We consider the distribution μ of numbers whose binary digits are generated from infinitely many tosses of a biased coin. It is shown that, if E has positive μ measure, then some n-fold sum of E with itself must contain an interval. This contrasts with the known result that all convolution powers of μ are singular.

Let ψ be a positive function defined near the origin such lim1 →0+ ψ(t) = 0. We consider the operator Tzƒ, defined as the pricipal value of the convolution of function ƒ and a kernel K(t) = eiy(t)t−z /ψ(t)1−z, where z is a complex number, 0 ≤ Re(z) ≤ 1, 0 < t ≤ 1 and γ is a real function. Assuming certain regularity conditions on ψ and γ and certain relations between ψ and γ we show that Tθ is a bounded operator on Lp (R) for 1/p = (1+ θ) /2 and 0 ≤ θ < 1, and T1 is bounded from H1 (R) to L1 (R).

Lipschitz spaces are important function spaces with relations to Hp spaces and Campanato spaces, the other two important function spaces in harmonic analysis. In this paper we give some characterizations for Lipschitz spaces on compact Lie groups, which are analogues of results in Euclidean spaces.

It is well known that c0(Z) is amenable and so its global dimension is zero. In this paper we will investigate the cyclic and Hochschild cohomology of Banach algebra c0 (Z, ω-1) and its unitisation with coefficients in its dual space, where ω is a weight on Z which satisfies inf {ω(i)} = 0.Moreover we show that the weak homological bi-dimension of c0 (Z, ω-1) is infinity.

In this paper we study commutative topological semigroups S admitting an absolutely continuous measure. When S is cancellative we show that S admits a weaker topology J with respect to which (S, J) is embeddable as a subsemigroup with non-empty interior in some locally compact topological group. As a consequence, we deduce certain results related to the existence of invariant measures on S and for a large class of locally compact topological semigroups S, we associate S with some useful topological subsemigroup of a locally compact group.

Let St = exp{−tH}, Tt = exp{−tK}, be C0-semigroups on a Banach space . For appropriate f one can define subordinate semigroups Sft = exp{−tf(H)}, Ttf = exp{−tf(K)}, on and examine order properties of the pairs S, T, and Sf, Tf. If , = Lp(X;dv) we define St≽ Tt ≽ 0 if St − Tt and Tt map non-negative functions into non-negative functions. Then for p fixed in the range 1 > p > ∞ we characterize the functions for which St ≽ Tt ≽ 0 implies Sft ≽ Tft ≽ 0 for each Lp(X;dv) and the converse is true for all Lp(X;dv). Further we give irreducibility criteria for the strict ordering of holomorphic semigroups. This extends earlier results for L2-spaces.

We develop a theory of ergodicity for unbounded functions ø: J → X, where J is a subsemigroup of a locally compact abelian group G and X is a Banach space. It is assumed that ø is continuous and dominated by a weight w defined on G. In particular, we establish total ergodicity for the orbits of an (unbounded) strongly continuous representation T: G → L(X) whose dual representation has no unitary point spectrum. Under additional conditions stability of the orbits follows. To study spectra of functions, we use Beurling algebras L1w(G) and obtain new characterizations of their maximal primary ideals, when w is non-quasianalytic, and of their minimal primary ideals, when w has polynomial growth. It follows that, relative to certain translation invariant function classes , the reduced Beurling spectrum of ø is empty if and only if ø ∈ . For the zero class, this is Wiener's tauberian theorem.

The aim of this paper is to study the spherical functions associated to an operator. These functions can be thought of abstractly as being eigenfunctions of the operator which can be expressed in terms of the operator. The meaning of these properties will be made precise as will a notion of boundedness. The results are obtained by studying a specific shift operator on the algebra of functionals on the complex polynomial ring. For the class studied, we obtain ellipses of eigenvalues for which there exist bounded spherical functions. As an application of the results, we study radial functions on discrete groups.

