Variations on uncertainty principles for integral operators

Ghobber, Saifallah

doi:10.1080/00036811.2013.816685

Cited by 23 publications

(20 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The continuous wavelet transform, which is a time-scale representation, was also shown to have non-compact support in [45]. Ghobber and Jaming [20,21] derived uncertainty principles for arbitrary integral operators (Fourier, Dunkl, Clifford transforms, etc) which have bounded kernels and satisfy a Plancherel theorem. A sharp version of the Beurling uncertainty principle was proven by B. Demange for the ambiguity function [12].…”

Section: Theorem 2 If Wρ ∈ W(r 2n ) Then Wρ Is Uniformly Continuous mentioning

confidence: 99%

“…The inequality between the first and the last term is, upon exponentiation, the Heinig-Smith uncertainty principle [25]. As a consequence of inequality (20) for the Wigner distribution and the refined RSUP (11), we derive the following Hirschman-Shannon inequality (Theorem 23):…”

Section: Introductionmentioning

confidence: 99%

“…This can be stated in the following terms: if | f | 2 and |Fh f | 2 have finite covariance matrices, then they have well defined entropies and inequality (20) holds. Moreover, we have equalities throughout if and only if f is a Gaussian.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Refinement of the Robertson–Schrödinger Uncertainty Principle and a Hirschman–Shannon Inequality for Wigner Distributions

Dias

Gosson

Prata

2018

J Fourier Anal Appl

View full text Add to dashboard Cite

We propose a refinement of the Robertson-Schrödinger uncertainty principle (RSUP) using Wigner distributions. This new principle is stronger than the RSUP. In particular, and unlike the RSUP, which can be saturated by many phase space functions, the refined RSUP can be saturated by pure Gaussian Wigner functions only. Moreover, the new principle is technically as simple as the standard RSUP. In addition, it makes a direct connection with modern harmonic analysis, since it involves the Wigner transform and its symplectic Fourier transform, which is the radar ambiguity function. As a by-product of the refined RSUP, we derive inequalities involving the entropy and the covariance matrix of Wigner distributions. These inequalities refine the Shanon and the Hirschman inequalities for the Wigner distribution of a mixed quantum state ρ. We prove sharp estimates which critically depend on the purity of ρ and which are saturated in the Gaussian case.

show abstract

Section: Theorem 2 If Wρ ∈ W(r 2n ) Then Wρ Is Uniformly Continuous mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Refinement of the Robertson–Schrödinger Uncertainty Principle and a Hirschman–Shannon Inequality for Wigner Distributions

Dias

Gosson

Prata

2018

J Fourier Anal Appl

View full text Add to dashboard Cite

show abstract

“…The aim of this paper is to continue the study of the uncertainty principle to a very general class of integral operators, which has been started in [11,12]. The transforms under consideration are integral operators T with bounded kernels K and for which there is a Parseval Theorem.…”

mentioning

confidence: 99%

“…The proof of Inequality (1.10) can be obtained by combining a Nash-type inequality[12, Proposition 2.2] and a Carlson-type inequality [12,Proposition 2.3], while the proof of Inequality (1.9) can be obtained from either the Faris-type local uncertainty inequalities [11,Theorem A], or from the fact that the Benedicks-Amrein-Berthier uncertainty inequality [11,Theorem B]. Theorem 1.1 can be refined for orthonormal sequence in L 2 (Ω, µ).…”

mentioning

confidence: 99%

Fourier-Like Multipliers and Applications for Integral Operators

Ghobber

2018

Complex Anal. Oper. Theory

Self Cite

View full text Add to dashboard Cite

Timelimited functions and bandlimited functions play a fundamental role in signal and image processing. But by the uncertainty principles, a signal cannot be simultaneously time and bandlimited. A natural assumption is thus that a signal is almost time and almost bandlimited. The aim of this paper is to prove that the set of almost time and almost bandlimited signals is not excluded from the uncertainty principles. The transforms under consideration are integral operators with bounded kernels for which there is a Parseval Theorem. Then we define the wavelet multipliers for this class of operators, and study their boundedness and Schatten class properties. We show that the wavelet multiplier is unitary equivalent to a scalar multiple of the phase space restriction operator. Moreover we prove that a signal which is almost time and almost bandlimited can be approximated by its projection on the span of the first eigenfunctions of the phase space restriction operator, corresponding to the largest eigenvalues which are close to one.

show abstract

A variation of the Lq,p‐uncertainty inequalities of Heisenberg‐type for symmetric q‐integral transforms

Nefzi

2022

Math Methods in App Sciences

View full text Add to dashboard Cite

The aim of this paper is to prove new uncertainty inequalities of Heisenberg-type for a q-integral operator  q with a bounded kernel. To do so, we prove a Nash and Carlson's inequalities for this transformation on  q,1 ∩ q,p (Ω, |𝜔(x)|d q x) for 1 < p ≤ 2, on  q,2 ∩ q,p (Ω, |𝜔(x)|d q x) for 1 < p < 2, and on  q,p 1 ∩ q,p 2 (Ω, |𝜔(x)|d q x) for 1 < p 1 < p 2 ≤ 2. Our results can be applied to the the q-Fourier-cosine transform, the q-Dunkl transform, and the q-Bessel-Fourier transform.

show abstract

Variations on uncertainty principles for integral operators

Cited by 23 publications

References 30 publications

A Refinement of the Robertson–Schrödinger Uncertainty Principle and a Hirschman–Shannon Inequality for Wigner Distributions

A Refinement of the Robertson–Schrödinger Uncertainty Principle and a Hirschman–Shannon Inequality for Wigner Distributions

Fourier-Like Multipliers and Applications for Integral Operators

A variation of the Lq,p‐uncertainty inequalities of Heisenberg‐type for symmetric q‐integral transforms

Contact Info

Product

Resources

About