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.     

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.     

Weak Uncertainty Principles on Fractals

We use the analytic tools such as the energy, and the Laplacians defined by Kigami for a class of post-critically finite (pcf) fractals which includes the Sierpinski gasket (SG), to establish some uncertainty relations for functions defined on these fractals. Although the existence of localized eigenfunctions on some of these fractals precludes an uncertainty principle in the vein of Heisenberg’s inequality, we prove in this article that a function that is localized in space must have high energy, and hence have high frequency components. We also extend our result to functions defined on products of pcf fractals, thereby obtaining an uncertainty principle on a particular type of non-pcf fractal.     

Symmetrization inequalities in the fractional case and Besov embeddings

We prove new extended forms of the Pólya-Szegö symmetrization principle in the fractional case. As a consequence we determine new results for rearrangement invariant hulls of generalized Besov spaces.     

Higher-order symmetrization inequalities and applications

We prove new extended forms of the Pólya-Szegö symmetrization principle. As a consequence new sharp embedding theorems for generalized Sobolev and Besov spaces are proved.     

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.     

Uncertainty principles for the continuous Hankel Wavelet transform

The aim of this paper is to prove Heisenberg-type uncertainty principles for the continuous Hankel wavelet transform. We also analyse the concentration of this transform on sets of finite measure. Benedicks-type uncertainty principle is given.     

Fixed Point Theorems for Mean Nonexpansive Mappings in CAT(0) Spaces

In this article, we prove the existence of fixed points and the demiclosed principle for mean nonexpansive mappings in Cartan, Alexandrov and Toponogov(0) spaces. We also obtain a Δ-convergence theorem and a strong convergence theorem of Ishikawa iteration for mean nonexpansive mappings in Cartan, Alexandrov and Toponogov(0) spaces.     

Symmetrization inequalities and Sobolev embeddings

We prove new extended forms of the Pólya-Szegö symmetrization principle. As a consequence new sharp embedding theorems for generalized Besov spaces are proved, including a sharpening of the limiting cases of the classical Sobolev embedding theorem. In particular, a surprising self-improving property of certain Sobolev embeddings is uncovered.

     

Quantitative uncertainty principles for the short time Fourier transform and the radar ambiguity function

Logarithmic uncertainty principle and Beckner’s uncertainty principle in terms of entropy are proved for the short time Fourier transform and the radar ambiguity function, also a Heisenberg inequality for generalized dispersion and Price’s local uncertainty principle are obtained.     

A Modified Uncertainty Principle for Two-Sided Quaternion Fourier Transform

This paper proposes a new uncertainty principle for the two-sided quaternion Fourier transform. This uncertainty principle describes that the spread of a quaternion-valued function and its two-sided quaternion Fourier transform (QFT) are inversely proportional. We obtain a tighter lower bound about the product of the spread of quaternion signal in the QFT domain. As a consequence, we show that the quaternionic Gabor filters minimize the uncertainty.     

Short-distance space-time structure and black holes in string theory: a short review of the present status

We briefly review the present status of string theory from the viewpoint of its implications on the short-distance space-time structure and black hole physics. Special emphases are given on two closely related issues in recent developments towards nonperturbative string theory, namely, the role of the space-time uncertainty relation as a qualitative but universal characterization of the short-distance structure of string theory and the microscopic formulation of black-hole entropies. We will also suggest that the space-time uncertainty relation can be an underlying principle for the holographic property of M theory, by showing that the space-time uncertainty relation naturally explains the UV/IR relation used in a recent derivation of the holographic bound for D3 brane by Susskind and Witten.     

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.     

Stochastic weighted variational inequalities in non-pivot Hilbert spaces with applications to a transportation model

A class of stochastic weighted variational inequalities in non-pivot Hilbert spaces is proposed. Existence and continuity results are proved. These theoretical results play a prominent role in order to introduce a new weighted transportation model with uncertainty. Moreover, they allow to establish the equivalence between the random weighted equilibrium principle and a suitable stochastic weighted variational inequality. At the end, a numerical model is discussed.     

Estimating maxima of generalized cross ambiguity functions,and uncertainty principles

In certain signal processing problems, it is customary to estimate parameters in distorted signals by approximating what is termed a cross ambiguity function and estimating where it attains its maximum modulus. To unify and generalize these procedures, we consider a generalized form of the cross ambiguity function and give error bounds for estimating the parameters, showing that these bounds are lower if we maximize the real part rather than the modulus. We also reveal a connection between these bounds and certain uncertainty principles, which leads to a new type of uncertainty principle.     

Uncertainty principles on two step nilpotent Lie groups

We extend an uncertainty principle due to Cowling and Price to two step nilpotent Lie groups, which generalizes a classical theorem of Hardy. We also prove an analogue of Heisenberg inequality on two step nilpotent Lie groups.     

Large deviation theory for a homogenized and “corrected” elliptic ODE

We study a one-dimensional elliptic problem with highly oscillatory random diffusion coefficient. We derive a homogenized solution and a so-called Gaussian corrector. We also prove a “pointwise” large deviation principle (LDP) for the full solution and approximate this LDP with a more tractable form. Applications to uncertainty quantification are considered.     

Rank-deficient submatrices of Fourier matrices

We consider the maximal rank-deficient submatrices of Fourier matrices with order a power of a prime number. We do this by considering a hierarchical subdivision of these matrices into low rank blocks. We also explore some connections with the fast Fourier transform (FFT), and with an uncertainty principle for Fourier transforms over finite Abelian groups.     

Support size restrictions on time-frequency representations of functions on finite abelian groups

We obtain uncertainty principles for finite abelian groups that relate the cardinality of the support of a function to the cardinality of the support of its short–time Fourier transform. These uncertainty principles are based on well–established uncertainty principles for the Fourier transform. In terms of applications, the uncertainty principle for the short–time Fourier transform implies the existence of a class of equal norm tight Gabor frames that are maximally robust to erasures. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Topological symmetry,background independence and matrix models

We illustrate a physical situation in which topological symmetry, its breakdown, space-time uncertainty principle, and background independence may play an important role in constructing and understanding matrix models. First, we show that the space-time uncertainty principle of string may be understood as a manifestation of the breakdown of the topological symmetry in the large N matrix model. Next, we construct a new type of matrix models which is a matrix model analog of the topological Chern-Simons and BF theories. It is of interest that these topological matrix models are not only completely independent of the background metric but also have nontrivial “p-brane” solutions as well as commuting classical space-time as the classical solutions. In this paper, we would like to point out some elementary and unsolved problems associated to the matrix models, whose resolution would lead to the more satisfying matrix model in future.