Mathematical Sciences: Square Summable Power Series

删除9622445 在1984年本调查获得了证明的猜想，在1916年提出的路德维希Bieberbach，估计系数的幂级数代表单射解析函数的单位磁盘。证明是包括在一份手稿平方求和电源系列的动机的论点和准备的方式应用。为了验证这一论点，需要对当时的苏联进行广泛的访问，在此期间，证据与其原始背景分开。对证据的模糊认识被传达出来，这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，这甚至掩盖了进行这种研究的必要性。由证明产生的研究机会被低估了。需要一个改进的治疗理论的平方和幂级数变得明显。本项目的目的是进行这种治疗所需的研究。 数学对全球经济的重要性很难评估，因为它通过科学和工程的中介进行。平方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密切相关。这种相互作用可以在工程期刊上出现的比伯巴赫猜想的证明中看到。一个相互作用在更高的水平被认为是证明比伯巴赫猜想出现在先进的数学文本。技术和科学界以这种方式表示需要有关证据的信息。但它同时也暴露了即使在专家中，对证明的现有知识的不足。需要证明作者作出巨大努力来克服这些困难。

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.