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Alexander Olevskii

Publications and source records attributed to Alexander Olevskii.

At least 37 records · Page 2Linked to original sources
arXiv2013

On multi-dimensional sampling and interpolation

The paper discusses sharp sufficient conditions for interpolation and sampling for functions of n variables with convex spectrum. When n=1, the classical theorems of Ingham and Beurling state that the critical values in the estimates from above (from below) for the distances between interpolation (sampling) nodes are the same. This is no longer true for n>1. While the critical value for sampling sets remains constant, the one for interpolation grows linearly with the dimension.

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arXiv2013

Approximation of discrete functions and size of spectrum

Let $Λ$ be a uniformly discrete set and $S$ be a compact set in $R$. We prove that if there exists a bounded sequence of functions in Paley--Wiener space $PW_S$, which approximates $δ-$functions on $Λ$ with $l^2-$error $d$, then measure($S$)$\geq 2π(1 - d^2)D^+(Λ)$. This estimate is sharp for every $d$. Analogous estimate holds when the norms of approximating functions have a moderate growth, and we find a sharp growth restriction.

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arXiv2013

Singular distributions and symmetry of the spectrum

This is a survey of the "Fourier symmetry" of measures and distributions on the circle in relation with the size of their support. Mostly it is based on our paper arxiv:1004.3631 and a talk given by the second author in the 2012 Abel symposium.

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arXiv2012

Perturbing PLA

We proved earlier that every measurable function on the circle, after a uniformly small perturbation, can be written as a power series (i.e. a series of exponentials with positive frequencies), which converges almost everywhere. Here we show that this result is basically sharp: the perturbation cannot be made smooth or even Hölder. We discuss also a similar problem for perturbations with lacunary spectrum.

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arXiv2011

On Beurling's sampling theorem in $\R^n$

We present an elementary proof of the classical Beurling sampling theorem which gives a sufficient condition for sampling of multi-dimensional band-limited functions.

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arXiv2011

Singular distributions, dimension of support, and symmetry of Fourier transform

We study the "Fourier symmetry" of measures and distributions on the circle, in relation with the size of their supports. The main results of this paper are: (1) A one-side extension of Frostman's theorem, which connects the rate of decay of Fourier transform of a distribution with the Hausdorff dimension of the support; (2) A construction of compacts of "critical" size, which support distributions (even pseudo-functions) with anti-analytic part belonging to l^2. We also give examples of non-symmetry which may occur for measures with "small" support. A number of open questions are stated.

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arXiv2008

Frame-type families of translates

We construct a uniformly discrete, and even sparse, sequence of real numbers $Λ=\{λ_n\}$ and a function g in $L^2(R)$, such that for every q>2, every function f in $L^2(R)$ can be approximated with arbitrary small error by a linear combination $\sum c_n g(t-λ_n)$ with an $l_q$ estimate of the coefficients: \|\{c_n\}\|_{l_q}\leq C(q)\|f\|. This can not be done for q=2, according to [2].

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arXiv2008

$L^2-$interpolation with error and size of spectra

Given a compact set $S$ and a uniformly discrete sequence $\La$, we show that "approximate interpolation" of delta--functions on $\La$ by a bounded sequence of $L^2-$functions with spectra in $S$ implies an estimate on measure of $S$ through the density of $\La$.

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arXiv2005

An "Analytic" Version of Menshov's Representation Theorem

Every measurable function f on the circle can be represented as a sum of harmonics with positive spectrum, converging in measure. For convergence almost everywhere this is not true. We discuss several other subsets of Z for which one might get a Menshov type representation converging almost everywhere or in measure.

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arXiv2005

Menshov representation spectra

A Menshov spectrum is a subset of the integers that is sufficient for representing every measurable function as an almost-everywhere converging trigonometric (non-Fourier) sum. In this language the celebrated "Menshov representation theorem" states that Z is a Menshov spectrum. In this paper we construct 1) Menshov spectra that are almost exponentially sparse 2) that are almost squares. Then we show that the positive integers are not a Menshov spectrum but are a Menshov spectrum in measure, and repeat 1) and 2) in the analytic settings.

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arXiv2005

Random Menshov spectra

We show that a spectrum of frequencies obtained by a random perturbation of the integers allows one to represent any measurable function on R by an almost everywhere converging sum of harmonics almost surely.

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arXiv2005

Maximal smoothness of the anti-analytic part of a trigonometric null series

We proved recently math.CA/0510403 that the anti-analytic part of a trigonometric series, converging to zero almost everywhere, may be square integrable on the circle. Here we prove that it can even be infinitely differentiable, and we characterize precisely the possible degree of smoothness in terms of the rate of decrease of the Fourier coefficients. This sharp condition might be viewed as a "new quasi-analyticity".

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arXiv2005

A null series with small anti-analytic part

We show that it is possible for a square integrable function on the circle, which is a sum of an almost everywhere convergent series of exponentials with positive frequencies, to not belong to the Hardy space. A consequence in the uniqueness theory is obtained.

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arXiv2005

Is PLA large?

We examine the class of functions representable by an analytic sum (by which we mean a trigonometric sum involving only positive frequencies) converging almost everywhere. We show that it is dense but that it is first category and has zero Wiener measure.

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