Analytical Methods and Special Functions – serie

Even though the theories of operational calculus and integral transforms are centuries old, these topics are constantly developing, due to their use in the fields of mathematics, physics, and electrical and radio engineering. Operational Calculus and Related Topics highlights the classical methods and applications as well as the recent advances in the field.Combining the best features of a textbook and a monograph, this volume presents an introduction to operational calculus, integral transforms, and generalized functions, the backbones of pure and applied mathematics. The text examines both the analytical and algebraic aspects of operational calculus and includes a comprehensive survey of classical results while stressing new developments in the field. Among the historical methods considered are Oliver Heaviside’s algebraic operational calculus and Paul Dirac’s delta function. Other discussions deal with the conditions for the existence of integral transforms, Jan Mikusiński’s theory of convolution quotients, operator functions, and the sequential approach to the theory of generalized functions. Benefits… · Discusses theory and applications of integral transforms · Gives inversion, complex-inversion, and Dirac’s delta distribution formulas, among others · Offers a short survey of actual results of finite integral transforms, in particular convolution theorems Because Operational Calculus and Related Topics provides examples and illustrates the applications to various disciplines, it is an ideal reference for mathematicians, physicists, scientists, engineers, and students.

The theory of distributions is most often presented as L. Schwartz originally presented it: as a theory of the duality of topological vector spaces. Although this is a sound approach, it can be difficult, demanding deep prior knowledge of functional analysis. The more elementary treatments that are available often consider distributions as limits of sequences of functions, but these usually present the theoretical foundations in a form too simplified for practical applications.Distributions, Integral Transforms and Applications offers an approachable introduction to the theory of distributions and integral transforms that uses Schwartz's description of distributions as linear continous forms on topological vector spaces. The authors use the theory of the Lebesgue integral as a fundamental tool in the proofs of many theorems and develop the theory from its beginnings to the point of proving many of the deep, important theorems, such as the Schwartz kernel theorem and the Malgrange-Ehrenpreis theorem. They clearly demonstrate how the theory of distributions can be used in cases such as Fourier analysis, when the methods of classical analysis are insufficient.Accessible to anyone who has completed a course in advanced calculus, this treatment emphasizes the remarkable connections between distributional theory, classical analysis, and the theory of differential equations and leads directly to applications in various branches of mathematics.

This volume presents the general theory of generalized functions, including the Fourier, Laplace, Mellin, Hilbert, Cauchy-Bochner and Poisson integral transforms and operational calculus, with the traditional material augmented by the theory of Fourier series, abelian theorems, and boundary values of helomorphic functions for one and several variables. The author addresses several facets in depth, including convolution theory, convolution algebras and convolution equations in them, homogenous generalized functions, and multiplication of generalized functions. This book will meet the needs of researchers, engineers, and students of applied mathematics, control theory, and the engineering sciences.

The theory of distributions is most often presented as L. Schwartz originally presented it: as a theory of the duality of topological vector spaces. Although this is a sound approach, it can be difficult, demanding deep prior knowledge of functional analysis. The more elementary treatments that are available often consider distributions as limits of sequences of functions, but these usually present the theoretical foundations in a form too simplified for practical applications.Distributions, Integral Transforms and Applications offers an approachable introduction to the theory of distributions and integral transforms that uses Schwartz's description of distributions as linear continous forms on topological vector spaces. The authors use the theory of the Lebesgue integral as a fundamental tool in the proofs of many theorems and develop the theory from its beginnings to the point of proving many of the deep, important theorems, such as the Schwartz kernel theorem and the Malgrange-Ehrenpreis theorem. They clearly demonstrate how the theory of distributions can be used in cases such as Fourier analysis, when the methods of classical analysis are insufficient.Accessible to anyone who has completed a course in advanced calculus, this treatment emphasizes the remarkable connections between distributional theory, classical analysis, and the theory of differential equations and leads directly to applications in various branches of mathematics.

Hypersingular integrals arise as constructions inverse to potential-type operators and are realized by the methods of regularization and finite differences. This volume develops these approaches in a comprehensive treatment of hypersingular integrals and their applications. The author is a renowned expert on the topic. He explains the basics before building more sophisticated ideas, and his discussions include a description of hypersingular integrals as they relate to functional spaces. Hypersingular Integrals and Their Applications also presents recent results and applications that will prove valuable to graduate students and researchers working in mathematical analysis.

This volume presents the general theory of generalized functions, including the Fourier, Laplace, Mellin, Hilbert, Cauchy-Bochner and Poisson integral transforms and operational calculus, with the traditional material augmented by the theory of Fourier series, abelian theorems, and boundary values of helomorphic functions for one and several variables. The author addresses several facets in depth, including convolution theory, convolution algebras and convolution equations in them, homogenous generalized functions, and multiplication of generalized functions. This book will meet the needs of researchers, engineers, and students of applied mathematics, control theory, and the engineering sciences.

Even though the theories of operational calculus and integral transforms are centuries old, these topics are constantly developing, due to their use in the fields of mathematics, physics, and electrical and radio engineering. Operational Calculus and Related Topics highlights the classical methods and applications as well as the recent advances in the field.Combining the best features of a textbook and a monograph, this volume presents an introduction to operational calculus, integral transforms, and generalized functions, the backbones of pure and applied mathematics. The text examines both the analytical and algebraic aspects of operational calculus and includes a comprehensive survey of classical results while stressing new developments in the field. Among the historical methods considered are Oliver Heaviside’s algebraic operational calculus and Paul Dirac’s delta function. Other discussions deal with the conditions for the existence of integral transforms, Jan Mikusiński’s theory of convolution quotients, operator functions, and the sequential approach to the theory of generalized functions. Benefits… · Discusses theory and applications of integral transforms · Gives inversion, complex-inversion, and Dirac’s delta distribution formulas, among others · Offers a short survey of actual results of finite integral transforms, in particular convolution theorems Because Operational Calculus and Related Topics provides examples and illustrates the applications to various disciplines, it is an ideal reference for mathematicians, physicists, scientists, engineers, and students.

Presenting a comprehensive theory of orthogonal polynomials in two real variables and properties of Fourier series in these polynomials, this volume also gives cases of orthogonality over a region and on a contour. The text includes the classification of differential equations which admits orthogonal polynomials as eigenfunctions and several two-dimensional analogies of classical orthogonal polynomials.

Three classes of Fourier transforms are presented: Fourier (Laplace) transforms on the halfline, Fourier transforms of measures with compact support and Fourier transforms of rapidly decreasing functions (on whole line). The focus is on the behaviour of Fourier transforms in the region of analyticity and the distribution of their zeros. Applications of results are presented: approximation by exponentials on the finite interval; behavior of the nonharmonic Fourier series; Müntz-Szasz's problem of approximation by powers on unit interval; approximation by weighted exponentials on whole line.

Presents some important classical and modern results of the series of Faber polynomials and their applications. Interest in this subject has increased rapidly over the last decade, although the presentation of research has, until now, been confined mainly to journal articles. Applications include theory of functions of complex variables, theory of analytic function approximation, and some aspects of numerical analysis.