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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
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
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
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
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
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
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
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
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