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 periodic decomposition of entire functions of several variables

Consider the difference equation on \({{\mathbb{C}}^n}\) : \({\Delta_{c_1}\Delta_{c_2}\cdots \Delta_{c_m}f(z)=0}\) , where \({c_1,c_2, \ldots, c_m \in \mathbb{C}^n\backslash \{0\}}\) and \({\Delta_{c_k}f(z)=f(z+c_k)-f(z)}\) . In this paper, we assume that c ₁, . . . , c _m are pairwise linearly independent over \({\mathbb{R}}\) , except for the case m = 2. Firstly, we establish a general representation of its entire solutions. Secondly, under the condition L ≤ n, we give a necessary and sufficient conditions for entire solutions to have representations as a sum of c _k -periodic entire functions. Here L is the maximum integer such that \({c_{k_1},c_{k_2}, \ldots,c_{k_L}}\) are pairwise linearly independent over \({\mathbb{C}}\) for some k ₁,k ₂, . . . ,k _L . Finally, we show that every entire solution has a representation as a sum of c _k -periodic meromorphic functions.     

Higher order monotonic functions of several variables

Multivariate functions with a specific degree of higher order monotonicity in each variable are introduced. When normalized, they turn out to form a simplex whose extreme points are precisely the tensor products of their univariate counterparts. Under natural conditions on the degrees these functions will be shown to operate on each other.     

Weighted approximation of entire functions of exponential type of several variables by trigonometric polynomials

Letf be an entire function (in Cⁿ) of exponential type for whichf(x)=0(?(x)) on the real subspace \(\mathbb{R}^w (\phi \geqslant 1,{\mathbf{ }}\mathop {\lim }\limits_{\left| x \right| \to \infty } \phi (x) = \infty )\) and ?δ>0?C_δ>0 $$\left| {f(z)} \right| \leqslant C_\delta \exp \left\{ {h_s (y) + S\left| z \right|} \right\},z = x + iy$$ where h, (x)=sup〈3, x〉, S being a convex set in ?ⁿ. Then for any ?, ?>0, the functionf can be approximated with any degree of accuracy in the form p→ \(\mathop {\sup }\limits_{x \in \mathbb{R}^w } \frac{{\left| {P(x)} \right|}}{{\varphi (x)}}\) by linear combinations of functions x→expi〈λx〉 with frequenciesX belonging to an ?-neighborhood of the set S.     

On approximation and generalized type of analytic functions of several complex variables

In the present paper, we study the polynomial approximation of analytic functions of several complex variables. The characterizations of generalized type of analytic functions of several complex variables have been obtained in terms of approximation and interpolation errors.     

A relationship between the maximum modulus and Taylor coefficients of entire functions of several complex variables

We present results on the relationship between the growth of the maximum modulus and the decay of Taylor coefficients of entire functions of several variables. The results are obtained by two different methods, the first of which had been proposed earlier by Oskolkov for the one-dimensional case, and the second is based on the use of the Legendre-Jung-Fenchel conjugates of the weight functions. Attention is mainly paid to the characterization of the growth of entire functions with respect to the conjunction of variables; however, some results are obtained for the case in which there is different growth with respect to different variables. Translated fromMatematicheskie Zametki, Vol. 62, No. 2, pp. 238–258, August, 1997. Translated by N. K. Kulman     

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.     

New generating functions involving several complex variables

In the present paper the authors derive a number of new classes of generating functions for certain multiple sequences of functions of several complex variables. These generating-function relationships involve either Taylor or Laurent series and provide extensions of several known results given earlier by H.M. Srivastava, R.G. Buschman, and D. Zeitlin. The authors also apply their main results to obtain the corresponding generating functions for the generalized (Lauricella's) hypergeometric functions of several complex variables, which were introduced elsewhere by H.M. Srivastava and M.C. Daoust.     

Properties of Bernstein functions of several complex variables

A multidimensional generalization of the class of Bernstein functions is introduced and the properties of functions belonging to this class are studied. In particular, a new proof of the integral representation of Bernstein functions of several variables is given. Examples are considered.     

Bounded universal functions in one and several complex variables

We show how to obtain functions that are universal for the ball of , where . The existence of our functions will follow from universality criteria, but we also show how to construct them. Then we study the connection between certain interpolating sequences, runaway automorphisms, and the existence of universal functions on domains in .      

Interpolation and basicity: Several variables entire functions of exponential type

The paper investigates the interpolation and basicity problem in Banach spaces of entire functions of exponential type in several variable. Some sufficient conditions for solvability of these problem and the explicit analytic form of the solutions are obtained.