The support of a function with thin spectrum

Kathryn Hare

Displaying similar documents to “The support of a function with thin spectrum”

On functions whose translates are independent

Ralph E. Edwards (1951)

Annales de l'institut Fourier

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Ce travail est l’étude de divers cas particuliers d’un problème nouveau, semble-t-il, concernant les translatées de fonctions ou de distributions sur un groupe. Soit $E$ un espace vectoriel topologique de fonctions ou de distributions sur un groupe abélien $G$ localement compact ; $E$ est supposé invariant par les translations $a \to f_{a} (x) = f (x + a) (f \in E, a \in G)$ . Si $f \in E$ et si $A$ est un sous-ensemble non vide de $G$ , $I (f, A) = I (f, A, E)$ désigne le sous-espace vectoriel fermé de $E$ engendré par les translatées $f_{a}$ de $f$ avec $a \in A$ . On dira qu’une $f \in E$ a ses...

A characterization of the minimal strongly character invariant Segal algebra

Viktor Losert (1980)

Annales de l'institut Fourier

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For a locally compact, abelian group $G$ , we study the space $S_{0} (G)$ of functions on $G$ belonging locally to the Fourier algebra and with $l^{1}$ -behavior at infinity. We give an abstract characterization of the family of spaces ${S_{0} (G) : G$ abelian $}$ by its hereditary properties.

Fourier coefficients of continuous functions and a class of multipliers

Serguei V. Kislyakov (1988)

Annales de l'institut Fourier

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If $x$ is a bounded function on $Z$ , the multiplier with symbol $x$ (denoted by $M_{x})$ is defined by $(M_{x} f) \hat{} = x \hat{f}$ , $f \in L^{2} (T)$ . We give some conditions on $x$ ensuring the “interpolation inequality” $∥ M_{x} f ∥_{L^{p}} \leq C ∥ f ∥_{L^{1}}^{α} ∥ M_{x} f ∥_{L^{q}}^{1 - α}$ (here $1 < p < q$ and $α = α (p, q, x)$ is between 0 and 1). In most cases considered $M_{x}$ fails to have stronger $L^{1}$ -regularity properties (e.g. fails to be of weak type (1,1)). The results are applied to prove that for many sets $E \subset Z$ every positive sequence in $ℓ^{2} (E)$ can be majorized by the sequence ${$ $| \hat{f} (n) |}_{n \in E}$ for some continuous funtion $f$ with spectrum...

Cellularity of a space of subgroups of a discrete group

Arkady G. Leiderman, Igor V. Protasov (2008)

Commentationes Mathematicae Universitatis Carolinae

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Given a discrete group $G$ , we consider the set $ℒ (G)$ of all subgroups of $G$ endowed with topology of pointwise convergence arising from the standard embedding of $ℒ (G)$ into the Cantor cube ${0, 1}^{G}$ . We show that the cellularity $c (ℒ (G)) \leq ℵ_{0}$ for every abelian group $G$ , and, for every infinite cardinal $τ$ , we construct a group $G$ with $c (ℒ (G)) = τ$ .

Sidon sets and $I^{0}$ -sets

David Grow (1987)

Colloquium Mathematicae

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On normal subgroups of compact groups

Nikolay Nikolov, Dan Segal (2014)

Journal of the European Mathematical Society

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Among compact Hausdorff groups $G$ whose maximal profinite quotient is finitely generated, we characterize those that possess a proper dense normal subgroup. We also prove that the abstract commutator subgroup $[H, G]$ is closed for every closed normal subgroup $H$ of $G$ .

On the Hausdorff-Young theorem for commutative hypergroups

Sina Degenfeld-Schonburg (2013)

Colloquium Mathematicae

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We study the Hausdorff-Young transform for a commutative hypergroup K and its dual space K̂ by extending the domain of the Fourier transform so as to encompass all functions in $L^{p} (K, m)$ and $L^{p} (K ̂, π)$ respectively, where 1 ≤ p ≤ 2. Our main theorem is that those extended transforms are inverse to each other. In contrast to the group case, this is not obvious, since the dual space K̂ is in general not a hypergroup itself.

Uncertainty principles for integral operators

Saifallah Ghobber, Philippe Jaming (2014)

Studia Mathematica

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The aim of this paper is to prove new uncertainty principles for integral operators with bounded kernel for which there is a Plancherel Theorem. The first of these results is an extension of Faris’s local uncertainty principle which states that if a nonzero function $f \in L ² (ℝ^{d}, μ)$ is highly localized near a single point then (f) cannot be concentrated in a set of finite measure. The second result extends the Benedicks-Amrein-Berthier uncertainty principle and states that a nonzero function $f \in L ² (ℝ^{d}, μ)$ and...

Rearrangement of coefficients of Fourier series on $S U_{2}$

Giancarlo Travaglini (1987)

Colloquium Mathematicae

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Metrization criteria for compact groups in terms of their dense subgroups

Dikran Dikranjan, Dmitri Shakhmatov (2013)

Fundamenta Mathematicae

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According to Comfort, Raczkowski and Trigos-Arrieta, a dense subgroup D of a compact abelian group G determines G if the restriction homomorphism Ĝ → D̂ of the dual groups is a topological isomorphism. We introduce four conditions on D that are necessary for it to determine G and we resolve the following question: If one of these conditions holds for every dense (or $G_{δ}$ -dense) subgroup D of G, must G be metrizable? In particular, we prove (in ZFC) that a compact abelian group determined...

Restriction theorems for the Fourier transform to homogeneous polynomial surfaces in ℝ³

E. Ferreyra, T. Godoy, M. Urciuolo (2004)

Studia Mathematica

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Let φ:ℝ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let Σ = (x,φ(x)): |x| ≤ 1 and let σ be the Borel measure on Σ defined by $σ (A) = \int_{B} χ_{A} (x, φ (x)) d x$ where B is the unit open ball in ℝ² and dx denotes the Lebesgue measure on ℝ². We show that the composition of the Fourier transform in ℝ³ followed by restriction to Σ defines a bounded operator from $L^{p} (ℝ ³)$ to $L^{q} (Σ, d σ)$ for certain p,q. For m ≥ 6 the results are sharp except for some border points.