On the Fourier transform on function algebras on locally compact Abelian groups

2017 ◽  
Vol 184 (4) ◽  
pp. 597-610
Author(s):  
R. Lakshmi Lavanya
Keyword(s):  
Fourier Transform ◽  
Abelian Groups ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Function Algebras ◽  
Compact Abelian Groups ◽  
The Fourier Transform

Pego everywhere

2016 ◽  
Vol 15 (04) ◽  
pp. 1650074 ◽  
Author(s):  
Przemysław Górka ◽  
Tomasz Kostrzewa
Keyword(s):  
Fourier Transform ◽  
Abelian Groups ◽  
General Version ◽  
Pontryagin Duality ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
The Fourier Transform ◽  
Compact Subsets

In this note we show the general version of Pego’s theorem on locally compact abelian groups. The proof relies on the Pontryagin duality as well as on the Arzela–Ascoli theorem. As a byproduct, we get the characterization of relatively compact subsets of [Formula: see text] in terms of the Fourier transform.

PEGO THEOREM ON LOCALLY COMPACT ABELIAN GROUPS

2014 ◽  
Vol 13 (04) ◽  
pp. 1350143 ◽  
Author(s):  
PRZEMYSłAW GÓRKA
Keyword(s):  
Fourier Transform ◽  
Abelian Groups ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
The Fourier Transform

In this paper, we show the version of Pego's theorem on locally compact abelian groups. This theorem, [R. L. Pego, Compactness in L2 and the Fourier transform, Proc. Amer. Math. Soc.95 (1985) 252–254], gives a characterization of precompact sets of L2 in terms of the Fourier transform.

scholarly journals Gordon's conjectures 1 and 2: Pontryagin-van Kampen duality in the hyperfinite setting

2021 ◽  
Vol 13 ◽  
Author(s):  
Pavol Jan Zlatos
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Nonstandard Analysis ◽  
Abelian Groups ◽  
Finite Abelian Groups ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Lca Groups

Using the ideas of E. I. Gordon we present and farther advancean approach, based on nonstandard analysis, to simultaneousapproximations of locally compact abelian groups and their dualsby (hyper)finite abelian groups, as well as to approximations ofvarious types of Fourier transforms on them by the discrete Fouriertransform. Combining some methods of nonstandard analysis andadditive combinatorics we prove the three Gordon's Conjectureswhich were open since 1991 and are crucial both in the formulationsand proofs of the LCA groups and Fourier transform approximationtheorems

Positive Definite Measures with Discrete Fourier Transform and Pure Point Diffraction

2011 ◽  
Vol 54 (3) ◽  
pp. 544-555 ◽  
Author(s):  
Nicolae Strungaru
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Abelian Groups ◽  
Positive Definite ◽  
Pure Point ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups

AbstractIn this paper we characterize the positive definite measures with discrete Fourier transform. As an application we provide a characterization of pure point diffraction in locally compact Abelian groups.

Fourier transforms of convolution operators on orlicz spaces

2021 ◽  
Vol 71 (2) ◽  
pp. 369-382
Author(s):  
Seyyed Mohammad Tabatabaie ◽  
AliReza Bagheri Salec
Keyword(s):  
Fourier Transform ◽  
Convolution Operator ◽  
Fourier Transforms ◽  
Orlicz Spaces ◽  
Locally Compact Group ◽  
Convolution Operators ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
The Fourier Transform

Abstract In this paper, we study convolution operators on an Orlicz space L Φ(G) commuting with left translations, where Φ is an N-function and G is a locally compact group. We also present some basic properties of the Fourier transform of a Φ-convolution operator in the context of locally compact abelian groups.

Scaling sets and generalized scaling sets on Cantor dyadic group

2020 ◽  
Vol 18 (04) ◽  
pp. 2050019
Author(s):  
Prasadini Mahapatra ◽  
Divya Singh
Keyword(s):  
Abelian Groups ◽  
Real Case ◽  
Locally Compact ◽  
Wavelet Sets ◽  
Locally Compact Abelian Groups ◽  
Dyadic Group ◽  
Compact Abelian Groups ◽  
Cantor Dyadic Group

Scaling and generalized scaling sets determine wavelet sets and hence wavelets. In real case, wavelet sets were proved to be an important tool for the construction of MRA as well as non-MRA wavelets. However, any result related to scaling/generalized scaling sets is not available in case of locally compact abelian groups. This paper gives a characterization of scaling sets and its generalized version along with relevant examples in dual Cantor dyadic group [Formula: see text]. These results can further be generalized to arbitrary locally compact abelian groups.

Singular measures with absolutely continuous convolution squares

1966 ◽  
Vol 62 (3) ◽  
pp. 399-420 ◽  
Author(s):  
Edwin Hewitt ◽  
Herbert S. Zuckerman
Keyword(s):  
Abelian Groups ◽  
Natural Order ◽  
Continuous Measure ◽  
Locally Compact ◽  
Absolutely Continuous ◽  
Character Group ◽  
Locally Compact Abelian Groups ◽  
Singular Measures ◽  
Compact Abelian Groups ◽  
Nikodym Derivative

Introduction. A famous construction of Wiener and Wintner ((13)), later refined by Salem ((11)) and extended by Schaeffer ((12)) and Ivašev-Musatov ((8)), produces a non-negative, singular, continuous measure μ on [ − π,π[ such thatfor every ∈ > 0. It is plain that the convolution μ * μ is absolutely continuous and in fact has Lebesgue–Radon–Nikodým derivative f such that For general locally compact Abelian groups, no exact analogue of (1 · 1) seems possible, as the character group may admit no natural order. However, it makes good sense to ask if μ* μ is absolutely continuous and has pth power integrable derivative. We will construct continuous singular measures μ on all non-discrete locally compact Abelian groups G such that μ * μ is a absolutely continuous and for which the Lebesgue–Radon–Nikodým derivative of μ * μ is in, for all real p > 1.

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