Vol 241, No 6 (2019)

We obtain a sufficient condition of the weak localizability of a principal submodule in the module of entire functions of exponential type and polynomial growth on the real line. Applications to the problem of the (weak) spectral synthesis in the Schwartz space C^∞ (a; b) are discussed.

We study the Dirichlet series whose domain of convergence is the half-plane, and the sequence of exponents can be extended to a certain “regular” sequence. We prove that the k-orders of the Dirichlet series are the same in all the half-strips whose widths are greater than a certain number called the special density of the distribution of exponents.

We prove that unconditional bases in a functional Hilbert space H have a generating function if and only if the space H is stable. Necessary and sufficient conditions for the stability of spaces adjoint to weighted spaces on an interval are obtained.

We develop an approach to the theory of growth of the class H(????ⁿ) of holomorphic functions in a multidimensional torus ????ⁿ based on the structure of elements of this class and well-known results of the heory of growth of entire functions of several complex variables. This approach is illustrated in the case where the growth of the function g ∈ H(????ⁿ) is compared with the growth of its maximum modulus on the skeleton of the polydisk. The properties of the corresponding characteristics of growth of the functions in the class H(????ⁿ) are studied with their relation to coefficients of the corresponding Laurent series. A comparative analysis of these results and similar assertions of the theory of growth of entire functions of several variables is given.

We describe closed invariant eigenspaces of the Pommmiez operator in the (LF)-space of entire functions of exponential type. This space is topologically equivalent (by means of the Laplace transform) to the strong dual space of all germs of functions that are analytic on a convex, locally closed subset of the complex plane.