Colin C. Graham

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.

The size of Lp-improving measures☆

Abstract A measure μ on the locally compact abelian group G is “Lp-improving” if for some 1⩽p p (G)⊆L r (G) . L-space properties of such μ are discussed: if μ is Lp-improving and non-negative, and f is a bounded Borel function, then fμ is Lp-improving. The non-negativity of μ and boundedness of f are essential, as is shown in several ways. In particular, there exists on each infinite compact abelian group an Lp-improving μ such that |μ| is not Lp-improving. The distribution function of every Lp-improving measure on the circle group is shown to satisfy a Lipschitz condition, but that is not sufficient for a measure to be Lp-improving, except in restricted cases. The Gelfand transform on Γ ‖Γ of an Lp-improving measure is shown to be “small” in a quantitative sense; from that the strong continuity of Lp-improving measures follows. The spectrum of an Lp-improving measure as an operator on Lp is shown to be precisely the closure of μ (Γ) . Related results and open problems are included.

Symbolic calculus for positive-definite functions

It is shown that if Γ is an infinite discrete abelian group of exponent strictly greater than two, and if F:{z∈C:|z|⩽1}→C is such that F ∘ μ is a Fourier-Stieltjes transform on Γ whenever \gm is positive-definite and \gm(0) ⩽ 1, then F(z) = Σm, n ⩾ 0 amnzmzn for z ϵ int ∪ {\gm(Γ) : \gm positive-definite, \gm(0) ⩽ 1}, and Σ ¦ amn ¦ < ∞. This result is extended to all noncompact LCA groups Γ such that Γ has a compact open subgroup, with ΓΛ of exponent strictly greater than two. The extended result is shown to be sharp, and is used to prove a theorem due to Herz and Rider: If F:{|z|⩽1}→C is such that F ∘ \gm is a continuous positive-definite function on Γ whenever \gm is a continuous positive-definite function on Γ, and Γ is not the product of a finite group with a group of exponent two, then F(z) = Σamnzmzn (¦ z ¦ ⩽ 1), amn ⩾ 0 and Σamn < ∞.

The algebraic radical of a normed ideal inL 1(G)

LetA be a proper normed ideal (in the sense ofCigler) insideL1(G), whereG is a non-discrete LCA group. This is proved: For each integern≧1 there existsf∈L1(G) such thatf, f2,..., fn∉A whilefn+1∈A.

The functions which operate on Fourier-Stieltjes transforms of L-subalgebras of T

Abstract An L-subalgebra OL of the algebra M T of Borel measures on the circle group T is a closed subalgebra OL ⊆M( T ) such that μ ∈ OL implies L1(μ)⊆ OL . A subalgebra OL is symmetric if μ∈ OL implies μ ∈ OL , where μ (−E) = μ(−E) , (E any Borel set). A function F : [−1, 1] → C operates on a set OL of Fourier-Stieltjes transforms if μ ∈ OL , μ ( Z )⊆[−1,1] imply F ∘ μ is a Fourier-Stieltjes transform. We have these results: If F : [−1, 1] → C operates on the Fourier-Stieltjes transforms of an infinite dimensional symmetric L-subalgebra OL , then F is analytic in a neighborhood of 0. If OL contains a measure concentrated on a Dirichlet set, then F is analytic in a neighborhood of [−1, 1]. If OL contains a continuous measure concentrated on a Kronecker set, then F is entire. Related results are given.

Classes of Trimeasures: Applications of Harmonic Analysis

Let X1,... ,X n be locally compact Hausdorff spaces and C 0 (X i ) the commutative C*-algebra of continuous complex functions on X i vanishing at infinity for i = 1,..., n. A bounded n-linear form Ф : C 0 (X 1 ) × × C 0 (X n ) → C will be called a polymeasure. (The term muliimeasure also appears in the literature as a synonym, but we avoid it in order not to conflict with its other uses.) The Banach space of such polymeasures equipped with the usual supremum norm ∥•∥ we denote by PM(X1,... ,Xn). The cases n = 2 (bimeasures) and especially n = 3 (trimeasures) are of particular interest to us.

Measures vanishing off the symmetric maximal ideals of M(G)

We show that if μ is a measure on the LCA group G whose Gelfand transform vanishes off the set Σ of symmetric maximal ideals, then μ µ M 0 (G) , that is, then the Fourier-Steiltjes transform of μ vanishes at infinity. This result is then used to show μ µ L 1 (G) ½.

The interior of the Šilov boundary of M(G) is trivial

Abstract Let G be a non-discrete locally compact abelian group, and let M(G) be the convolution algebra of regular bounded Borel measures on G. Let Γ denote the dual group of G. Then the interior of the Silov boundary of M(G) is exactly Γ. The proof uses generalized Riesz products for the compact metrizable case and standard liftings from that case.

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