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Multiresolution analysis and Radon measures on a locally compact Abelian group

Félix Galindo, Javier Sanz (2001)

Czechoslovak Mathematical Journal

A multiresolution analysis is defined in a class of locally compact abelian groups G . It is shown that the spaces of integrable functions ℒ p ( G ) and the complex Radon measures M ( G ) admit a simple characterization in terms of this multiresolution analysis.

Recovery of band-limited functions on locally compact Abelian groups from irregular samples

H. G. Feichtinger, S. S. Pandey (2003)

Czechoslovak Mathematical Journal

Using the techniques of approximation and factorization of convolution operators we study the problem of irregular sampling of band-limited functions on a locally compact Abelian group G . The results of this paper relate to earlier work by Feichtinger and Gröchenig in a similar way as Kluvánek’s work published in 1969 relates to the classical Shannon Sampling Theorem. Generally speaking we claim that reconstruction is possible as long as there is sufficient high sampling density. Moreover, the iterative...

The dual group of a dense subgroup

William Wistar Comfort, S. U. Raczkowski, F. Javier Trigos-Arrieta (2004)

Czechoslovak Mathematical Journal

Throughout this abstract, G is a topological Abelian group and G ^ is the space of continuous homomorphisms from G into the circle group 𝕋 in the compact-open topology. A dense subgroup D of G is said to determine G if the (necessarily continuous) surjective isomorphism G ^ ↠ D ^ given by h ↦ h | D is a homeomorphism, and G is determined if each dense subgroup of G determines G . The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is...

The existence of initially ω 1 -compact group topologies on free Abelian groups is independent of ZFC

Artur Hideyuki Tomita (1998)

Commentationes Mathematicae Universitatis Carolinae

It was known that free Abelian groups do not admit a Hausdorff compact group topology. Tkachenko showed in 1990 that, under CH, a free Abelian group of size ℭ admits a Hausdorff countably compact group topology. We show that no Hausdorff group topology on a free Abelian group makes its ω -th power countably compact. In particular, a free Abelian group does not admit a Hausdorff p -compact nor a sequentially compact group topology. Under CH, we show that a free Abelian group does not admit a Hausdorff...

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