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Alexander Ulanovskii

Publications and source records attributed to Alexander Ulanovskii.

24 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.

math.CA↗
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.

math.CA↗
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.

math.CA↗
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 $\La$.

math.CA↗
arXiv2003

Completeness in $L^1(R)$ of discrete translates

We characterize, in terms of the Beurling-Malliavin density, the discrete spectra $Λ\subset\R$ for which a generator exists, that is a function $ϕ\in L^1(\R)$ such that its $Λ$-translates $ϕ(x-λ), λ\inΛ$, span $L^1(\R)$. It is shown that these spectra coincide with the uniqueness sets for certain analytic classes. We also present examples of discrete spectra $Λ\subset\R$ which do not admit a single generator while they admit a pair of generators.

math.CA↗