Equivalent conditions of strong incompleteness of exponential system

Abstract

We study interpolating sequences in the Pavlov — Korevaar  — Dixon sense ($\Omega$-interpolation sequences) and generalizations, as well as the approximative properties of exponential systems with corresponding exponents. For instance, the interpolation problem is of interest in the class of entire functions of exponential type determined by some growing majorant in the convergent class (non-quasianalytic weight). In a narrower class, when the majorant possessed the concavity property, a similar problem was completely solved Berndtsson, but in the case when the interpolation nodes are natural numbers. He obtained the solvability criterion of this interpolation problem. The corresponding criterion for an arbitrary increasing sequence of positive nodes was recently obtained by R.A.Gaisin. In 2021 he also proved the criterion of the interpolation ($W$-interpolating) in the case of an arbitrary non-quasianalytic weight. As in works by A.I.Pavlov, J.Korevaar and M.Dixon, we found a close relation between the interpolation property of sequences and Macintyre problem. It was also shown that if the sequence of real numbers is $\Omega$-interpolating, then the corresponding exponential system is strongly incomplete (minimal) with respect to the rectangles; in the case of the $W$-interpolation property the strong incompleteness (minimality) holds with respect to the vertical strips. However the conditions of $\Omega$-interpolation property proposed by A.M.Gaisin in 1991 were a bit unsatisfactory since there were not visual enough.