Beckner type of the logarithmic Sobolev and a new type of Shannon’s inequalities and an application to the uncertainty principle

Kubo, Hideo; Ogawa, Takayoshi; Suguro, Takeshi

doi:10.1090/proc/14350

Cited by 17 publications

(9 citation statements)

References 14 publications

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“…Firstly, we show the Kubo-Ogawa-Suguro inequality and as an application, we derive the Shannon inequality. The following inequality was shown to hold on R n in [KOS19]. Here we extend it to the anisotropic setting, also allowing a choice of any homogeneous quasi-norm.…”

Section: Resultsmentioning

confidence: 93%

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Anisotropic Shannon inequality

Chatzakou¹,

Kassymov²,

Ruzhansky³

2021

Preprint

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Section: Resultsmentioning

confidence: 93%

“…We give two proofs of the Shannon inequality: a direct proof with the best constant, and as a consequence of the Kubo-Ogawa-Suguro inequality [KOS19] that we first extend to the anisotropic setting of homogeneous groups, also allowing a choice of any homogeneous quasi-norm.…”

Section: Introductionmentioning

confidence: 99%

Anisotropic Shannon inequality

Chatzakou¹,

Kassymov²,

Ruzhansky³

2021

Preprint

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“…The inequality () shares a bond with Heisenberg's uncertainty principle, so it is often labeled as the logarithmic uncertainty principle 14 . Over the couple of decades, this inequality has garnered significant attention, which leads to numerous generalizations and refinements 14,16,19,20 …”

Section: Logarithm Local and Entropy‐based Uncertainty Principlesmentioning

confidence: 99%

“…In the sequel, Kubo et al 20 derived another logarithmic Sobolev‐type inequality admitting a dual relation with the Beckner's inequality. () For

f \in {scriptW}_{s}^{p} ((), ℝ)

, the inequality reads

- \int_{ℝ} (||, f false(t false)) \ln ((), \frac{(||, f false(t false))}{false‖ f {false‖}_{1}}) d t \leq \int_{ℝ} (||, f false(t false)) \ln ((), C_{s} ((), 1 + (||, t)^{s})) d t,

where

C_{s} = ({}, \frac{2 \sqrt{π} normalΓ false(1 false/ s false) normalΓ false(1 false/ s^{'} false)}{s normalΓ false(1 false) normalΓ false(1 false/ 2 false)}), \frac{1}{s} + \frac{1}{s^{'}} = 1 .

…”

Section: Logarithm Local and Entropy‐based Uncertainty Principlesmentioning

confidence: 99%

Uncertainty principles for the quadratic‐phase Fourier transforms

Shah

Nisar

Lone

et al. 2021

Math Methods in App Sciences

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The quadratic‐phase Fourier transform (QPFT) is a recent addition to the class of Fourier transforms and embodies a variety of signal processing tools including the Fourier, fractional Fourier, linear canonical, and special affine Fourier transform. In this article, we formulate several classes of uncertainty principles for the QPFT. Firstly, we formulate the Heisenberg's uncertainty principle governing the simultaneous localization of a signal and the corresponding QPFT. Secondly, we obtain some logarithmic and local uncertainty inequalities such as Beckner and Sobolev inequalities for the QPFT. Thirdly, we study several concentration‐based uncertainty principles, including Nazarov's, Amrein–Berthier–Benedicks's, and Donoho–Stark's uncertainty principles. Finally, we conclude the study with the formulation of Hardy's and Beurling's uncertainty principles for the QPFT.

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“…To date, several generalizations, modifications and variations of the harmonic based uncertainty principles have appeared in the open literature, for instance, the logarithmic uncertainty principles (Beckner-type uncertainty principles), entropy-based uncertainty relations, Benedick's uncertainty principles, Nazarov's uncertainty principles, local uncertainty principles and much more [15,29,30,31,32,35]. However, to the best of our knowledge, no such work has been explicitly carried out yet for the continuous shearlet transforms.…”

Section: Introductionmentioning

confidence: 99%

Uncertainty Principles for the Continuous Shearlet Transforms in Arbitrary Space Dimensions

Shah¹,

Tantary²

2019

Preprint

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The aim of this article is to formulate some novel uncertainty principles for the continuous shearlet transforms in arbitrary space dimensions. Firstly, we derive an analogue of the Pitt's inequality for the continuous shearlet transforms, then we formulate the Beckner's uncertainty principle via two approaches: one based on a sharp estimate from Pitt's inequality and the other from the classical Beckner's inequality in the Fourier domain. Secondly, we consider a logarithmic Sobolev inequality for the continuous shearlet transforms which has a dual relation with Beckner's inequality. Thirdly, we derive Nazarov's uncertainty principle for the shearlet transforms which shows that it is impossible for a non-trivial function and its shearlet transform to be both supported on sets of finite measure. Towards the culmination, we formulate local uncertainty principles for the continuous shearlet transforms in arbitrary space dimensions.

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Beckner type of the logarithmic Sobolev and a new type of Shannon’s inequalities and an application to the uncertainty principle

Cited by 17 publications

References 14 publications

Anisotropic Shannon inequality

Anisotropic Shannon inequality

Uncertainty principles for the quadratic‐phase Fourier transforms

Uncertainty Principles for the Continuous Shearlet Transforms in Arbitrary Space Dimensions

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