A converse to Kluvánek’s theorem

M.G. Beaty; M.M. Dodson; S.P. Eveson

doi:10.1007/s00041-006-6025-x

A converse to Kluvánek’s theorem

M.G. Beaty, M.M. Dodson, S.P. Eveson

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

I. Kluvánek extended the Whittaker-Kotel’nikov-Shannon (WKS) theorem to the abstract harmonic analysis setting. To do this, the ‘band limited’ condition on the spectrum of a continuous square-integrable function (analogue signal) required for classical WKS theorem is replaced by an ‘almost disjoint’ translates condition arising from the Fourier transform of the function vanishing almost everywhere outside a transversal of a compact quotient group. A converse of Kluvánek’s theorem is established, i.e., if the representation given by the abstract WKS theorem holds for a continuous square-integrable function with support of its Fourier transform essentially A, then A is a subset of a transversals of Gamma/Lambda

Original language	English
Pages (from-to)	187-196
Number of pages	10
Journal	Journal of Fourier Analysis and Applications
Volume	13
Issue number	2
DOIs	https://doi.org/10.1007/s00041-006-6025-x
Publication status	Published - Apr 2007

Keywords

Whittaker-Kotel'nikov-Shannon theorem
Plancherel's formula
locally compact abelian groups
discrete subgroups
transversals

Access to Document

10.1007/s00041-006-6025-x

http://dx.doi.org/10.1007/s00041-006-6025-x

Cite this

@article{d243f5c033984f9b854bdaabec1af0c7,

title = "A converse to Kluv{\'a}nek{\textquoteright}s theorem",

abstract = "I. Kluv{\'a}nek extended the Whittaker-Kotel{\textquoteright}nikov-Shannon (WKS) theorem to the abstract harmonic analysis setting. To do this, the {\textquoteleft}band limited{\textquoteright} condition on the spectrum of a continuous square-integrable function (analogue signal) required for classical WKS theorem is replaced by an {\textquoteleft}almost disjoint{\textquoteright} translates condition arising from the Fourier transform of the function vanishing almost everywhere outside a transversal of a compact quotient group. A converse of Kluv{\'a}nek{\textquoteright}s theorem is established, i.e., if the representation given by the abstract WKS theorem holds for a continuous square-integrable function with support of its Fourier transform essentially A, then A is a subset of a transversals of Gamma/Lambda",

keywords = "Whittaker-Kotel'nikov-Shannon theorem, Plancherel's formula, locally compact abelian groups, discrete subgroups, transversals",

author = "M.G. Beaty and M.M. Dodson and S.P. Eveson",

year = "2007",

month = apr,

doi = "10.1007/s00041-006-6025-x",

language = "English",

volume = "13",

pages = "187--196",

journal = "Journal of Fourier Analysis and Applications",

issn = "1531-5851",

publisher = "Birkhauser Boston",

number = "2",

}

TY - JOUR

T1 - A converse to Kluvánek’s theorem

AU - Beaty, M.G.

AU - Dodson, M.M.

AU - Eveson, S.P.

PY - 2007/4

Y1 - 2007/4

N2 - I. Kluvánek extended the Whittaker-Kotel’nikov-Shannon (WKS) theorem to the abstract harmonic analysis setting. To do this, the ‘band limited’ condition on the spectrum of a continuous square-integrable function (analogue signal) required for classical WKS theorem is replaced by an ‘almost disjoint’ translates condition arising from the Fourier transform of the function vanishing almost everywhere outside a transversal of a compact quotient group. A converse of Kluvánek’s theorem is established, i.e., if the representation given by the abstract WKS theorem holds for a continuous square-integrable function with support of its Fourier transform essentially A, then A is a subset of a transversals of Gamma/Lambda

AB - I. Kluvánek extended the Whittaker-Kotel’nikov-Shannon (WKS) theorem to the abstract harmonic analysis setting. To do this, the ‘band limited’ condition on the spectrum of a continuous square-integrable function (analogue signal) required for classical WKS theorem is replaced by an ‘almost disjoint’ translates condition arising from the Fourier transform of the function vanishing almost everywhere outside a transversal of a compact quotient group. A converse of Kluvánek’s theorem is established, i.e., if the representation given by the abstract WKS theorem holds for a continuous square-integrable function with support of its Fourier transform essentially A, then A is a subset of a transversals of Gamma/Lambda

KW - Whittaker-Kotel'nikov-Shannon theorem

KW - Plancherel's formula

KW - locally compact abelian groups

KW - discrete subgroups

KW - transversals

U2 - 10.1007/s00041-006-6025-x

DO - 10.1007/s00041-006-6025-x

M3 - Article

SN - 1531-5851

VL - 13

SP - 187

EP - 196

JO - Journal of Fourier Analysis and Applications

JF - Journal of Fourier Analysis and Applications

IS - 2

ER -