No abstract

Abstract.The Dunkl operators involve a multiplicity function as parameter. For generic values of this function the simultaneous kernel of these operators, acting on polynomials, is equal to the constants. For special values, however, this kernel is larger. We determine these singular values completely and give partial results on the representations of G that occur in this kernel.

We prove in a direct fashion that a multidimensional probability measure µ is determinate if the higher dimensional analogue of Carleman's condition is satisfied. In that case, the polynomials, as well as certain proper subspaces of the trigonometric functions, are dense in all associated Lp-spaces for 1 ≤ p < ∞. In particular these three statements hold if the reciprocal of a quasi-analytic weight has finite integral under µ. We give practical examples of such weights, based on their classification.As in the one dimensional case, the results on determinacy of measures supported on R n lead to sufficient conditions for determinacy of measures supported in a positive convex cone, i.e. the higher dimensional analogue of determinacy in the sense of Stieltjes.1991 Mathematics Subject Classification. Primary 44A60; Secondary 41A63, 41A10, 42A10, 46E30, 26E10.

In this paper, we describe the commutant of an arbitrary subalgebra A of the algebra of functions on a set X in a crossed product of A with the integers, where the latter act on A by a composition automorphism defined via a bijection of X. The resulting conditions which are necessary and sufficient for A to be maximal abelian in the crossed product are subsequently applied to situations where these conditions can be shown to be equivalent to a condition in topological dynamics. As a further step, using the Gelfand transform, we obtain for a commutative completely regular semi-simple Banach algebra a topological dynamical condition on its character space which is equivalent to the algebra being maximal abelian in a crossed product with the integers.

Abstract. We conjecture a geometrical form of the Paley-Wiener theorem for the Dunkl transform and prove three instances thereof, by using a reduction to the one-dimensional even case, shift operators, and a limit transition from Opdam's results for the graded Hecke algebra, respectively. These PaleyWiener theorems are used to extend Dunkl's intertwining operator to arbitrary smooth functions.Furthermore, the connection between Dunkl operators and the Cartan motion group is established. It is shown how the algebra of radial parts of invariant differential operators can be described explicitly in terms of Dunkl operators. This description implies that the generalized Bessel functions coincide with the spherical functions. In this context of the Cartan motion group, the restriction of Dunkl's intertwining operator to the invariants can be interpreted in terms of the Abel transform. We also show that, for certain values of the multiplicities of the restricted roots, the Abel transform is essentially inverted by a differential operator.