Essays in Commutative 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.- 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.- 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 $$/tilde{A}$$(E).- 13 Unsolved Problems.- 13.1 Dichotomy.- 13.2 Finite Sets.- 13.3 Isomorphisms between Quotients of A(G).- 13.4 The Rearrangements Question of N. N. Lusin.- 13.5 Continuity of Linear Operators on L1(R).- 13.6 p-Helson Sets.- 13.7 Questions of Atzmon on Translation Invariant Subspaces of LP and Co.- 13.8 Questions on Subsets E of the Integer Group.- 13.9 Questions on Sets of Synthesis.- 13.10 Characterizing Sidon Sets in Certain Groups.- 13.11 Subalgebras of L1.- 13.12 ?(p)-Sets and Multipliers.- 13.13 Identifying the Maximal Ideal Spaces of Certain L-Subalgebras of M(G).- 13.14 A Question of Katznelson on Measures with Real Spectra.- 13.15 The Support Group ofa Tame Measure.- 13.16 The Šilov Boundary of M(G).- 13.17 Taylor’s Theorems.- 13.18 Two Factorization Questions.- 13.19 Questions about Tensor Algebras.- 13.20 Other Question Lists.- A.1. Riesz Products in Brief.- A.3. A Proof That Singletons Obey Synthesis, and How.- A.4. S. Bernstein’s Inequality.- A.S. Triangles and Trapezoids.- A.6. Convolution and Relative Absolute Continuity.- A.7. An Extension Theorem for Fourier-Stieltjes Transforms.- References.