Essays in Commutative Harmonic Analysis: (238 Grundlehren der mathematischen Wissenschaften)

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 spaces and algebras. A mathematician needs to know only a little about Fourier analysis on the commutative groups, and then may go many ways within the large subject of harmonic analysis-into the beautiful theory of Lie group representations, for example. But this book represents the tendency to linger on the line, and the other abelian groups, and to keep asking questions about the structures thereupon. That tendency, pursued since the early days of analysis, has defined a field of study that can boast of some impressive results, and in which there still remain unanswered questions of compelling interest. We were influenced early in our careers by the mathematicians Jean-Pierre Kahane, Yitzhak Katznelson, Paul Malliavin, Yves Meyer, Joseph Taylor, and Nicholas Varopoulos. They are among the many who have made the field a productive meeting ground of probabilistic methods, number theory, diophantine approximation, and functional analysis. Since the academic year 1967-1968, when we were visitors in Paris and Orsay, the field has continued to see interesting developments. Let us name a few. Sam Drury and Nicholas Varopoulos solved the union problem for Helson sets, by proving a remarkable theorem (2.1.3) which has surely not seen its last use.

Table of Contents:
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.