Fourier Analysis on Groups
In the late 1950s, many of the more refined aspects of Fourier analysis were transferred from their original settings (the unit circle, the integers, the real line) to arbitrary locally compact abelian (LCA) groups. Rudin's book, published in 1962, was the first to give a systematic account of these developments and has come to be regarded as a classic in the field. The basic facts concerning Fourier analysis and the structure of LCA groups are proved in the opening chapters, in order to make the treatment relatively self-contained.
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Chapter 2 The Structure of Locally Compact Abelian Groups
Chapter 3 Idempotent Measures
Chapter 4 Homomorphisms of Group Algebras
Chapter 5 Measures and Fourier Transforms on Thin Sets
Chapter 6 Functions of Fourier Transforms
Chapter 7 Closed Ideals in L1 G
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abelian group affine map Appendix Banach algebra Banach space belongs Borel function Borel set bounded linear functional Cantor set characteristic function closed ideal closed subgroup closure compact abelian group compact neighborhood compact set compact support completes the proof complex homomorphism complex numbers constant converges coset coset-ring countable dense direct sum disjoint dual group element exists F operates fe L1(G finite order Fourier transform Fourier-Stieltjes transforms function f G is compact Haar measure Hausdorff space Helson set Hence F holds implies inequality infinite isomorphism Kronecker set L1 G LCA group G Lemma locally compact Math maximal ideal metric non-negative norm open set open subgroup piecewise affine polynomial on G proof is complete proof of Theorem proved real number S-set Section sequence shows Sidon set subgroup of G subspace Suppose G topology translates trigonometric polynomial union