The aim of this book is to provide the reader with a basic understanding of Fourier series, Fourier transforms and Laplace transforms. The book is an expanded and polished version of the authors' notes for a one semester course, for students of mathematics, electrical engineering, physics and computer science. Prerequisites for readers of this book are a basic course in both calculus and linear algebra. Otherwise the material is self-contained with numerous exercises and various examples of applications.

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We show that every analytic filter is generated by a prefilter, every filter is generated by a prefilter, and if is a prefilter then the filter generated by it is also . The last result is unique for the Borel classes, as there is a -complete prefilter P such that the filter generated by it is -complete. Also, no complete coanalytic filter is generated by an analytic prefilter. The proofs use König's infinity lemma, a normal form theorem for monotone analytic sets, and Wadge reductions.

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