## Fourier Analysis on GroupsIn 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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abelian group affine map Appendix Banach algebra Banach space belongs Borel function Borel set bounded linear functional C-set CA(G Cantor set characteristic function closed ideal closed subgroup closure compact neighborhood compact set compact support completes the proof complex homomorphism constant converges coset coset-ring countable defined dense direct sum disjoint dual group element exists F operates Fourier transform Fourier-Stieltjes transform G is compact Haar measure Hausdorff space Helson set Hence G holds implies inequality infinite order isomorphism Kronecker set LCA group G Lemma locally compact maximal ideal metric neighborhood of y0 non-negative norm open set open subgroup piecewise affine polynomial on G proof is complete proof of Theorem proved real number S-set Section semigroup sequence shows Sidon set subgroup of G subspace Suppose G topology translates trigonometric polynomial union x e G y e r