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.
Generalized Functions and Their Properties. Test and Generalized Functions. Differentiation of Generalized Functions. Direct Product of Generalized Functions. The Convolution of Generalized Functions. Tempered Generalized Functions. Integral Transformations of Generalized Functions. The Fourier Transform of Tempered Generalized Functions. Fourier Series of Periodic Generalized Functions. Positive Definite Generalized Functions. The Laplace Transform of Tempered Generalized Functions. The Cauchy Kernel and the Transforms of Cauchy-Bochner and Hilbert. Poisson Kernel and Poisson Transform. Algebras of Holomorphic Functions. Equations in Convolution Algebras. Tauberian Theorems for Generalized Functions. Some Applications in Mathematical Physics. Differential Operators with Constant Coefficients. The Cauchy Problem. Holomorphic Functions with Nonnegative Imaginary Part in T^Oc. Holomorphic Functions with Nonnegative Imaginary Part in T^On. Positive Real Matrix Functions in T^Oc. Linear Passive Systems. Abstract Scattering Operator.