We prove an analogue of Gutzmer's formula for Hermite expansions. As a consequence we obtain a new proof of a characterisation of the image of L²(ℝⁿ) under the Hermite semigroup. We also obtain some new orthogonality relations for complexified Hermite functions.
We observe that the classical theorem of Hardy on Fourier transform pairs can be reformulated in terms of the heat kernel associated with the Laplacian on the Euclidean space. This leads to an interesting version of Hardy's theorem for the sublaplacian on the Heisenberg group. We also consider certain Rockland operators on the Heisenberg group and Schrödinger operators on ℝⁿ related to them.
We study the action of elementary shift operators on spherical functions on ordered symmetric spaces of Cayley type, where denotes the multiplicity of the short roots and the rank of the symmetric space. For even we apply this to prove a Paley-Wiener theorem for the spherical Laplace transform defined on by a reduction to the rank 1 case. Finally we generalize our notions and results to type root systems.
We give an elementary proof for the case of the circle group of the theorem of O. Hatori and E. Sato, which states that every measure on a compact abelian group G can be decomposed into a sum of two measures with a natural spectrum and a discrete measure.
There is constructed a compactly generated, separable, locally compact group G and a continuous irreducible unitary representation π of G such that the image π(C*(G)) of the group C*-algebra contains the algebra of compact operators, while the image of the -group algebra does not containany nonzero compact operator. The group G is a semidirect product of a metabelian discrete group and a “generalized Heisenberg group”.
The first explicit example of a positive semidefinite double sequence which is not a moment sequence was given by Friedrich. We present an example with a simpler definition and more moderate growth as .
We study a method of approximating representations of the group by those of the group . As a consequence we establish a version of a theorem of DeLeeuw for Fourier multipliers of that applies to the “restrictions” of a function on the dual of to the dual of .
We generalize the classical F. and M. Riesz theorem to metrizable compact groups whose center contains a copy of the circle group. Important examples of such groups are the isotropy groups of the bounded symmetric domains.The proof uses a criterion for absolute continuity involving spaces with : A measure on a compact metrisable group is absolutely continuous with respect to Haar measure on if for some a certain subspace of which is related to has sufficiently many continuous linear...