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Essays in Commutative Harmonic Analysis

This book considers various spaces and algebras made up of functions, measures, and other objects-situated always on one or another locally compact abelian group, and studied in the light of the Fourier transform. The emphasis is on the objects themselves, and on the structure-in-detail of the space...

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Bibliographic Details
Main Authors: Graham, C. C., McGehee, O. C. (Author)
Format: eBook
Language:English
Published: New York, NY Springer New York 1979, 1979
Edition:1st ed. 1979
Series:Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics
Subjects:
Mathematical Analysis
Analysis
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 7.6 Small Subsets of Z That Are Dense in bZ
  • 7.7 Non-trivial Idempotents in B(E) for E ? Z
  • 8 The Šilov Boundary, Symmetric Ideals, and Gleason Parts of ?M(G)
  • 8.1 Introduction
  • 8.2 The Šilov Boundary of M(G)
  • 8.3 Some Translation Theorems
  • 8.4 Non-symmetric Maximal Ideals in M(G)
  • 8.5 Point Derivations and Strong Boundary Points for M(G)
  • 8.6 Gleason Parts for Convolution Measure Algebras
  • 9 The Wiener-Lévy Theorem and Some of Its Converses
  • 9.1 Introduction
  • 9.2 Proof of the Wiener-Lévy Theorem and Marcinkiewicz’s Theorem
  • 9.3 Converses to the Wiener-Lévy Theorem
  • 9.4 Functions Operating in B(?)
  • 9.5 Functions Operating in Bo(?)
  • 9.6 Functions Operating on Norm One Positive-Definite Functions
  • 10 The Multiplier Algebras Mp(?), and the Theorem of Zafran
  • 10.1 Introduction
  • 10.2 The Basic Theory of the Algebras Mp(?)
  • 10.3 Zafran’s Theorem about the Algebra Mpo(Z)
  • 11 Tensor Algebras and Harmonic Analysis
  • 1 The Behavior of Transforms
  • 1.1 Introduction
  • 1.2 The Idempotents in the Measure Algebra
  • 1.3 Paul Cohen’s Theorem on the Norms of Idempotents
  • 1.4 Transforms of Continuous Measures
  • 1.5 The Two Sides of a Fourier Transform
  • 1.6 Transforms of Rudin-Shapiro Type
  • 1.7 A Separable Banach Space That Has No Basis
  • 1.8 Restrictions of Fourier-Stieltjes Transforms to Sets of Positive Haar Measure
  • 2 A Proof That the Union of Two Helson Sets Is a Helson Set
  • 2.1 Introduction
  • 2.2 Definition of the Functions ?N
  • 2.3 Transfering the Problem from One Group to Another
  • 2.4 Proof of Theorem 2.1.3
  • 2.5 Remarks and Credits
  • 3 Harmonic Synthesis
  • 3.1 Introduction
  • 3.2 When Synthesis Succeeds
  • 3.3 When Synthesis Fails
  • 1.2 Transducers for long term hemodynamic signals monitoring
  • 4 Sets of Uniqueness, Sets of Multiplicity
  • 4.1 Introduction
  • 4.2 The Support of a Pseudomeasure
  • 4.3 The Weak * Closure of I(E)
  • 4.4 An M1-Set That Is Not an Mo-Set
  • 4.5 Results about Helson Sets and Kronecker Sets
  • 4.6 M-Sets Whose Helson Constant Is One
  • 4.7 Independent Mo-Sets
  • 5 A Brief Introduction to Convolution Measure Algebras
  • 5.1 Elementary Properties
  • 5.2 L-Subalgebras and L-Ideals
  • 5.3 Critical Point Theory and a Proof of the Idempotent Theorem
  • 5.4 A Guide for Further Study
  • 6 Independent Power Measures
  • 6.1 Introduction and Initial Results
  • 6.2 Measures on Algebraically Scattered Sets
  • 6.3 Measures on Dissociate Sets
  • 6.4 Infinite Product Measures
  • 6.5 General Results on Infinite Convolutions
  • 6.6 Bernoulli Convolutions
  • 6.7 Coin Tossings
  • 6.8 Mo(G) Contains Tame i.p. Measures
  • 7 Riesz Products
  • 7.1 Introduction and Initial Results
  • 7.2 Orthogonality Relations for Riesz Products
  • 7.3 Most Riesz Products Are Tame
  • 7.4 A Singular Measure in Mo(G)That Is Equivalent to Its Square
  • 7.5 A Multiplier Theorem and the Support of Singular Fourier-Stieltjes Transforms
  • 11.1 Introduction and Initial Results
  • 11.2 Transfer Methods: Harmonic Synthesis and Non-finitely Generated Ideals in L1(G)
  • 11.3 Sets of Analyticity and Tensor Algebras
  • 11.4 Infinite Tensor Products and the Saucer Principle
  • 11.5 Continuity Conditions for Membership in V(T,T)
  • 11.6 Sidon Constants of Finite Sets for Tensor Algebras and Group Algebras
  • 11.7 Automorphisms of Tensor Algebras
  • 11.8 V-Sidon and V-Interpolation Sets
  • 11.9 Tilde Tensor Algebras
  • 12 Tilde Algebras
  • 12.1 Introduction
  • 12.2 Subsets of Discrete Groups
  • 12.3 The Connection with Synthesis
  • 12.4 Sigtuna Sets
  • 12.5 An Example in which A(E) Is a Dense Proper Subspace of

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