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The asymptotic connection between approximation by cardinal splines and entire functions of exponential type

机译：基数样条逼近与指数型所有函数之间的渐近联系

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摘要

Entire functions of exponential type and cardinal splines are both fundamental approximating tools for the classes of functions defined on line R. On the one hand, in the 1930s, Krein, Akhieser, Berstein studied the extremal properties of entire functions of exponential type and obtained many outstanding results. On the other hand, in the 1970s, Schoenberg and others investigated the properties of cardinal splines and their research gave another new and powerful tool for the approximation of classesof functions defined on R. Recently, after the concept of average width was introduced by Tikhomirov, much work demonstrated that both of entire functions of exponential type and cardinal splines are optimal in the sense of average width for some fundamental classes of functions defined on R. It is a purpose of the present note to establish some connection between approximating properties of the entire functions of exponential type and cardinal splines.
机译：指数型和基数样条的全部函数都是R行上定义的函数类别的基本近似工具。一方面，在1930年代，Krein，Akhieser，Berstein研究了指数型整个函数的极值性质，并获得了许多出色的结果。另一方面，在1970年代，Schoenberg等人研究了基数样条的特性，他们的研究为逼近R上定义的函数类提供了另一个新的强大工具。最近，在Tikhomirov引入平均宽度的概念之后，许多工作表明，对于R上定义的某些基本功能类别，指数类型和基数样条曲线的全部功能在平均宽度的意义上都是最佳的。本说明的目的是在整体的近似性质之间建立某种联系指数型和基数样条函数。

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