346 research outputs found
Duality in spaces of finite linear combinations of atoms
In this note we describe the dual and the completion of the space of finite
linear combinations of -atoms, on . As
an application, we show an extension result for operators uniformly bounded on
-atoms, , whose analogue for is known to be false.
Let and let be a linear operator defined on the space of finite
linear combinations of -atoms, , which takes values in a
Banach space . If is uniformly bounded on -atoms, then
extends to a bounded operator from into .Comment: The paper has appeared as Ricci, F., & Verdera, J. (2011). Duality in
spaces of finite linear combinations of atoms. Transactions of the American
Mathematical Society, 363(3), 1311-132
Paley--Wiener Theorems for the U(n)--spherical transform on the Heisenberg group
We prove several Paley--Wiener-type theorems related to the spherical
transform on the Gelfand pair , where is
the -dimensional Heisenberg group.
Adopting the standard realization of the Gelfand spectrum as the Heisenberg
fan in , we prove that spherical transforms of --invariant functions and distributions with compact support in
admit unique entire extensions to , and we find real-variable
characterizations of such transforms. Next, we characterize the inverse
spherical transforms of compactly supported functions and distributions on the
fan, giving analogous characterizations
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