Support Dependent Weighted Norm Estimates for Fourier Transforms

Paneah, Boris

doi:10.1006/jmaa.1995.1035

Cited by 7 publications

(5 citation statements)

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“…which provides conclusion (40). Next, to prove that g k 's are mutually independent, we denote J k the range of g k on T. So it suffices to prove that for every (a 1 , • • • , a n ) ∈ n+1 k=1 J k , the probability of the random vector (46) holds by the fact that P is a Haar measure.…”

Section: G(x/hmentioning

confidence: 97%

“…Uncertainty principles à la Amrein-Berthier or Logvinenko-Sereda: These assert that if the energy of f is largely concentrated in a "small" set E and that of f is largely concentrated in a "small" set F , then f itself must have small energy (L 2 -norm). See, e.g., [31,21,39,40,28] for statements where "small" means compact, and [1,37,25] for statements where "small" means of finite Lebesgue measure.…”

Section: Introductionmentioning

confidence: 99%

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The norm of the Fourier transform on compact or discrete abelian groups

Madiman,

2016

Preprint

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We calculate the norm of the Fourier operator from L p (X) to L q ( X) when X is an infinite locally compact abelian group that is, furthermore, compact or discrete. This subsumes the sharp Hausdorff-Young inequality on such groups. In particular, we identify the region in (p, q)-space where the norm is infinite, generalizing a result of Fournier, and setting up a contrast with the case of finite abelian groups, where the norm was determined by Gilbert and Rzeszotnik. As an application, uncertainty principles on such groups expressed in terms of Rényi entropies are discussed.

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Section: G(x/hmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

The norm of the Fourier transform on compact or discrete abelian groups

Madiman,

2016

Preprint

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“…This notion has been extensively studied in the case S is a compact set by Logvinenko and Sereda [LS], Paneah [Pa1,Pa2], Havin and Jöricke [HJ] and Kovrijkine [Ko], see also [HJ]. In this case the class of all Σ's for which (S, Σ) is a strong a-pair is characterized.…”

Section: Definitionmentioning

confidence: 99%

“…This notion has been extensively studied in the case S is a compact set by Havin and Jöricke [9], Kovrijkine [10] Logvinenko and Sereda [11], and Paneah [13][14][15], see also [8] for detailed results and the history of the subject. In this case the class of all 's for which (S, ) is a strong a-pair is characterized.…”

Section: Introductionmentioning

confidence: 99%

Nazarov's uncertainty principles in higher dimension

Jaming¹

2007

Journal of Approximation Theory

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In this paper we prove that there exists a constant $C$ such that, if $S,\Sigma$ are subsets of $\R^d$ of finite measure, then for every function $f\in L^2(\R^d)$, $$\int_{\R^d}|f(x)|^2 dx \leq C e^{C \min(|S||\Sigma|, |S|^{1/d}w(\Sigma), w(S)|\Sigma|^{1/d})} (\int_{\R^d\setminus S}|f(x)|^2 dx + \int_{\R^d\setminus\Sigma}|\hat{f}(x)|^2 dx) $$ where $\hat{f}$ is the Fourier transform of $f$ and $w(\Sigma)$ is the mean width of $\Sigma$. This extends to dimension $d\geq 1$ a result of Nazarov \cite{pp.Na} in dimension $d=1$

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“…(i) for all elements f L,n the estimate (1) Ilwfll g(f)llfll is valid with a constant K(f) which does not depend on f;…”

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confidence: 99%