Existence of Borel transversals in groups

Feldman, Jacob J.; Greenleaf, Frederick P.

doi:10.2140/pjm.1968.25.455

Cited by 61 publications

(44 citation statements)

References 3 publications

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“…The Haar measure m on will be normalized so that the Weil coset decomposition formula The quotient group / has a complete set of coset representatives, also referred to as a transversal, the terminology we will use. Transversals are not unique and there is always a measurable one [29] which we will choose. Note that while the quotient group is compact, the transversal is only of finite measure; indeed unbounded transversals are possible (see [5,Sect.…”

Section: Abstract Sampling Theorymentioning

confidence: 99%

Seven pivotal theorems of Fourier analysis, signal analysis, numerical analysis and number theory: their interconnections

Butzer¹,

Dodson²,

Ferreira³

et al. 2014

Bull. Math. Sci.

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The present paper deals mainly with seven fundamental theorems of mathematical analysis, numerical analysis, and number theory, namely the generalized Parseval decomposition formula (GPDF), introduced 15 years ago, the well-known approximate sampling theorem (ASF), the new approximate reproducing kernel theorem, the basic Poisson summation formula, already known to Gauß, a newer version of the GPDF having a structure similar to that of the Poisson summation formula, namely, the Parseval decomposition-Poisson summation formula, the functional equation of Riemann's zeta function, as well as the Euler-Maclaurin summation formula. It will in fact be shown that these seven theorems are all equivalent to one another, in the sense that each is a corollary of the others. Since these theorems can all be deduced from each other, one of them has to be proven independently in order to verify all. It is convenient to choose the ASF, introduced in 1963. The epilogue treats possible extensions to the more general contexts of reproducing kernel theory and of abstract harmonic analysis, using locally compact abelian groups. This paper is expository in the sense that it treats a number of mathematical theorems, their interconnections, their equivalence to one another. On the other hand, the proofs of the many intricate interconnections among these theorems are new in their essential steps and conclusions.

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Section: Abstract Sampling Theorymentioning

confidence: 99%

Seven pivotal theorems of Fourier analysis, signal analysis, numerical analysis and number theory: their interconnections

Butzer¹,

Dodson²,

Ferreira³

et al. 2014

Bull. Math. Sci.

View full text Add to dashboard Cite

show abstract

“…The quotient group / has a complete set of coset representatives, sometimes referred to as a fundamental domain [31] or a transversal, the terminology we will use. Transversals are not unique and there is always a measurable one [21]. Note that while the quotient group is compact, the transversal is only of finite measure.…”

Section: Definitions and Notationmentioning

confidence: 99%

On the approximate form of Kluvánek's theorem

Beaty

Dodson

Eveson

et al. 2009

Journal of Approximation Theory

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An abstract form of the classical approximate sampling theorem is proved for functions on a locally compact abelian group that are continuous, square-integrable and have integrable Fourier transforms. An additional hypothesis that the samples of the function are square-summable is needed to ensure the convergence of the sampling series. As well as establishing the representation of the function as a sampling series plus a remainder term, an asymptotic formula is obtained under mild additional restrictions on the group. In conclusion a converse to Kluvánek's theorem is established.

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“…The quotient group / has a measurable transversal, say [10]. The transversal will not in general be compact or even bounded.…”

Section: Quotient Groups Transversals Discrete Subgroups and Disjoinmentioning

confidence: 99%

A Converse to Kluvánek’s Theorem

Beaty

Dodson

Eveson

2007

J Fourier Anal Appl

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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 ofKluvánek'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 / .

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Existence of Borel transversals in groups

Cited by 61 publications

References 3 publications

Seven pivotal theorems of Fourier analysis, signal analysis, numerical analysis and number theory: their interconnections

Seven pivotal theorems of Fourier analysis, signal analysis, numerical analysis and number theory: their interconnections

On the approximate form of Kluvánek's theorem

A Converse to Kluvánek’s Theorem

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