arXiv subjects
Publications and source records attributed to Alexander Olevskii.
We survey our recent result that for every continuous function there is an absolutely continuous homeomorphism such that the composition has a uniformly converging Fourier expansion. We mention the history of the problem, orginally stated by Luzin, and some details of the proof.
We show that for every continuous function there exists an absolutely continuous homeomorphism of the circle such that the Fourier series of the composition converges uniformly. This resolves a problem set by N. N. Luzin.
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.
Every set $Λ\subset R$ such that the sum of $δ$-measures sitting at the points of $Λ$ is a Fourier quasicrystal, is the zero set of an exponential polynomial with imaginary frequencies.
We prove that every pair of exponential polynomials with imaginary frequencies generates a Poisson-type formula.
The classical Szegö--Kolmogorov Prediction Theorem gives necessary and sufficient condition on a weight $w$ on the unite cirlce $T$ so that the exponentials with positive integer frequences span the weighted space $L^2(T,w)$. We consider the problem how many of these exponentials can be removed while still keeping the completeness property.
We construct a trigonometric series converging to zero everywhere on a subsequence, with coefficients tending to zero. We show that any such series must satisfy that the subsequence is very sparse, and that the support of the related distribution is quite large.
We consider the algebras $M_p$ of Fourier multipliers and show that every bounded continuous function $f$ on $\mathbb R^d$ can be transformed by an appropriate homeomorphic change of variable into a function that belongs to $M_p(\mathbb R^d)$ for all $p$, $1<p<\infty$. Moreover, under certain assumptions on a family $K$ of continuous functions, one change of variable will suffice for all $f\in K$. A similar result holds for functions on the torus $\mathbb T^d$. This may be contrasted with the known result on the Wiener algebra, related to Luzin's rearrangement problem.
One says that a pair of sets $(S,Q)$ in $\mathbb{R}$ is 'annihilating' if no function can be concentrated on $S$ while having its Fourier transform concentrated on $Q$. One uses to distinguish between weak and strong annihilation types. It is well known that if both sets $S$ and $Q$ are of finite measure then they are strongly annihilating. In this paper we prove that if $S$ is a set of finite measure with periodic gaps, and $Q$ is a set of density zero, then weak annihilation holds. On the other hand a counter-example is constructed, showing that strong annihilation, in general, does not.
We prove that a positive-definite measure in $\mathbb{R}^n$ with uniformly discrete support and discrete closed spectrum, is representable as a finite linear combination of Dirac combs, translated and modulated. This extends our recent results where we proved this under the assumption that also the spectrum is uniformly discrete. As an application we obtain that Hof's quasicrystals with uniformly discrete diffraction spectra must have a periodic diffraction structure.
We construct a Schwartz function $φ$ such that for every exponentially small perturbation of integers $Λ$, the set of translates $\{φ(t-λ), λ\inΛ\}$ spans the space $L^p(R)$, for every $p > 1$. This result remains true for more general function spaces $X$, whose norm is "weaker" than $L^1$ (on bounded functions).
We construct a function on the real line supported on a set of finite measure whose spectrum has density zero.
It is well-known that entire functions whose spectrum belongs to a fixed bounded set $S$ admit real uniformly discrete uniqueness sets $Λ$. We show that the same is true for much wider spaces of continuous functions. In particular, Sobolev spaces have this property whenever $S$ is a set of infinite measure having "periodic gaps". The periodicity condition is crucial. For sets $S$ with randomly distributed gaps, we show that the uniformly discrete sets $Λ$ satisfy a strong non-uniqueness property: Every discrete function $c(λ)\in l^2(Λ)$ can be interpolated by an analytic $L^2$-function with spectrum in $S$.
Given a set $\mathcal{S}$ of positive measure on the circle and a set of integers $Λ$, one may consider the family of exponentials $E\left(Λ\right):=\left\{ e^{iλt}\right\}_{λ\inΛ}$ and ask whether it is a Riesz sequence in the space $L^{2}\left(\mathcal{S}\right)$. We focus on this question in connection with some arithmetic properties of the set of frequencies. Improving a result of Bownik and Speegle, we construct a set $\mathcal{S}$ such that $E\left(Λ\right)$ is never a Riesz sequence if $Λ$ contains arbitrary long arithmetic progressions of length $N$ and step $\ell=O\left(N^{1-\varepsilon}\right)$. On the other hand, we prove that every set $\mathcal{S}$ admits a Riesz sequence $E\left(Λ\right)$ such that $Λ$ does contain arbitrary long arithmetic progressions of length $N$ and step $\ell=O\left(N\right)$.
We present a simple method based on the stability and duality of the properties of sampling and interpolation, which allows one to substantially simplify the proofs of some classical results.
We proved recently that a measure on R, whose support and spectrum are both uniformly discrete sets, must have a periodic structure. Here we show that this is not the case if the support and the spectrum are just discrete closed sets.
We characterize the measures on R which have both their support and spectrum uniformly discrete. A similar result is obtained in R^n for positive measures.
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 $L^2(S)$