Approximation by Multivariate Singular Integrals

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.

First monograph to deal exclusively with the study of the approximation of multivariate singular integrals to the identity-unit operator Quantitatively studies the basic approximation properties of the general multivariate singular integral operators, special cases of which are the multivariate Picard, Gauss-Weierstrass, Poisson-Cauchy and trigonometric singular integral operators, etc. Results presented are expected to find applications in many areas of pure and applied mathematics, such as mathematical analysis, probability, statistics and partial differential equations, etc.

Zusammenfassung

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.

Inhalt
-1. Uniform Approximation by General Multivariate Singular Integral Operators(Introduction, Main Results, Applications, References). -2. Approximation by General Multivariate Singular Integral Operators(Introduction, Main Results, Applications, References). -3. Global Smoothness Preservation and Simultaneous Approximation by Multivariate General Singular Integrals(Introduction, Main Results, Applications, References). -4. Multivariate Voronovskaya Asymptotic Expansions for General Singular Integrals (Introduction, Main Results, Applications, References).-5. Simultaneous Approximation by Multivariate Complex General Singular Integrals (Introduction, Main Results, Applications, References). -6. Approximation of Functions of Two Variables via Almost Convergence of Double Sequences (Introduction and Preliminaries, Korovkin type approximation theorem, Some consequences, References).