Mathematical Sciences: Square Summable Power Series

本研究者在1984年证明了Ludwig Bieberbach在1916年提出的关于单位圆盘上表示内射解析函数的幂级数的系数的估计。这个证明包含在一个关于平方可和幂级数的手稿中，它激发了这个论证，并为应用铺平了道路。对这一论点的核查需要对当时的苏联进行广泛的访问，在此期间，证明与原来的内容分开了。对证据的模糊认识不足以作为进一步研究的指导。人们普遍认为一个时代已经结束，这种观点甚至掩盖了进行这种研究的必要性。证明所产生的研究机会被低估了。对平方可和幂级数理论的改进处理的必要性变得明显起来。这种治疗所需的研究是本项目的目的。数学对全球经济的重要性很难评估，因为它是通过科学和工程的中介进行的。平方可和幂级数是复分析的一个方面，也是泛函分析的一个方面，它通过线性系统理论与工程最密切相关。这种相互作用可以在比伯巴赫猜想的证明中看到，这些证明出现在工程期刊上。在高等数学文本中出现的比伯巴赫猜想的证明中可以看到更高层次的相互作用。技术和科学界以这种方式表达了对证据信息的需要。但与此同时，它也暴露出即使在专家中，现有的证明知识也是不足的。为了克服这些困难，证明的作者需要作出很大的努力。

deBranges 9622445 In 1984 the present investigator obtained a proof of a conjecture, made in 1916 by Ludwig Bieberbach, estimating the coefficients of power series which represent injective analytic functions in the unit disk. The proof was included in a manuscript on square summable power series which motivated the argument and prepared the way for applications. The verification on the argument required an extensive visit to the then Soviet Union during which the proof was separated from its original context. A dim perception of the proof was conveyed which was insufficient as a guide for further research. The need for such research was even obscured by the commonly accepted view that an era had come to a close. The research opportunities generated by the proof were underestimated. The need of an improved treatment of the theory of square summable power series became evident. The research required for such a treatment is the aim of the present project. The significance of mathematics for the global economy is difficult to assess because it proceeds through intermediaries in science and engineering. Square summable power series are an aspect of complex analysis, and also of functional analysis, which is most closely related to engineering through the theory of linear systems. The interplay is seen in proofs of the Bieberbach conjecture which appear in engineering journals. An interplay at a higher level is seen in proofs of the Bieberbach conjecture which appear in advanced mathematical texts. The technical and scientific community has in this way expressed its need for information about the proof. But it has at the same time exposed the inadequacy of existing knowledge of the proof even among experts. A substantial effort is required from the author of the proof to surmount these difficulties.