A dispersion inequality in the Hankel setting

The aim of this paper is to prove a quantitative version of Shapiro’s uncertainty principle for orthonormal sequences in the setting of Gabor-Hankel theory.     

Uncertainty Principles for the Segal-Bargmann Transform

We recall some properties of the Segal-Bargmann transform; and we establish for this transform qualitative uncertainty principles: local uncertainty principle, Heisenberg uncertainty principle, Donoho-Stark''s uncertainty principle and Matolcsi-Sz\"ucs uncertainty principle.     

On the Prolate Spheroidal Wave Functions and Hardy’s Uncertainty Principle

We prove a weak version of Hardy’s uncertainty principle using properties of the prolate spheroidal wave functions. We describe the eigenvalues of the sum of a time limiting operator and a band limiting operator acting on \(L^2(\mathbb {R})\) . A weak version of Hardy’s uncertainty principle follows from the asymptotic behavior of the largest eigenvalue as the time limit and the band limit approach infinity. An asymptotic formula for this eigenvalue is obtained from its well-known counterpart for the prolate integral operator.     

Energy-Time Uncertainty Principle and Lower Bounds on Sojourn Time

One manifestation of quantum resonances is a large sojourn time, or autocorrelation, for states which are initially localized. We elaborate on Lavine’s time-energy uncertainty principle and give an estimate on the sojourn time. For the case of perturbed embedded eigenstates the bound is explicit and involves Fermi’s Golden Rule. It is valid for a very general class of systems. We illustrate the theory by applications to resonances for time-dependent systems including the AC Stark effect as well as multistate systems.     

Pitt’s Inequality and Logarithmic Uncertainty Principle for the Dunkl Transform on {\mathbb{R}}

We establish Pitt’s inequality and deduce Beckner’s logarithmic uncertainty principle for the Dunkl transform on \({\mathbb{R}}\) . Also, we prove Stein–Weiss inequality for the Dunkl–Bessel potentials.     

Uncertainty principles associated with the offset linear canonical transform

As a time‐shifted and frequency‐modulated version of the linear canonical transform (LCT), the offset linear canonical transform (OLCT) provides a more general framework of most existing linear integral transforms in signal processing and optics. To study simultaneous localization of a signal and its OLCT, the classical Heisenberg's uncertainty principle has been recently generalized for the OLCT. In this paper, we complement it by presenting another two uncertainty principles, ie, Donoho‐Stark's uncertainty principle and Amrein‐Berthier‐Benedicks's uncertainty principle, for the OLCT. Moreover, we generalize the short‐time LCT to the short‐time OLCT. We likewise present Lieb's uncertainty principle for the short‐time OLCT and give a lower bound for its essential support.     

On Hardy’s Uncertainty Principle for Solvable Locally Compact Groups

We establish analogues of Hardy’s uncertainty principle for the Fourier transform on ? for locally compact Abelian groups and for some classes of solvable Lie groups, such as diamond groups and exponential Lie groups with non-trivial centre.     

A Generalized Beurling Theorem for Some Lie Groups

Mathematical Notes - For any real numbers p,q ≥ 1, we present in this paper a (p, q)-generalized version of Beurling’s uncertainty principle for ?n, which largely extends the...     

Uncertainty principles for the weinstein transform

The Weinstein transform satisfies some uncertainty principles similar to the Euclidean Fourier transform. A generalization and a variant of Cowling-Price theorem, Miyachi’s theorem, Beurling’s theorem, and Donoho-Stark’s uncertainty principle are obtained for the Weinstein transform.     

Uncertainty principles for locally compact quantum groups

In this paper, we prove the Donoho–Stark uncertainty principle for locally compact quantum groups and characterize the minimizer which are bi-shifts of group-like projections. We also prove the Hirschman–Beckner uncertainty principle for compact quantum groups and discrete quantum groups. Furthermore, we show Hardy's uncertainty principle for locally compact quantum groups in terms of bi-shifts of group-like projections.     

Uncertainty principles on certain Lie groups

There are several ways of formulating the uncertainty principle for the Fourier transform on ? ⁿ . Roughly speaking, the uncertainty principle says that if a functionf is ‘concentrated’ then its Fourier transform $\tilde f$ cannot be ‘concentrated’ unlessf is identically zero. Of course, in the above, we should be precise about what we mean by ‘concentration’. There are several ways of measuring ‘concentration’ and depending on the definition we get a host of uncertainty principles. As several authors have shown, some of these uncertainty principles seem to be a general feature of harmonic analysis on connected locally compact groups. In this paper, we show how various uncertainty principles take form in the case of some locally compact groups including ? ⁿ , the Heisenberg group, the reduced Heisenberg groups and the Euclidean motion group of the plane.     

