Introduction to Abstract Harmonic AnalysisThis classic monograph is the work of a prominent contributor to the field of harmonic analysis. Geared toward advanced undergraduates and graduate students, it focuses on methods related to Gelfand's theory of Banach algebra. Prerequisites include a knowledge of the concepts of elementary modern algebra and of metric space topology. The first three chapters feature concise, self-contained treatments of measure theory, general topology, and Banach space theory that will assist students in their grasp of subsequent material. An in-depth exposition of Banach algebra follows, along with examinations of the Haar integral and the deduction of the standard theory of harmonic analysis on locally compact Abelian groups and compact groups. Additional topics include positive definite functions and the generalized Plancherel theorem, the Wiener Tauberian theorem and the Pontriagin duality theorem, representation theory, and the theory of almost periodic functions. |
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Baire functions Banach space bounded linear Cartesian product character closed ideal closed set closure commutative Banach algebra compact set complex numbers contains continuous function converges convolution Corollary coset defined dense direct sum disjoint element equation equivalent exists f Q L1 find finite first fixed follows Fourier transform func function algebra function f given group algebra group G Haar measure Hausdorff space hence Hilbert space homomorphism hull idempotent identified includes inequality integral intersection inverse involution isomorphic kernel L1 fl left invariant Lemma linear functional locally compact group mapping minimal closed multiplication neighborhood non-negative non-zero normed linear space null open set operator orthogonal Plancherel theorem positive definite Proof proved regular maximal ideal representation satisfies scalar self-adjoint semi-simple sequence subset subspace summable Tauberian theorem theory tion uniform norm unique unitary vanishing at infinity weak topology zero