It is proved that on certain kinds of homogeneous spaces, the only Lp function, 1≤ p < ∞, satisfying the mean value property is the zero function.

W. Rudin has proved that the union of the Riesz set N ⊆ R with a Λ(l)-subset of Z is again a Riesz set. In this note we generalize his result to compact groups whose contains a circle group, thereby extending an earlier F. and M. Riesz theorem for such groups by the author. We also investigate the possibility of constructing Λ(p)-sets for these groups, departing from Λ(p)-sets for the circle group in center.

In this paper we make use of semigroup methods on the space of compactly supported measures to obtain a Bochner representation for α-bounded positive-definite functions on a commutative hypergroup.

In this paper we investigate when negative definite functions on commutative hypergroups satisfy the Schoenberg criterion.

The Plancherel formula for the horocycle space, and several generalizations, is derived within the framework of quasi-regular representations which have monomial spectrum. The proof uses only machinery from the Penney-Fujiwara distribution-theoretic technique; no special semisimple harmonic analysis is needed. The Plancherel formulas obtained include the spectral distributions and the intertwining operators that effect the direct integral decomposition of the quasi-regular representation.

Let µ be Radon measure on Rd which may be non doubling. The only condition that µ must satisfy is the size condition µ(B(x, r)) ≤ Crn for some fixed n є (0, d). Recently, Tolsa introduced the spaces RBMO(µ) and Hatb1.∞ (µ), which, in some ways, play the role of the classical spaces BMO and H1 in case u is a doubling measure. In this paper, the author considers the local versions of the spaces RBMO(µ) and Hatb1.∞ (µ) in the sense of Goldberg and establishes the connections between the spaces RBMO(µ) and Hatb1.∞ (µ) with their local versions. An interpolation result of linear operators is also given.

Let K1, K2 be locally compact hypergroups. It is shown that every isometric isomorphism between their measure algebras restricts to an isometric isomorphism between their L1-algebras. This result is used to relate isometries of the measure algebras to homeomorphisms of the underlying locally compact spaces.

Let G be a locally compact abelian group, 1 < p < ∞, and A be a commutative Banach algebra. In this paper we study the space of multipliers on Lp (G, A) and characterize it as the space of multipliers of certain banach algebra. We also study the multipliers space on L1 (G, A) ∩ Lp (G, A).

In this paper we consider pseudo differential operators on local Hardy spaces hp (0 < p ≤ 1) on Chébli-Trimèche hypergroups of exponential growth.

In this paper we prove a new version of the Cowling-Price theorem for Fourier transforms on Rn. Using this we formulate and prove an uncertainty principle for operators. This leads to an analogue of the Cowling-Price theorem for nilpotent Lie groups. We also prove an exact analogue of the Cowling-Price theorem for the Heisenberg group.

The special Denjoy-Bochner integral (the D*B-integral) which are generalisations of Lebesgue-Bochner integral are discussed in [7, 6, 5]. Just as the concept of numerical almost periodicity was extended by Burkill [3] to numerically valued D*- or D-integrable function, we extend the concept of almost periodicity for Banach valued function to Banach valued D*B-integrable function. For this purpose we introduce as in [3] a distance in the space of all D*B-integrable functions with respect to which the D*B-almost periodicity is defined. It is shown that the D*B-almost periodicity shares many of the known properties of the almost periodic Banach valued function [1, 4].

A weighted generalization of a p-Sidon set, called an (a, p)-Sidon set, is introduced and studied for infinite, non-abelian, connected, compact groups G. The entire dual object Ĝ is shown never to be central (p − 1, p)-Sidon for 1 ≦ p < 2, nor central (1 + ε, 2)-Sidon for ε > 0. Local (p, p)-Sidon sets are shown to be identical to local Sidon sets. Examples are constructed of infinite non-Sidon sets which are central (2p − 1, p)-Sidon, or (p − 1, p)-Sidon, for 1 < p < 2. Full m-fold FTR sets are proved not to be central (a, 2m/(m + 1))-Sidon for any a > 1.