O. Carruth McGehee

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 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.

A Brief Introduction to Convolution Measure Algebras

We have two objectives in this chapter: to introduce the general theory of convolution measure algebras and to give examples and applications pertinent to measure algebras on groups. We shall thus make clear the setting in which the action of Chapters 6 through 8 takes place. We shall state without proof some results (the most important ones are Theorems 5.1.1 and 5.3.6). It is not necessary to read their proofs in order to appreciate that action. We begin with some useful terminology.

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

The question, whether the union of two Helson sets is a Helson set, resisted answering for some time. S. W. Drury and N. Th. Varopoulos solved the problem in 1970, and we now know that if H = H1 ∪ H2 where H1 and H2 are Helson subsets of G, then

Tensor Algebras and Harmonic Analysis

\alpha \left( H \right) \leqslant \frac{{{3^{{3/2}}}}}{2}(\alpha {({H_{1}})^{3}} + \alpha {({H_{2}})^{3}}).

The Wiener-Lévy Theorem and Some of Its Converses

One may still hope for simpler proofs and better inequalities.

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

Since the action in this Chapter takes place on the circle group T, we write A for A(T), PF for PF(T), and so forth.

The Multiplier Algebras M p (Γ), and the Theorem of Zafran

Let U and W be Banach spaces. The complete tensor product with greatest cross norm U \( \hat{ \otimes } \) W is the space obtained as follows. The usual tensor product U ⊗ W is given the norm