On weighted spaces of holomorphic functions of several variables

We generalize the results of [11] and [12] for the unit ball $ \mathbb{B}_d $ \mathbb{B}_d of ℂ ^d . In particular, we show that under the weight condition (B) the weighted H ^∞-space on $ \mathbb{B}_d $ \mathbb{B}_d is isomorphic to ℓ^∞ and thus complemented in the corresponding weighted L ^∞-space. We construct concrete, generalized Bergman projections accordingly. We also consider the case where the domain is the entire space ℂ ^d . In addition, we show that for the polydisc $ \mathbb{D}^d $ \mathbb{D}^d ^d , the weighted H ^∞-space is never isomorphic to ℓ^∞.     

Hayman conjecture for p-adic meromorphic functions in several variables

In this paper we prove a generalized version of the Hayman Conjecture for p-adic meromorphic functions in several variables.     

The star function for meromorphic functions of several complex variables

We define an analogue of the Baernstein star function for a meromorphic function f in several complex variables. This function is subharmonic on the upper half-plane and encodes some of the main functionals attached to f. We then characterize meromorphic functions admitting a harmonic star function.     

Some boundary properties of holomorphic functions of several complex variables

A local uniqueness theorem and analogs of the theorem on removable singularities under the hypothesis of boundedness are proved for functions satisfying the tangential Cauchy-Riemann conditions on hypersurfaces in Cⁿ. The results can be interpreted as giving certain boundary properties of holomorphic functions of several complex variables.Translated from Matematicheskie Zametki, Vol. 23, No. 6, pp. 839–844, June, 1978.In conclusion I express my thanks to B. V. Shabat for his contant interest in this work.     

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.     

Approximation of classes of smooth functions of several variables

The paper deals with imbedding theorems and harmonic approximation for classes of periodic functions of several variables. Necessary and sufficient conditions for imbedding are found for various classes of smooth functions. The paper contains asymptotic estimates for fractional order derivatives of multidimensional Dirichlet and Favard kernels, for Fourier best approximation and for the Kolmogorov diameter of smooth function classes.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 10, pp. 207–226, 1984.