Distribution Theory : Convolution, Fourier Transform, and Laplace Transform.

Preface; 1 Introduction; 2 Definition and First Properties of Distributions; 2.1 Test Functions; 2.2 Distributions; 2.3 Support of a Distribution; 3 Differentiating Distributions; 3.1 Definition and Properties; 3.2 Examples; 3.3 The Distributions x+?-1(??0,-1,-2 ...)*; 3.4 Exercises; 3.5 Green's Formula and Harmonic Functions; 3.6 Exercises; 4 Multiplication and Convergence of Distributions; 4.1 Multiplication with a C8 Function; 4.2 Exercises; 4.3 Convergence in D'; 4.4 Exercises; 5 Distributions with Compact Support; 5.1 Definition and Properties; 5.2 Distributions Supported at the Origin.

5.3 Taylor's Formula for Rn5.4 Structure of a Distribution*; 6 Convolution of Distributions; 6.1 Tensor Product of Distributions; 6.2 Convolution Product of Distributions; 6.3 Associativity of the Convolution Product; 6.4 Exercises; 6.5 Newton Potentials and Harmonic Functions; 6.6 Convolution Equations; 6.7 Symbolic Calculus of Heaviside; 6.8 Volterra Integral Equations of the Second Kind; 6.9 Exercises; 6.10 Systems of Convolution Equations*; 6.11 Exercises; 7 The Fourier Transform; 7.1 Fourier Transform of a Function on R; 7.2 The Inversion Theorem; 7.3 Plancherel's Theorem.

7.4 Differentiability Properties7.5 The Schwartz Space S(R); 7.6 The Space of Tempered Distributions S'(R); 7.7 Structure of a Tempered Distribution*; 7.8 Fourier Transform of a Tempered Distribution; 7.9 Paley Wiener Theorems on R*; 7.10 Exercises; 7.11 Fourier Transform in Rn; 7.12 The Heat or Diffusion Equation in One Dimension; 8 The Laplace Transform; 8.1 Laplace Transform of a Function; 8.2 Laplace Transform of a Distribution; 8.3 Laplace Transform and Convolution; 8.4 Inversion Formula for the Laplace Transform; 9 Summable Distributions*; 9.1 Definition and Main Properties.

9.2 The Iterated Poisson Equation9.3 Proof of the Main Theorem; 9.4 Canonical Extension of a Summable Distribution; 9.5 Rank of a Distribution; 10 Appendix; 10.1 The Banach Steinhaus Theorem; 10.2 The Beta and Gamma Function; 11 Hints to the Exercises; References; Index.

The theory of distributions has numerous applications and is extensively used in mathematics, physics and engineering. There is however relatively little elementary expository literature on distribution theory. This book is intended as an introduction. Starting with the elementary theory of distributions, it proceeds to convolution products of distributions, Fourier and Laplace transforms, tempered distributions, summable distributions and applications. The theory is illustrated by several examples, mostly beginning with the case of the real line and then followed by examples in higher dimensi.

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Includes bibliographical references and index.

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