arXiv subjects
Publications and source records attributed to Shahaf Nitzan.
We revisit Pavlov's characterization for Riesz bases of exponentials and study the corresponding lower Riesz basis bounds. In particular, this approach allows us to improve on known estimates for the bounds in Avdonin's theorem regarding average perturbations, and Levin's theorem regarding zeroes of sine-type functions.
A.Olevskii and A.Ulanovskii obtained a scale of density results, which correspond to how well an exponential system approximates a uniformly minimal system over a compact set. We extend their result in several directions. First, we show that it holds for any set of positive finite measure. Next, we consider a relaxed version of frames, which we term 'uniformly complete systems', and obtain an analogues scale of density results for such systems.
We obtain sharp embeddings from the Sobolev space $W^{k,2}_0(-1,1)$ into the space $L^1(-1,1)$ and determine the extremal functions. This improves on a previous estimate of the sharp constants of these embeddings due to Kalyabin.
We conjecture a new correlation-like inequality for percolation probabilities and support our conjecture with numerical evidence and a few special cases which we prove. This inequality, if true, implies that there is no percolation at criticality on the Euclidean lattice, for any dimension bigger than one.
We show that there exists a bounded subset of R such that no system of exponentials can be a Riesz basis for the corresponding Hilbert space. An additional result gives a lower bound for the Riesz constant of any putative Riesz basis of the two dimensional disk.
We apply a new approach to the study of the density of Gabor systems, and obtain a simple and straightforward proof of Ramanathan and Steger's well known result regarding the density of Gabor frames and Gabor Riesz sequences. Moreover, this point of view allows us to extend this result in several directions. The approach we use was first observed by A. Olevskii and the third author in their study of exponential systems, here we develop and simplify it further.
We study the persistence probability of a centered stationary Gaussian process on $\mathbb{Z}$ or $\mathbb{R}$, that is, its probability to remain positive for a long time. We describe the delicate interplay between this probability and the behavior of the spectral measure of the process near zero and infinity.
Let $f$ be a zero-mean continuous stationary Gaussian process on ${\mathbb R}$ whose spectral measure vanishes in a $δ$-neighborhood of the origin. Then the probability that $f$ stays non-negative on an interval of length $L$ is at most $e^{-cδ^2 L^2}$ with some absolute $c>0$ and the result is sharp without additional assumptions.
We formulate and prove finite dimensional analogs for the classical Balian-Low theorem, and for a quantitative Balian-Low type theorem that, in the case of the real line, we obtained in a previous work. Moreover, we show that these results imply their counter-parts on the real line.
We prove that every finite union of rectangles in $R^d$ admits a Riesz basis of exponentials.
For every set $S$ of finite measure in $\mathbb{R}$ we construct a discrete set of real frequencies $Λ$ such that the exponential system $\{\exp(iλt),λ\inΛ\}$ is a frame in $L^2(S)$
We show that any finite union of intervals supports a Riesz basis of exponentials
We study functions generating Gabor Riesz bases on the integer lattice. The classical Balian-Low theorem restricts the simultaneous time and frequency localization of such functions. We obtain a quantitative estimate that extends both this result and other related theorems.
Given a set $S$ of small measure, we discuss existence of sampling sequences for the Paley-Wiener space $PW_S$, which have both densities and sampling bounds close to the optimal ones.
The de Branges spaces of entire functions generalise the classical Paley-Wiener space of square summable bandlimited functions. Specifically, the square norm is computed on the real line with respect to weights given by the values of certain entire functions. For the Paley-Wiener space, this can be chosen to be an exponential function where the phase increases linearly. As our main result, we establish a natural geometric characterisation, in terms of densities, for real sampling and interpolating sequences in the case when the derivative of the phase function merely gives a doubling measure on the real line. Moreover, a consequence of this doubling condition, is that the spaces we consider are one component model spaces. A novelty of our work is the application to de Branges spaces of techniques developed by Marco, Massaneda and Ortega-Cerdá for Fock spaces satisfying a doubling condition analogue to ours.
We look at the time-frequency localisation of generators of lattice Gabor systems. For a generator of a Riesz basis, this localisation is described by the classical Balian-Low theorem. We establish Balian-Low type theorems for complete and minimal Gabor systems with a frame-type approximation property. These results describe how the best possible localisation of a generator is limited by the degree of control over the coefficients in approximations given by the system, and provide a continuous transition between the classical Balian-Low conditions and the corresponding conditions for generators of complete and minimal systems. Moreover, this holds for the non-symmetric generalisations of these theorems as well.