Uniqueness sets for unbounded spectra

For every set $S?\mathbb{R}$ of finite measure, we construct a system of exponentials ${\{e^{i\lambda t}\}}_{\lambda \in \Lambda }$ which is complete in $L^2(S)$ and such that the set of frequencies Λ has the critical density $D(\Lambda )=\mathrm{mes}(S)/2\pi$.     

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 PW _S , which have both densities and sampling bounds close to the optimal ones.     

The Sturm-Hurwitz Theorem and its Extensions

The following principle is well-known in Harmonic Analysis: If a real function has a spectral gap at the origin then it must have many sign changes. We obtain some sharp estimates showing that the set of positivity of such functions cannot be too small. We also extend the principle above to complex functions: If a complex function has a spectral gap at the origin then the variation of argument of this function must be large.     

Paley�CWiener Functions with a Generalized Spectral Gap

It is well known that real functions whose Fourier transform vanishes around the origin must have many sign changes. We show that a similar phenomenon occurs for real Paley–Wiener functions whose Fourier transform is “smoother” at the origin than elsewhere. We also show that if the Fourier transform of a function is “less smooth” in a neighborhood of the origin than elsewhere, then the function cannot have too many real zeros.     

Interpolation in Bernstein and Paley-Wiener spaces

We consider interpolation of discrete functions by continuous ones with restriction on the size of spectra. We discuss a sharp contrast between the cases of compact and unbounded spectra. In particular we construct ‘universal’ spectra of small measure which deliver positive solution of the interpolation problem in Bernstein spaces for every discrete sequence of knots.     

On the Lévy–Raikov–Marcinkiewicz theorem

Let μ be a finite nonnegative Borel measure. The classical Lévy–Raikov–Marcinkiewicz theorem states that if its Fourier transform $\text{μ}\text{?}$ can be analytically continued to some complex half-neighborhood of the origin containing an interval (0,iR) then $\text{μ}\text{?}$ admits analytic continuation into the strip $\text{{t:}\text{0<}\text{I}\text{t<R}}$. 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).