Abstract Let $$g\ne 0$$ be an entire function of exponential type in the complex plane $$\mathbb C$$ , and let $${\mathsf Z}=\{{\mathsf z}_k\}_{k=1,2,\dots}$$ be a sequence of points in $$\mathbb C$$ . We give a criterion for the existence of an entire function $$f\ne 0$$ of exponential type which vanishes on $${\mathsf Z}$$ and satisfies the constraint $$ \ln |f(iy)|\le \ln |g(iy)|+o(|y|),\qquad y\to \pm\infty. $$ Our results generalize and develop joint results of P. Malliavin and L. A. Rubel. Applications to multipliers for entire functions of exponential type, to analytic functionals and their convolutions in the complex plane, and to the completeness problem for exponential systems in spaces of locally analytic functions on compact spaces in terms of the widths of these spaces are given.

中文翻译：

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具有增长约束的指数型全函数的零点分布沿线

摘要 令 $$g\ne 0$$ 为复平面 $$\mathbb C$$ 中指数型的完整函数，令 $${\mathsf Z}=\{{\mathsf z}_k\}_ {k=1,2,\dots}$$ 是 $$\mathbb C$$ 中的点序列。我们给出了一个完整的指数类型函数 $$f\ne 0$$ 的存在标准，该函数在 $${\mathsf Z}$$ 上消失并满足约束 $$ \ln |f(iy)|\le \ln |g(iy)|+o(|y|),\qquad y\to \pm\infty。$$ 我们的结果概括并发展了 P. Malliavin 和 LA Rubel 的联合结果。给出了对指数型整函数的乘法器、复平面中的解析泛函及其卷积的应用，以及对紧空间上局部解析函数空间中指数系统的完备性问题的应用，这些问题就这些空间的宽度而言是这样的。 11． ． ．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．