Volume 7 Issue #4 (December 2015)

  • Abanin A.V. Sampling sets for the space of holomorphic functions of polynomial growth in a ball. Pp. 3 - 14
    Abstract: We develop a new approach to study sampling sets in the space of holomorphic functions of polynomial growth in a ball in the sense of Horowitz, Korenblum, and Pinchuk (Michigan Math. J., 44:2, 1997). It is based on involving weakly sufficient sets for intermediate inductive limits. By means of this approach we obtain a complete topological description of such sets and, as an application of this description, some new properties of sampling sets of general and special type are established. In particular, the main result of the above mentioned paper on sampling sequences of circles is extended to the multi-dimensional case.
  • Bërdëllima A. About a conjecture regarding plurisubharmonic functions. Pp. 154 - 165
    Abstract: In this work we present Khabibullin's conjecture in its different equivalent forms. Applying the concept of the integral operator, we transform the original conjecture into a new form which proves to be helpful in studying it by means of the Laplace transform. Using Laplace transform of integral inequalities, we are able to show the uniqueness of a solution that satisfies both inequalities with identity. Furthermore we provide a new proof of Khabibullin's theorem by methods of the Laplace transform and contour integration from complex analysis. However, this method of transform fails to prove the conjecture and a brief reasoning is provided.