A guide to distribution theory and Fourier transforms
This book provides a concise exposition of the basic ideas of the theory of distribution and Fourier transforms and its application to partial differential equations. The author clearly presents the ideas, precise statements of theorems, and explanations of ideas behind the proofs. Methods in which techniques are used in applications are illustrated, and many problems are included. The book also introduces several significant recent topics, including pseudo-differential operators, wave front sets, wavelets, and quasicrystals. Background mathematical prerequisites have been kept to a minimum, with only a knowledge of multidimensional calculus and basic complex variables needed to fully understand the concepts in the book.in applied analysis and mathematical physics.
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The Calculus of Distributions
Fourier Transforms of Tempered Distributions
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adjoint identity analytic function argument boundary bounded boundedness Cauchy change of variable compact support condition continuous function converges convolution cotangent curve cut-off function d/dx derivatives differential equation differential operator dimensions distribution of compact distribution theory elliptic operators estimate example expansion fact ffi1 ffi2 ffin finite number Fourier inversion formula Fourier series Fourier transform Gaussian homogeneous of degree hyperbolic implies integrable function linear locally integrable means microlocally smooth multiplication neighborhood nonnegative obtain open set P(ft Paley-Wiener theorems parametrix Plancherel formula Poisson summation formula polynomial positive definite positive distribution probability measure problem proof radial rapidly decreasing restriction Riemann-Lebesgue lemma satisfies sense Show sing supp singularities Sobolev inequality Sobolev spaces solve sphere structure theorem Suppose tempered distribution test functions top-order symbol uncertainty principle vanishes at infinity vector wave equation wave front set wavelet write zero