Let $l$ be a continuous function on $\mathbb{R}$ increasing to $+\infty$, and $\varphi$ be a positive function on $\mathbb{R}$. We proved that the condition$$\varliminf_{x\to+\infty}\frac{\varphi(\ln[x])}{\ln x}>0$$is necessary and sufficient in order that for any complex sequence $(\zeta_n)$ with $n(r)\ge l(r)$, $r\ge r_0$, and every set $E\subset\mathbb{R}$ which is unbounded from above there exists an entire function $f$ having zeros only at the points $\zeta_n$ such that$$\varliminf_{r\in E,\ r\to+\infty}\frac{\ln\ln M_f(r)}{\varphi(\ln n_\zeta(r))\ln l^{-1}(n_\zeta(r))}=0.$$Here $n(r)$ is the counting function of $(\zeta_n)$, and $M_f(r)$ is the maximum modulus of $f$.
С помощью конформных отображений односвязной области на единичный круг построены системы биортогональных функций, которые являются базисами в пространствах аналитических функций. Ключевые слова: биортогональная система функций, конформные отображения, уравнения Гельмгольца.
SYSTEMS OF BIORTHOGONAL NONLINEAR COMBINATIONS OF EXPONENTIAL FUNCTIONSBy applying conforming mappings of simply connected domain onto the unit disk, the systems of biorthogonal functions are constructed that are the bases in the spaces of analytic functions.
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