Wegner estimates and localization for continuum Anderson models with some singular distributions

We give a simple geometric proof of Wegner's estimate which leads to a variety of new results on localization for multi-dimensional random operators.     

Localization near fluctuation boundaries via fractional moments and applications

We present a short, new, self-contained proof of localization properties of multi-dimensional continuum random Schödinger operators in the fluctuation boundary regime. Our method is based on the recent extension of the fractional moment method to continuum models in [2] but does not require the random potential to satisfy a covering condition. Applications to random surface potentials and potentials with random displacements are included.     

An uncertainty principle for finite frames

We consider the notion of uncertainty for finite frames. Using a difference operator inspired by the Gauss-Hermite differential equation we obtain a time-frequency measure for finite frames. We then find the minimizer of the measure over all equal norm Parseval frames, dependent on the dimension of the space and the number of elements in the frame. Next we show that given a frame one can find the dual frame that minimizes this time-frequency measure, generalizing some work of Daubechies, Landau and Landau to the finite case and extending some recent work on Sobolev duals for finite frames.     

An uncertainty principle for Hankel transforms

There exists a generalized Hankel transform of order on , which is based on the eigenfunctions of the Dunkl operator

For this transform coincides with the usual Fourier transform on . In this paper the operator replaces the usual first derivative in order to obtain a sharp uncertainty principle for generalized Hankel transforms on . It generalizes the classical Weyl-Heisenberg uncertainty principle for the position and momentum operators on ; moreover, it implies a Weyl-Heisenberg inequality for the classical Hankel transform of arbitrary order on

     

An uncertainty principle on homogeneous trees

Let be a homogeneous tree of degree . We prove an uncertainty principle in this setting regarding ``exponentially decreasing' functions on trees whose Fourier transforms have a ``deep zero'.

     

An uncertainty principle for function fields

In a recent paper, Granville and Soundararajan (2007) [5] proved an “uncertainty principle” for arithmetic sequences, which limits the extent to which such sequences can be well-distributed in both short intervals and arithmetic progressions. In the present paper we follow the methods of Granville and Soundararajan (2007) [5] and prove that a similar phenomenon holds in Fq[t].     

An inequality associated with the uncertainty principle

We prove \(\left\| F \right\|_{2,\Omega } \leqslant c({\rm T} \Omega )\left\| f \right\|_{A{}_T} \) , whereF is the Fourier transform off,||F||_2,Ω is theL ²-norm ofF on \([ - \Omega ,\Omega ],\left\| f \right\|_{A{}_T} \) is the absolutely convergent Fourier series norm for 2T-periodic functions, and $$c(T\Omega ) = (\frac{1}{\pi }\int\limits_{ - T\Omega }^{T\Omega } {\frac{{\sin ^2 \gamma }}{{\gamma ^2 }}d\gamma } )^{1/2} $$ Analogous inequalities, depending on prolate spheroidal wave functions, are more difficult to prove and their constants are less explicit.     

An uncertainty principle for the basic Bessel transform

The aim of this paper is to prove an uncertainty principle for the basic Bessel transform of order . In order to obtain a sharp uncertainty principle, we introduce and study a generalized q-Bessel-Dunkl transform which is based on the q-eigenfunctions of the q-Dunkl operator newly given by:

An uncertainty principle for convolution operators on discrete groups

Consider a discrete group and a bounded self-adjoint convolution operator on ; let be the spectrum of . The spectral theorem gives a unitary isomorphism between and a direct sum , where , and is a regular Borel measure supported on . Through this isomorphism corresponds to multiplication by the identity function on each summand. We prove that a nonzero function and its transform cannot be simultaneously concentrated on sets , such that and the cardinality of are both small. This can be regarded as an extension to this context of Heisenberg's classical uncertainty principle.

     

An entropy-based uncertainty principle for a locally compact abelian group

We classify all functions on a locally compact, abelian group giving equality in an entropy inequality generalizing the Heisenberg Uncertainty Principle. In particular, for functions on a real line, we proof a conjecture of Hirschman published in 1957.     

Signal representation,uncertainty principles and localization measures

The following collection of articles addresses one of the most basic problems in signal and image processing, namly the search for function systems (basis, frames, dictionaries) which allow efficient representations of certain classes of signals/images. Such representations are essential for decomposition and synthesis of signals, hence they are at the core of almost any application (coding, compression, pattern matching, feature extraction, classification, etc.) in this field. Accordingly, this is one of the best-studied topics in data analysis and a multitude of different concepts also addressing discretization/algorithmic issues has been investigated in this context. The starting point for reviving activities in this field was a recently rediscovered inconsistency in the concept of constructing optimally localized basis functions by minimizing uncertainty principles. In this short introductory note, we shortly sketch the basic dilemma, which was the starting point for this research approximately three years ago. However, the subsequent investigations presented in this collection of papers cover a much wider range of more general localization measures, discretization concepts as well as discussing algorithmic efficiency and stability.     

Spectral gaps,additive energy,and a fractal uncertainty principle

We obtain an essential spectral gap for n-dimensional convex co-compact hyperbolic manifolds with the dimension \({\delta}\) of the limit set close to \({{n-1\over 2}}\). The size of the gap is expressed using the additive energy of stereographic projections of the limit set. This additive energy can in turn be estimated in terms of the constants in Ahlfors–David regularity of the limit set. Our proofs use new microlocal methods, in particular a notion of a fractal uncertainty principle.     

The Kadison-Singer problem and the uncertainty principle

We compare and contrast the Kadison-Singer problem to the Uncertainty Principle via exponential frames. Our results suggest that the Kadison-Singer problem, if true, is in a sense a stronger version of the Uncertainty Principle.

     

Weak uncertainty principle for fractals, graphs and metric measure spaces

We develop a new approach to formulate and prove the weak uncertainty inequality, which was recently introduced by Okoudjou and Strichartz. We assume either an appropriate measure growth condition with respect to the effective resistance metric, or, in the absence of such a metric, we assume the Poincaré inequality and reverse volume doubling property. We also consider the weak uncertainty inequality in the context of Nash-type inequalities. Our results can be applied to a wide variety of metric measure spaces, including graphs, fractals and manifolds.

     

Spacetime non-commutativity,generalized uncertainty principle and the fine structure constant

Quantum gravitational effects and spacetime non-commutativity should affect the value of the fine structure constant. In this paper, using generalized uncertainty principle, we calculate the modified fine structure constant in non-commutative spacetime.     

An uncertainty principle on the group of rigid motions of the plane

Let M(2) be the group of rigid motions of the plane. The Fourier transform and the Plancherel formula on M(2) can be explicitly given by the general group representation theory. Using this fact, we establish a kind of uncertainty principle on M(2). The result can easily be generalized to higher dimensional cases. An application of the result yields an uncertainty principle on the Euclidean spaces obtained by R. S. Strichartz.