51,927 research outputs found
Eisenstein integrals and induction of relations
I give a survey of joint work with Henrik Schlichtkrull on the induction of
certain relations among (partial) Eisenstein integrals for the minimal
principal series of a reductive symmetric space. I explain the application of
this principle of induction to the proofs of a Fourier inversion formula and a
Paley-Wiener theorem. Finally, the relation with the Plancherel decomposition
is discussed.Comment: Latex2e, 22 pp, Proc. Conf. `Analyse Harmonique Non Commutative
(colloque en l'honneur de Jacques Carmona)' CIRM, Luminy, 20-24 Mai, 200
Transference Principles for Semigroups and a Theorem of Peller
A general approach to transference principles for discrete and continuous
operator (semi)groups is described. This allows to recover the classical
transference results of Calder\'on, Coifman and Weiss and of Berkson, Gillespie
and Muhly and the more recent one of the author. The method is applied to
derive a new transference principle for (discrete and continuous) operator
semigroups that need not be groups. As an application, functional calculus
estimates for bounded operators with at most polynomially growing powers are
derived, culminating in a new proof of classical results by Peller from 1982.
The method allows a generalization of his results away from Hilbert spaces to
\Ell{p}-spaces and --- involving the concept of -boundedness --- to
general Banach spaces. Analogous results for strongly-continuous one-parameter
(semi)groups are presented as well. Finally, an application is given to
singular integrals for one-parameter semigroups
Co-compact Gabor systems on locally compact abelian groups
In this work we extend classical structure and duality results in Gabor
analysis on the euclidean space to the setting of second countable locally
compact abelian (LCA) groups. We formulate the concept of rationally
oversampling of Gabor systems in an LCA group and prove corresponding
characterization results via the Zak transform. From these results we derive
non-existence results for critically sampled continuous Gabor frames. We obtain
general characterizations in time and in frequency domain of when two Gabor
generators yield dual frames. Moreover, we prove the Walnut and Janssen
representation of the Gabor frame operator and consider the Wexler-Raz
biorthogonality relations for dual generators. Finally, we prove the duality
principle for Gabor frames. Unlike most duality results on Gabor systems, we do
not rely on the fact that the translation and modulation groups are discrete
and co-compact subgroups. Our results only rely on the assumption that either
one of the translation and modulation group (in some cases both) are co-compact
subgroups of the time and frequency domain. This presentation offers a unified
approach to the study of continuous and the discrete Gabor frames.Comment: Paper (v2) shortened. To appear in J. Fourier Anal. App
Density and duality theorems for regular Gabor frames
We investigate Gabor frames on locally compact abelian groups with
time-frequency shifts along non-separable, closed subgroups of the phase space.
Density theorems in Gabor analysis state necessary conditions for a Gabor
system to be a frame or a Riesz basis, formulated only in terms of the index
subgroup. In the classical results the subgroup is assumed to be discrete. We
prove density theorems for general closed subgroups of the phase space, where
the necessary conditions are given in terms of the "size" of the subgroup. From
these density results we are able to extend the classical Wexler-Raz
biorthogonal relations and the duality principle in Gabor analysis to Gabor
systems with time-frequency shifts along non-separable, closed subgroups of the
phase space. Even in the euclidean setting, our results are new
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