Walter R. Bloom

Harmonic analysis of probability measures on hypergroups

A systematic presentation of the applications of the hypergroup method to problems in probability theory that deals exclusively with topological hypergroups, focusing on those that are commutative. It considers hypergroups as locally compact spaces with a group-like structure on which the bounded measures convolve in a similar way to that on a locally compact group. The volume covers hypergroups and their measure algebras, the dual of a commutative hypergroup, some special classes of hypergroups, positive and negative definite functions and measures, convolution semigroups and divisibility of measures, transience of convolution semigroups, and randomized sums of hypergroup-valued random variables...

Fourier Transforms of Schwartz Functions on Chébli-Trimèche Hypergroups.

The Fourier transform for Schwartz spaces on Chébli-Trimèche hypergroups is studied, leading to results on approximation to the identity for functions and distributions on the half-line. In particular it is shown that the heat and Poisson kernels on the half-line form approximate units in various function spaces. A characterization of the convolution of a tempered distribution and a Schwartz function is also given.

Fourier multipliers for Lp on Chébli-Trimèche hypergroups

In this paper we consider Fourier multipliers for

Bernstein's inequality for locally compact Abelian groups

L^p

Positive definite and related functions on hypergroups

(p>1)

Isomorphisms of hypergroups

on Chebli-Trimeche hypergroups and establish a version of Hormanders multiplier theorem. As applications we give some results concerning the Riesz potentials and oscillating multipliers.

Hypergroup structures on the set of natural numbers

This paper is concerned with version of Bernsteins inequality for Hausdroff locally compact Abelian groups. The ideas used are suggested by Exercise 12, p. 17 of Katznelsons book [4].

Jackson's Theorem for locally compact abelian groups

Due to the changing nature of learning and teaching in universities, there is a growing need for professional development for lecturers and tutors teaching in disciplines in the mathematical sciences. Mathematics teaching staff receive some training in learning and teaching but many of the courses running at university level are not tailored to the mathematical sciences. This article reports on a collaborative research project aimed at investigating the type of professional development that Australian tertiary mathematics teachers need and their preference for delivery modes. Effective teaching promotes effective learning in our students and discipline-specific professional development will enhance outcomes for teachers, students, and mathematics.

Negative definite and Schoenberg functions on commutative hypergroups

In this paper we make use of semigroup methods on the space of compactly supported probability measures to obtain a complete Levy-Khinchin representation for negative definite functions on a commutative hypergroup. In addition we obtain representation theorems for completely monotone and completely alternating functions. The techniques employed here also lead to considerable simplification of the proofs of known results on positive definite and negative definite functions on hypergroups.