth ods are optimal. Various moduli of smoothness are intensively involved in Part II. Therefore, methods from Part I can be used to calculate exactly the error of global smoothness preservation. It is the first time in the literature that a book has studied GSPP.

Author : George A. Anastassiou
Publisher : Springer Nature
Page : 523 pages
File Size : 30,35 MB
Release : 2021-04-01
Category : Technology & Engineering
ISBN : 3030714810

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This book includes constructive approximation theory; it presents ordinary and fractional approximations by positive sublinear operators, and high order approximation by multivariate generalized Picard, Gauss–Weierstrass, Poisson–Cauchy and trigonometric singular integrals. Constructive and Computational Fractional Analysis recently is more and more in the center of mathematics because of their great applications in the real world. In this book, all presented is original work by the author given at a very general level to cover a maximum number of cases in various applications. The author applies generalized fractional differentiation techniques of Riemann–Liouville, Caputo and Canavati types and of fractional variable order to various kinds of inequalities such as of Opial, Hardy, Hilbert–Pachpatte and on the spherical shell. He continues with E. R. Love left- and right-side fractional integral inequalities. They follow fractional Landau inequalities, of left and right sides, univariate and multivariate, including ones for Semigroups. These are developed to all possible directions, and right-side multivariate fractional Taylor formulae are proven for the purpose. It continues with several Gronwall fractional inequalities of variable order. This book results are expected to find applications in many areas of pure and applied mathematics. As such this book is suitable for researchers, graduate students and seminars of the above disciplines, also to be in all science and engineering libraries.

Author : E. W. Cheney
Publisher : SIAM
Page : 74 pages
File Size : 33,63 MB
Release : 1986-10-01
Category : Mathematics
ISBN : 0898712076

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This monograph deals with the development of algorithms or the derivation of approximations from linear projections.

Author : George A Anastassiou
Publisher : World Scientific
Page : 347 pages
File Size : 40,84 MB
Release : 2016-09-29
Category : Mathematics
ISBN : 9813145854

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In this monograph, we present the authors' recent work of the last seven years in Approximation Theory. Chapters are self-contained and can be read independently and advanced courses can be taught out of this book. Here our generalized discrete singular operators are of the following types: Picard, Gauss-Weierstrass and Poisson-Cauchy operators. We treat both the unitary and non-unitary, univariate and multivariate cases of these operators, which are not necessarily positive operators. The book's results are expected to find applications in many areas of pure and applied mathematics, and statistics. As such, it is suitable for researchers, graduate students, and seminars of related subjects, and serves well as an invaluable resource for all science libraries.

Author : N. Dyn
Publisher : Cambridge University Press
Page : 300 pages
File Size : 41,52 MB
Release : 2001-05-17
Category : Mathematics
ISBN : 0521800234

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Approximation theory in the multivariate setting has many applications including numerical analysis, wavelet analysis, signal processing, geographic information systems, computer aided geometric design and computer graphics. This advanced introduction to multivariate approximation and related topics consists of nine articles written by leading experts surveying many of the new ideas and their applications. Each article takes the reader to the forefront of research and ends with a comprehensive bibliography.

Author : C. K. Chui
Publisher : Elsevier
Page : 346 pages
File Size : 36,49 MB
Release : 2014-05-10
Category : Mathematics
ISBN : 1483271005

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Topics in Multivariate Approximation contains the proceedings of an international workshop on multivariate approximation held at the University of Chile in Santiago, Chile, on December 15-19, 1986. Leading researchers in the field discussed several problem areas related to multivariate approximation and tackled topics ranging from multivariate splines and fitting of scattered data to tensor approximation methods and multivariate polynomial approximation. Numerical grid generation and finite element methods were also explored, along with constrained interpolation and smoothing. Comprised of 22 chapters, this book first describes the application of Boolean methods of approximation in combination with the theory of right invertible operators to bivariate Fourier expansions. The reader is then introduced to ill-posed problems in multivariate approximation; interpolation of scattered data by radial functions; and shape-preserving surface interpolation. Subsequent chapters focus on approximation by harmonic functions; numerical generation of nested series of general triangular grids; triangulation methods; and inequalities arising from best local approximations in rectangles. A bibliography of multivariate approximation concludes the book. This monograph will be of interest to mathematicians.