Alexander Ulanovskii

Completeness in L^1(R) of discrete translates

We characterize, in terms of the Beurling-Malliavin density, the discrete spectra Λ ⊂ R for which a generator exists, that is a function φ ∈ L1(R) such that its Λ-translates φ(x − λ), λ ∈ Λ, span L1(R). It is shown that these spectra coincide with the uniqueness sets for certain analytic classes. We also present examples of discrete spectra Λ ⊂ R which do not admit a single generator while they admit a pair of generators.

A few remarks on sampling of signals with small spectrum

Given a bounded set S of small measure, we discuss the existence of sampling sequences for the Paley-Wiener space PWS, which have both densities and sampling bounds close to the optimal ones.

On the Lévy-Raikov-Marcinkiewicz theorem

Abstract Let μ be a finite nonnegative Borel measure. The classical Levy–Raikov–Marcinkiewicz theorem states that if its Fourier transform μ can be analytically continued to some complex half-neighborhood of the origin containing an interval (0,i R ) then μ admits analytic continuation into the strip {t: 0 I t . We extend this result to general classes of measures and distributions, assuming non-negativity only on some ray and allowing temperate growth on the whole line. To cite this article: I. Ostrovskii, A. Ulanovskii, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

On the duality between sampling and interpolation

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.

Non-oscillating Paley-Wiener functions

A non-oscillating Paley-Wiener function is a real entire functionf of exponential type belonging toL 2(R) and such that each derivativef (n),n=0, 1, 2,…, has only a finite number of real zeros. It is established that the class of such functions is non-empty and contains functions of arbitrarily fast decay onR allowed by the convergence of the logarithmic integral. It is shown that the Fourier transform of a non-oscillating Paley-Wiener function must be infinitely differentiable outside the origin. We also give close to best possible asymptotic (asn→∞) estimates of the number of real zeros of then-th derivative of a functionf of the class and the size of the smallest interval containing these zeros.

On a problem of H. Shapiro

Let µ be a real measure on the line such that its Poisson integral M(z) converges and satisfies |M(x+iy)|≤Ae-cyα, y → + ∞, for some constants A, c > 0 and 0 < α ≤ 1. We show that for 1/2 < α ≤ 1 the measure µ must have many sign changes on both positive and negative rays. For 0 < α ≤ 1/2 this is true for at least one of the rays, and not always true for both rays. Asymptotical bounds for the number of sign changes are given which are sharp in some sense.