Polynomiality criterion for entire functions of several complex variables

The “radial” polynomiality criterion for entire functions of several complex variables is proved. Translated fromMatermaticheskie Zametki, Vol. 66, No. 4, pp. 500–502, October, 1999.     

On the decomposition of infinitely divisible characteristic functions of several variables

Summary We prove here some theorems describing infinitely divisible characteristic functions defined on R ⁿ which have positive Poisson spectrum and belong to the class I ₀ of characteristic functions without indecomposable factor. These theorems are generalizations to the case of several variables of results due to I.V. Ostrovskiy in the case of one variable.This work was supported by the National Science Foundation under grant NSF-GP-6175.     

Fractional integrals of periodic functions of several variables

In this paper it has been systematically studied the imbedding properties of fractional integral operators of periodic functions of several variables, and isomorphic properties of fractional integral operators in the spaces of Lipschitz continuous functions. It has also been proved that the space of fractional integration, the space of Lipschitz continuous functions and the Sobolev space are identical in L²-norm. Results obtained here are not true for fractional integrals (or Riesz potentials) in ℝ ⁿ . Supported by NNSFC     

Two related extremal problems for entire functions of several variables

We establish a relationship between the Logan problem for functions whose Fourier transform is supported in a centrally symmetric convex closed subset of ℝ ^m and whose mean value on ℝ ^m is nonnegative and the Chernykh problem on the optimal point for the Jackson inequality inL ₂(ℝ ^m ), which relates the best approximation of a function by the class of entire functions of exponential type to the first modulus of continuity. Both problems are solved exactly in several cases. Translated fromMatematicheskie Zametki, Vol. 66, No. 3, pp. 336–350, September, 1999.     

Approximation of classes of periodic functions of several variables

We study classes of periodic functions of several variables with bounded generalized derivative in the metric of the space L_p. We obtain order estimates of deviations of Fourier sums, which are constructed depending on the behavior of functions that define the operator of generalized differentiation. We find estimates of the Kolmogorov widths, which are realized by the Fourier sums that are constructed.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 5, pp. 662–672, May, 1992.     

Proximate order and type of entire functions of several complex variables

Letp be an odd prime,K a field, andG _K its absolute Galois group. It is shown thatK isp-adically closed if and only ifG _K is isomorphic to an open subgroup of . It is also shown that if , withq=p ^r, thenK has a non-trivial henselian valuation.     

Approximation of classes of periodic functions of several variables by nuclear operators

Translated from Matematicheskii Zametki, Vol. 47, No. 3, pp. 32–41, March, 1990.     

On uniform approximations of almost periodic functions by entire functions of finite degree

We give a new proof of the well-known Bernshtein statement that, among entire functions of degree which realize the best uniform approximation (of degree ) of a periodic function on (–,), there is a trigonometric polynomial of degree . We prove an analog of the mentioned Bernshtein statement and the Jackson theorem for uniform almost periodic functions with arbitrary spectrum.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 9, pp. 1274–1279, September, 1995.