## Abstract Harmonic Analysis, Volume 2This book is a continuation of vol. I (Grundlehren vol. 115, also available in softcover), and contains a detailed treatment of some important parts of harmonic analysis on compact and locally compact abelian groups. From the reviews: "This work aims at giving a monographic presentation of abstract harmonic analysis, far more complete and comprehensive than any book already existing on the subject...in connection with every problem treated the book offers a many-sided outlook and leads up to most modern developments. Carefull attention is also given to the history of the subject, and there is an extensive bibliography...the reviewer believes that for many years to come this will remain the classical presentation of abstract harmonic analysis." Publicationes Mathematicae |

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apply approximate unit arbitrary Banach algebra belongs Borel measurable character group closed ideal closed subset compact Abelian group compact subset complex numbers conjugation consider contains continuous function converges convolution Corollary cosets countable defined definition denote dense direct sum dual object eigenvalues element equal equivalent finite-dimensional follows Fourier transform function f functions on G group G Haar measure Hence Hilbert space holds homomorphism identity implies inequality infinite irreducible unitary representations isometry Lemma Let f Let G linear space linear subspace locally compact Abelian locally compact group mapping Math matrix nonnegative nonvoid nonzero norm normal subgroup notation obtain obvious one-to-one orthogonal orthonormal basis Plainly PLANCHEREL'S theorem positive integer positive number positive-definite functions Proof prove RAAA representation of G representation space sequence shows Sidon set space H spectral set spectral synthesis subgroup of G subset of G Suppose tensor product theorem trivial two-sided ideal unitary representation write zero

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Page 735 - On the expression of a given function as the Fourier-Stieltjes transform of a distribution function whose spectrum is confined to a given set, J. London Math. Soc., 12, 253-257 (1937).

Page 740 - Lp Fourier Transforms on Locally Compact Unimodular Groups," Trans. Amer. Math. Soc. 89 (1958), 519.

Page 749 - John [1], Ideals in a class of commutative Banach algebras, Duke Math. J., 20, 273-278 (1953). — [2], Maximal subalgebras of group-algebras, Proc.