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BL
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| Record Number
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860331
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| Main Entry
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Liflyand, Elijah
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| Title & Author
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Functions of bounded variation and their Fourier transforms /\ Elijah Liflyand.
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| Publication Statement
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Cham, Switzerland :: Birkhäuser,, 2019.
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| Series Statement
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Applied and numerical harmonic analysis,
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| Page. NO
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1 online resource (xxiv, 194 pages)
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| ISBN
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3030044297
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: 9783030044299
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3030044289
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9783030044282
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| Bibliographies/Indexes
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Includes bibliographical references and index.
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| Contents
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Intro; ANHA Series Preface; Contents; Foreword; Introduction; Transforms; Functions; Basic theorem; Advancement; Tools; Sharp versions; Multivariate case; Picture and details; Part I: One-dimensional Case; Chapter 1 A toolkit; 1.1 Functions of bounded variation; 1.2 Fourier transform; 1.3 Hilbert transform; 1.3.1 Fourier transform weakly generates Hilbert transform; 1.3.2 Existence almost everywhere; 1.3.3 Integrability of the Hilbert transform; 1.3.4 Special cases of the Hilbert transform; 1.3.5 Conditions for the integrability of the Hilbert transform; 1.4 Hardy spaces and subspaces
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1.4.1 Atomic characterization1.4.2 Molecular characterization; 1.4.3 Integrability spaces; 1.4.4 A Paley-Wiener theorem; 1.5 Balance integral operator; Chapter 2 Functions with derivative in a Hardy space; 2.1 First steps; 2.2 Derivative in H1 o (R+); 2.3 Derivative in H1 e (R+); 2.4 Derivative in a subspace of H1 o (R+) or H1 e (R+); 2.5 Functions on the whole axis; 2.6 Absolute continuity, integrability of the Fourier transform and a Hardy-Littlewood theorem; Chapter 3 Integrability spaces: wide, wider and widest; 3.1 Widest integrability spaces; 3.2 The sine Fourier transform
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3.3 Intermediate spaces3.3.1 Embeddings; 3.3.2 A counterexample; 3.3.3 Intermediate spaces between H1 0 and H1Q; 3.4 Fourier-Hardy type inequalities; Chapter 4 Sharper results; 4.1 The Fourier transform of a convex function; 4.1.1 General representation of the Fourier transform; 4.1.2 Convex functions; 4.2 Generalizations of Theorems 2.8 and 2.20; 4.3 The sine Fourier transform revisited; 4.4 A Szökefalvi-Nagy type theorem; Part II: Multi-dimensional Case; Chapter 5 A toolkit for several dimensions; 5.1 Indicator notation; 5.2 Multidimensional variations; 5.2.1 Vitali's and Hardy's variations
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5.2.2 Tonelli's variation5.3 Fourier transform; 5.3.1 L1-theroy; 5.3.2 L2- and Lp-theory; 5.3.3 Poisson summation formula; 5.4 Multidimensional spaces; 5.5 Absolute continuity; 5.6 Integration by parts; Chapter 6 Integrability of the Fourier transforms; 6.1 Functions with derivatives in the Hardy type spaces; 6.2 Absolute continuity, integrability of the Fourier transform and a Hardy-Littlewood theorem; 6.2.1 Commutativity; 6.2.2 Conditions for absolute continuity; 6.2.3 Hardy-Littlewood type theorems; Chapter 7 Sharp results; 7.1 Convexity type results; 7.1.1 Functions of convex type
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7.2 Equalities7.2.1 (Even) more general cases; 7.2.2 The most general situation; 7.3 Szökefalvi-Nagy type theorem; 7.3.1 Auxiliary results; 7.3.2 Proof of Theorem 7.17; Chapter 8 Bounded variation and sampling; 8.1 Bridge; 8.1.1 One-dimensional bridge; 8.1.2 Temporary bridge; 8.1.3 Stable bridge; 8.2 On the Poisson summation formula; 8.2.1 Background; 8.2.2 A version of the Poisson summation formula; 8.2.3 Concluding remarks and an example; Chapter 9 Multidimensional case: radial functions; 9.1 Fractional derivative and MV Classes; 9.2 Necessary conditions; 9.3 Radial extensions; Afterword
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| Abstract
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Functions of bounded variation represent an important class of functions. Studying their Fourier transforms is a valuable means of revealing their analytic properties. Moreover, it brings to light new interrelations between these functions and the real Hardy space and, correspondingly, between the Fourier transform and the Hilbert transform. This book is divided into two major parts, the first of which addresses several aspects of the behavior of the Fourier transform of a function of bounded variation in dimension one. In turn, the second part examines the Fourier transforms of multivariate functions with bounded Hardy variation. The results obtained are subsequently applicable to problems in approximation theory, summability of the Fourier series and integrability of trigonometric series.
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| Subject
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Fourier transformations.
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| Subject
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Functions of bounded variation.
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| Subject
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Fourier transformations.
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| Subject
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Functions of bounded variation.
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| Dewey Classification
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515/.8
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| LC Classification
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QA331.5
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