Convergence and summability of Gabor expansions at the Nyquist density

It is well known that Gabor expansions generated by a lattice of Nyquist density are numerically unstable, in the sense that they do not constitute frame decompositions. In this paper, we clarify exactly how bad such Gabor expansions are, we make it clear precisely where the edge is between enough and too little, and we find a remedy for their shortcomings in terms of a certain summability method. This is done through an investigation of somewhat more general sequences of points in the time-frequency plane than lattices (all of Nyquist density), which in a sense yields information about the uncertainty principle on a finer scale than allowed by traditional density considerations. An important role is played by certain Hilbert scales of function spaces, most notably by what we call the Schwartz scale and the Bargmann scale, and the intrinsically interesting fact that the Bargmann transform provides a bounded invertible mapping between these two scales. This permits us to turn the problems into interpolation problems in spaces of entire functions, which we are able to treat.     

Analogues of the V. I. Smirnov spaces for nonintegral exponents

Questions regarding the completeness and the minimality of systems of exponentials on curves are investigated. The analogues of the classical Hardy spaces are considered.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 50, pp. 115–127, 1988.     

Direct and inverse multichannel scattering problems

We consider small oscillations of a system of pairwise interacting particles in an external field near a stable equilibrium. The system is assumed to consist of finitely many channels, i.e., semi-infinite linear chains of particles, attached to a scatterer, which is a finite system of interacting particles. Direct and inverse scattering problems are considered. In particular, an algorithm finding characteristics of the channels on the basis of scattering data is given.     

Mass-spectrometric investigation of solid lubricating coatings subjected to friction in a deep vacuum

The life of solid lubricating coatings of the VNII NP type, based on molybdenum disulfides and various binders, has been experimentally investigated under deep vacuum conditions (10^–8–5 · 10^–9 torr) together with the composition of the gas released in the friction process. It is shown that both under atmospheric conditions and in a deep vacuum the life of the coatings depends on the chemical nature of the film-former. The depth of the vacuum also has an important influence on the life of the coatings, both the mechanism and the end result of this effect depending to a large extent on the physicochemical properties of the bind. On the interval 10^–1–10^–2 torr there is a sudden change in the life of the coating.Physicotechnical Institute of Low Temperatures, Academy of Sciences of the Ukrainian SSR, Khar'kov. Translated from Mekhanika Polimerov, No. 6, pp. 1070–1075, November–December, 1970.     

Effect of proton-electron irradiation on the strength of a glass-reinforced plastic and its components

The action of protons and electrons on the properties of glass strands under vacuum conditions has been experimentally investigated. The experimental procedure is described. The effects of irradiation, strain rate, and reduced temperature and pressure on the properties of glass reinforcement and glass-reinforced plastics are estimated.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev; Physicotechnical Institute of Low Temperatures, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 4, pp. 755–760, July–August, 1970.     

On Summation of Nonharmonic Fourier Series

Let a sequence \(\Lambda \subset {\mathbb {C}}\) be such that the corresponding system of exponential functions \({\mathcal {E}}(\Lambda ):=\left\{ {\text {e}}^{i\lambda t}\right\} _{\lambda \in \Lambda }\) is complete and minimal in \(L^2(-\pi ,\pi )\), and thus each function \(f\in L^2(-\pi ,\pi )\) corresponds to a nonharmonic Fourier series in \({\mathcal {E}}(\Lambda )\). We prove that if the generating function \(G\) of \(\Lambda \) satisfies the Muckenhoupt \((A_2)\) condition on \({\mathbb {R}}\), then this series admits a linear summation method. Recent results show that the \((A_2)\) condition cannot be omitted.