Essays in Commutative Harmonic Analysis

1. Front Matter

   Pages i-xxi

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2. The Behavior of Transforms

   • Colin C. Graham, O. Carruth McGehee

   Pages 1-47

3. A Proof That the Union of Two Helson Sets Is a Helson Set

   • Colin C. Graham, O. Carruth McGehee

   Pages 48-67

4. Harmonic Synthesis

   • Colin C. Graham, O. Carruth McGehee

   Pages 68-90

5. Sets of Uniqueness, Sets of Multiplicity

   • Colin C. Graham, O. Carruth McGehee

   Pages 91-121

6. A Brief Introduction to Convolution Measure Algebras

   • Colin C. Graham, O. Carruth McGehee

   Pages 122-137

7. Independent Power Measures

   • Colin C. Graham, O. Carruth McGehee

   Pages 138-195

8. Riesz Products

   • Colin C. Graham, O. Carruth McGehee

   Pages 196-227

9. The Šilov Boundary, Symmetric Ideals, and Gleason Parts of ΔM(G)

   • Colin C. Graham, O. Carruth McGehee

   Pages 228-250

10. The Wiener-Lévy Theorem and Some of Its Converses

    • Colin C. Graham, O. Carruth McGehee

    Pages 251-280

11. The Multiplier Algebras M _p (Γ), and the Theorem of Zafran

    • Colin C. Graham, O. Carruth McGehee

    Pages 281-307

12. Tensor Algebras and Harmonic Analysis

    • Colin C. Graham, O. Carruth McGehee

    Pages 308-361

13. Tilde Algebras

    • Colin C. Graham, O. Carruth McGehee

    Pages 362-401

14. Unsolved Problems

    • Colin C. Graham, O. Carruth McGehee

    Pages 402-423

15. Back Matter

    Pages 425-466

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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.

• Algebra
• Derivation
• Division
• Finite
• Harmonische Analyse
• Invariant
• Morphism
• calculus
• commutative property
• function
• proof
• theorem

• Department of Mathematics, Northwestern University, Evanston, USA

  Colin C. Graham

• Department of Mathematics, Louisiana State University, Baton Rouge, USA

  O. Carruth McGehee

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