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This textbook is an application-oriented introduction to the theory of distributions, a powerful tool used in mathematical analysis. The treatment emphasizes applications that relate distributions to linear partial differential equations and Fourier analysis problems found in mechanics, optics, quantum mechanics, quantum field theory, and signal analysis. Throughout the book, methods are developed to deal with formal calculations involving functions, series, and integrals that cannot be mathematically justified within the classical framework. Key features: • Many examples, exercises, hints, and solutions guide the reader throughout the text. • Includes an introduction to distributions, differentiation, convergence, convolution, the Fourier transform, and spaces of distributions having special properties. • Original proofs, which may be difficult to locate elsewhere, are given for many well-known results. • The Fourier transform is transparently treated and applied to provide a new proof of the Kernel Theorem, which in turn is used to efficiently derive numerous important results. • The systematic use of pullback and pushforward introduces concise notation. • Emphasizes the role of symmetry in obtaining short arguments and investigates distributions that are invariant under the actions of various groups of transformations. Distributions: Theory and Applications is aimed at advanced undergraduates and graduate students in mathematics, theoretical physics, and engineering, who will find this textbook a welcome introduction to the subject, requiring only a minimal mathematical background. The work may also serve as an excellent self-study guide for researchers who use distributions in various fields.

Electronic books. -- local. --- Theory of distributions (Functional analysis). --- Theory of distributions (Functional analysis) --- Fourier transformations --- Civil & Environmental Engineering --- Mathematics --- Physical Sciences & Mathematics --- Engineering & Applied Sciences --- Operations Research --- Calculus --- Distribution (Functional analysis) --- Distributions, Theory of (Functional analysis) --- Functions, Generalized --- Generalized functions --- Mathematics. --- Approximation theory. --- Fourier analysis. --- Functional analysis. --- Differential equations. --- Partial differential equations. --- Applied mathematics. --- Engineering mathematics. --- Functional Analysis. --- Approximations and Expansions. --- Applications of Mathematics. --- Partial Differential Equations. --- Fourier Analysis. --- Ordinary Differential Equations. --- Engineering --- Engineering analysis --- Mathematical analysis --- Partial differential equations --- 517.91 Differential equations --- Differential equations --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Analysis, Fourier --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Math --- Science --- Differential equations, partial. --- Differential Equations. --- Differential equations, Partial.

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The rich subject matter in this book brings in mathematics from different domains, especially from the theory of elliptic surfaces and dynamics.The material comes from the author's insights and understanding of a birational transformation of the plane derived from a discrete sine-Gordon equation, posing the question of determining the behavior of the discrete dynamical system defined by the iterates of the map. The aim of this book is to give a complete treatment not only of the basic facts about QRT maps, but also the background theory on which these maps are based. Readers with a good knowledge of algebraic geometry will be interested in Kodaira's theory of elliptic surfaces and the collection of nontrivial applications presented here. While prerequisites for some readers will demand their knowledge of quite a bit of algebraic- and complex analytic geometry, different categories of readers will be able to become familiar with any selected interest in the book without having to make an extensive journey through the literature.

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• Reference Manager
• EndNote
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This textbook is an application-oriented introduction to the theory of distributions, a powerful tool used in mathematical analysis. The treatment emphasizes applications that relate distributions to linear partial differential equations and Fourier analysis problems found in mechanics, optics, quantum mechanics, quantum field theory, and signal analysis. Throughout the book, methods are developed to deal with formal calculations involving functions, series, and integrals that cannot be mathematically justified within the classical framework. Key features: ¢ Many examples, exercises, hints, and solutions guide the reader throughout the text. ¢ Includes an introduction to distributions, differentiation, convergence, convolution, the Fourier transform, and spaces of distributions having special properties. ¢ Original proofs, which may be difficult to locate elsewhere, are given for many well-known results. ¢ The Fourier transform is transparently treated and applied to provide a new proof of the Kernel Theorem, which in turn is used to efficiently derive numerous important results. ¢ The systematic use of pullback and pushforward introduces concise notation. ¢ Emphasizes the role of symmetry in obtaining short arguments and investigates distributions that are invariant under the actions of various groups of transformations. Distributions: Theory and Applications is aimed at advanced undergraduates and graduate students in mathematics, theoretical physics, and engineering, who will find this textbook a welcome introduction to the subject, requiring only a minimal mathematical background. The work may also serve as an excellent self-study guide for researchers who use distributions in various fields.

Measure theory. Mathematical integration