## Classical Fourier transformsThis book gives a thorough introduction on classical Fourier transforms in a compact and self-contained form. Chapter I is devoted to the L1-theory: basic properties are proved as well as the Poisson summation formula, the central limit theorem and Wiener's general tauberian theorem. As an illustraiton of a Fourier transformation of a function "not" belonging to L1 (-, ) an integral due to Ramanujan is given. Chapter II is devoted to the L2-theory, including Plancherel's theorem, Heisenberg's inequality, the Paley-Wiener theorem, Hardy's interpolation formula and two inequalities due to Bernstein. Chapter III deals with Fourier-Stieltjes transforms. After the basic properties are explained, distribution functions, positive-definite functions and the uniqueness theorem of Offord are treated. The book is intended for undergraduate students and requires of them basic knowledge in real and complex analysis. |

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### Contents

Fourier transforms on 1 Basic properties and examples | 1 |

2 The Lialgebra | 16 |

3 Differentiability properties | 18 |

Copyright | |

29 other sections not shown

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Bochner and Chandrasekharan bounded function bounded variation Cauchy principal value Ch.I characteristic function complex-valued continuity of f continuous function convolution Corollary cosh defined denotes distribution function dominated convergence e"iaxda entire function everywhere Example exists a function exists a sequence f is bounded f is continuous f(x+t finite interval fn(x following Theorem formula Fourier coefficient Fourier series Fourier transform Fubini's theorem function f functions belonging given gx(t implies inequality infinitely differentiable functions ip(u ip(x ipn(x kernel L1-norm Lebesgue Lebesgue's theorem Lemma Let f linear Math measurable function Mellin transform non-decreasing obtain OO OO OO orthonormal Paley-Wiener theorem Plancherel's theorem points of continuity Proof of Theorem prove R+oo real number Remarks Riesz-Fischer theorem Schwartz's space summability tends to zero Theorem 11 Theorem 9 theorem on dominated transform of f trigonometric uniformly vanishes variable Wiener x)dx Zygmund