Qualitative uncertainty principles for the hypergeometric Fourier transform

We prove an L ^p version of the Donoho–Stark’s uncertainty principle for the hypergeometric Fourier transform on \({\mathbb{R}^d}\). Next, using the ultracontractive properties of the semigroups generated by the Heckman–Opdam Laplacian operator, we obtain an L ^p Heisenberg–Pauli–Weyl uncertainty principle for the hypergeometric Fourier transform on \({\mathbb{R}^d}\).     

Uncertainty principles for the continuous wavelet transform in the Hankel setting

Shapiro’s dispersion and Umbrella theorems are proved for the continuous Hankel wavelet transform. As a side results, we extend local uncertainty principles for set of finite measure to the latter transform.     

On the Connection of Uncertainty Principles for Functions on the Circle and on the Real Line

An uncertainty principle for 2-periodic functions and the classical Heisenberg uncertainty principle are shown to be linked by a limit process. Dependent on a parameter, a function on the real line generates periodic functions either by periodization or sampling. It is proven that under certain smoothness conditions, the periodic uncertainty products of the generated functions converge to the real-line uncertainty product of the original function if the parameter tends to infinity. These results are used to find asymptotically optimal sequences for the periodic uncertainty principle, based either on Theta functions or trigonometric polynomials obtained by sampling B-splines.     

A mesh-independence principle for operators equations and the Steffensen method

In this study we prove the mesh-independence principle via Steffensen’s method. This principle asserts that when Steffensen’s method is applied to a nonlinear equation between some Banach spaces, as well as to some finite-dimensional discretization of that equation, then the behavior of the discretized process is asymptotically the same as that for the original iteration. Local and semilocal convergence results as well as an error analysis for Steffensen’s method are also provided.     

Uncertainty Principles for the Generalized Fourier Transform Associated with Spherical Mean Operator

The aim of this paper is to establish an extension of qualitative and quantitative uncertainty principles for the Fourier transform connected with the spherical mean operator.     

Two geometric interpretations of the multidimensional Hardy uncertainty principle

Hardy's uncertainty principle says that a square integrable function and its Fourier transform cannot be simultaneously arbitrarily sharply localized. We show that a multidimensional version of this uncertainty principle can be best understood in geometrical terms using the fruitful notion of symplectic capacity, which was introduced in the mid-eighties following unexpected advances in symplectic topology (Gromov's non-squeezing theorem). In this geometric formulation, the notion of Fourier transform is replaced with that of polar duality, well-known from convex geometry.     

Radar ambiguity function inL p frame

In modern radar techniques, pulse signal can be replaced by the chirps, which is taken to be signals inL _p space. Wigner’s function and ambiguity functions are constructed in anL _p frame and a partial generalization of the uncertainty principle is given by means of the notion of coherent dual.     

Modifications of the Hurwicz’s decision rule

The Hurwicz’s criterion is one of the classical decision rules applied in decision making under uncertainty as a tool enabling to find an optimal pure strategy both for interval and scenarios uncertainty. The interval uncertainty occurs when the decision maker knows the range of payoffs for each alternative and all values belonging to this interval are theoretically probable (the distribution of payoffs is continuous). The scenarios uncertainty takes place when the result of a decision depends on the state of nature that will finally occur and the number of possible states of nature is known and limited (the distribution of payoffs is discrete). In some specific cases the use of the Hurwicz’s criterion in the scenarios uncertainty may lead to quite illogical and unexpected results. Therefore, the author presents two new procedures combining the Hurwicz’s pessimism-optimism index with the Laplace’s approach and using an additional parameter allowing to set an appropriate width for the ranges of relatively good and bad payoffs related to a given decision. The author demonstrates both methods on the basis of an example concerning the choice of an investment project. The methods described may be used in each decision making process within which each alternative (decision, strategy) is characterized by only one criterion (or one synthetic measure).     

Dunkl Multiplier Operators on a Class of Reproducing Kernel Hilbert Spaces

We study some class of Dunkl multiplier operators; and we establish for them the Heisenberg-Pauli-Weyl uncertainty principle and the Donoho-Stark''s uncertainty principle. For these operators we give also an application of the theory of reproducing kernels to the Tikhonov regularization on the Sobolev-Dunkl spaces.