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Free interpolation by entire functions. (Russian) Zbl 0577.30031

In the paper four conditions on the divisor \(D=\{s_ k;q_ k\}(\in {\mathbb{C}}\times {\mathbb{N}})\) are given each being equivalent to D being interpolatory for the class [\(\rho\) (r),h(\(\theta)\)] of entire functions with indicator \(h_ f(\theta)\leq h(\theta)\) under proximate order \(\rho (r)\to \rho >0\), that is, for each sequence \(\lambda_{k,j}\), \(j=1,...,q_ k\), \(k=1,2,..\). such that \[ \overline{\lim}_{k\to \infty,(s_ k;q_ k)\in D}[| s_ k|^{-\rho (| s_ k|)}\ln \sup_{\quad 1\leq j\leq q_ k}(| \lambda_{k,j}| /(j-1)!)-h(\arg s_ k)]\leq 0, \] there exists an \(F\in [\rho (r)\), h(\(\theta)\)] with the property \[ F^{(j-1)}(s_ k)=\lambda_{k,j},\quad j=1,...,q_ k,\quad k=1,2,... \] The conditions are of both analytic and geometric form.

MSC:

30E05 Moment problems and interpolation problems in the complex plane
30D20 Entire functions of one complex variable (general theory)
30D15 Special classes of entire functions of one complex variable and growth estimates
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