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.

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.

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

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.

On a Stein And Weiss Property of the Conjugate Function

1990 ◽  
Vol 42 (1) ◽  
pp. 109-125
Author(s):  
Nakhlé Asmar
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Conjugate Function ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Character Group ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Image Position ◽  
Almost All

(1.1) The conjugate function on locally compact abelian groups. Let G be a locally compact abelian group with character group Ĝ. Let μ denote a Haar measure on G such that μ(G) = 1 if G is compact. (Unless stated otherwise, all the measures referred to below are Haar measures on the underlying groups.) Suppose that Ĝ contains a measurable order P: P + P ⊆P; PU(-P)= Ĝ; and P⋂(—P) =﹛0﹜. For ƒ in ℒ2(G), the conjugate function of f (with respect to the order P) is the function whose Fourier transform satisfies the identity for almost all χ in Ĝ, where sgnP(χ)= 0, 1, or —1, according as χ =0, χ ∈ P\\﹛0﹜, or χ ∈ (—P)\﹛0﹜.

scholarly journals A COMPACT QUALITATIVE UNCERTAINTY PRINCIPLE FOR SOME NONUNIMODULAR GROUPS

2018 ◽  
Vol 99 (1) ◽  
pp. 114-120
Author(s):  
WASSIM NASSERDDINE
Keyword(s):  
Regular Representation ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compactly Supported ◽  
Finite Dimensional ◽  
Left Regular Representation ◽  
Left Regular ◽  
Compact Abelian Groups ◽  
Qualitative Uncertainty ◽  
The Fourier Transform

Let $G$ be a separable locally compact group with type $I$ left regular representation, $\widehat{G}$ its dual, $A(G)$ its Fourier algebra and $f\in A(G)$ with compact support. If $G=\mathbb{R}$ and the Fourier transform of $f$ is compactly supported, then, by a classical Paley–Wiener theorem, $f=0$. There are extensions of this theorem for abelian and some unimodular groups. In this paper, we prove that if $G$ has no (nonempty) open compact subsets, $\hat{f}$, the regularised Fourier cotransform of $f$, is compactly supported and $\text{Im}\,\hat{f}$ is finite dimensional, then $f=0$. In connection with this result, we characterise locally compact abelian groups whose identity components are noncompact.

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.

scholarly journals Integrability of Fourier transforms under an ergodic hypothesis

1978 ◽  
Vol 26 (2) ◽  
pp. 129-153
Author(s):  
Gavin Brown
Keyword(s):  
Fourier Transforms ◽  
General Setting ◽  
Locally Compact ◽  
Ergodic Hypothesis ◽  
Locally Compact Abelian Groups ◽  
Discrete Random Variables ◽  
Ergodic Condition ◽  
Compact Abelian Groups ◽  
Plancherel’S Theorem ◽  
Underlying Theme

AbstractThe object is to unify and complement some recent theorems of Hewitt and Ritter on the integrability of Fourier transforms, but the underlying theme is the ancient one that Plancherel's theorem is the “only” integrability constraint on Fourier transforms. The distinguishing feature of the results is that we restrict attention to positive measures (or functions) which satisfy an ergodic condition and whose transforms are positive. (In fact we employ sums of discrete random variables, a technique which seems to have been largely ignored in context.) The general setting is that of locally compact abelian groups but we are chiefly interested in the line or the circle, and it appears that the theorems are new for these classical groups.

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