Nina Bari

Nina Bari的父母是Olga Eduardovna Seligson和Karl Adolfovich Bari。Karl Adolfovich是一名医生。Nina Karlovna在莫斯科的L O Vyazemska女子中学接受教育，在那里她展现出巨大的数学潜力。当时俄国为女孩提供的教育质量低于男孩，因此女孩的毕业考试标准也低于男孩。然而，Bari想展示她在数学上的能力，于是她采取了极不寻常的一步，参加了男孩的毕业考试。1917年俄国十月革命后，布尔什维克党推行了重大教育改革，其中与Bari最相关的是大学首次向女性和男性同时开放。革命前，想要继续深造的女性就读于莫斯科女子学院等院校。1918年是女性被允许进入莫斯科国立大学的第一年，这与Bari获得入学资格恰好吻合。事实上，该大学在革命造成的动荡期间曾关闭，1918年是它重新开放的第一年。她进入莫斯科国立大学数学物理系时，该系正成为一个极具活力的研究环境。

在莫斯科数学学派中，她受到了尼古拉·卢津的影响，但他只是当时在大学任教的多位世界级数学家之一，包括谢尔盖·阿列克谢耶维奇·恰普雷金、德米特里·叶戈罗夫、维亚切斯拉夫·瓦西里耶维奇·斯捷潘诺夫和尼古拉·叶戈罗维奇·茹科夫斯基。在本科生中，帕维尔·萨穆伊洛维奇·乌雷松正处于学习的最后一年，并仍然是数学学派从事研究的一部分。这个强大的数学团体，学生们称之为“Luzitania”，于1920年加入了帕维尔·亚历山德罗夫[4]：-

Nina Bari的学生时代恰逢莫斯科数学学派的迅速发展，这对苏联整个数学的后续繁荣产生了非常大的影响。这是莫斯科实变函数学派密集发展的时期，由尼古拉·卢津领导，在他的指导下，Nina Bari还在本科时就开始自己的数学工作。事实上，正是在那时，她首次开始研究三角级数的唯一性问题。

要了解为什么“Luzitania”如此成功，我们只需看看帕维尔·亚历山德罗夫当时所作的评论：-

在那些年里见到尼古拉·卢津，就是见到一种所谓与科学灵感相通的关系的展现。我从他那里不仅学到了数学，还受到了一课：什么造就了真正的学者，以及一位大学教授能够并且应该是什么样的人。此外，我还看到，对科学的追求和对年轻人的科学训练是同一活动的两个方面——即学者的活动。

Bari于1921年毕业，她仅用三年就完成了四年的课程，成为莫斯科国立大学第一位女毕业生。她开始在若干机构教授数学，包括莫斯科林学院、莫斯科工业学院和斯维尔德洛夫共产主义学院。然而，她的目标是成为大学教师，并且在本科期间体验过“卢西塔尼亚”的兴奋之后，她申请了研究奖学金。她赢得了为数不多的奖学金之一，加入了莫斯科国立大学力学与数学研究所。Bari在该研究所开始研究，尼古拉·卢津是她的学位论文导师，但她继续保留她的教职。她从事三角级数理论的研究，她的主要结果发表在她的第一篇论文Sur l'unicite du developpement trigonometrique Ⓣ（论三角展开的唯一性）中，该文由巴黎的Académie des Sciences于1923年发表。同年，她在莫斯科数学会的一次演讲中报告了这些结果（成为第一位在该学会演讲的女性）。除了来自尼古拉·卢津的支持外，她还受到迪米特里·缅绍夫的强烈影响，后者曾与尼古拉·卢津一起进行研究，但于1922年回到莫斯科国立大学担任讲师。Bari的学位论文On the uniqueness of trigonometric expansions于1925年提交，学位在她于1926年1月答辩后授予。这篇学位论文的卓越质量使她获得了Glavnauk奖。此后，Bari成为莫斯科数学与力学研究所的研究助理。她在1926年发表了更多论文，即Sur la representation analytique d'une classe de fonctions continues Ⓣ（论一类连续函数的解析表示）和（与迪米特里·缅绍夫合作）Sur l'integrale de Lebesgue-Stieltjes et les functions absolument continues de fonctions absolument continues Ⓣ（论勒贝格-汤姆斯·斯蒂尔吉斯积分和绝对连续函数的绝对连续函数的积分），这两篇论文都荣幸地也出现在Académie des Sciences的出版物中。1927年，她发表了她学位论文结果的详细证明，这些结果她已在第一篇论文[4]中宣布：-

Nina Bari这第一项工作已经证明了她的巨大数学才能，因为它包含了三角级数理论中几个非常困难问题的解答，这些问题近来一直吸引着许多杰出数学家的注意。

在1927-29年间，她在巴黎的索邦大学和法兰西学院度过了一段时间，听雅克·阿达马的讲座。在利沃夫参加波兰数学大会后，她还参加了1928年在博洛尼亚举行的国际数学家大会，并在会上作了邀请报告Sur la structure analytique d'une fonction continue arbitraire Ⓣ（论任意连续函数的解析框架）。此后，她获得了洛克菲勒奖学金，资助了第二次为期一年的巴黎访问。1929年，她回到莫斯科国立大学，1932年成为正教授。三年后，她被授予相当于物理-数学科学博士的学位。这一学位通常是在提交学位论文后授予的，但Bari被免除了学位论文要求[4]：-

……因为她已经被公认为专门研究实变函数理论的领先数学家之一。

Bari从莫斯科国立大学毕业的那一年，维克托·弗拉基米罗维奇·涅梅茨基进入该大学攻读数学。他们成为亲密的朋友，不仅分享数学兴趣，还热爱在山中徒步。他们最终结婚了。由尼古拉·卢津领导的莫斯科国立大学的活跃学派在20世纪20年代的最后几年开始失去动力，因为他专注于撰写专著。他在20世纪30年代初离开大学，前往苏联科学院工作，并从1935年起领导弗拉基米尔·安德烈耶维奇·斯捷克洛夫研究所的实变函数理论系。这使得Bari和迪米特里·缅绍夫接管了他领导函数论学派的角色，他们在20世纪40年代做到了这一点。

Bari 是一位杰出的研究型数学家，撰写了五十多篇研究论文。[4] 的作者们写道：-

Nina Bari 是苏联实变函数学派公认的领袖之一，也是这一学科在莫斯科大学的杰出代表，而该学科与莫斯科大学联系如此紧密。她的众多论文对函数论的诸如三角级数理论、正交级数与基理论等基本分支的发展产生了巨大影响。她的若干研究被公认为经典，例如她关于三角级数唯一性理论以及关于函数叠加的论文。……她的许多论文，例如关于唯一性理论和关于基的论文，已成为函数论中新研究方向的起点，这些方向后来得到了深入发展……

让我们特别提及她于1949年发表的论文 The uniqueness problem of the representation of functions by trigonometric series（俄文）。Antoni Zygmund 对它描述如下：-

……对用三角级数表示函数的唯一性理论中已有结果的一份详尽综述，许多情况下附有完整证明。

这篇论文被译成英文，并于1951年由 美国数学会 出版为一部90页的著作。在 [1] 中，她最后一部出版物——一部936页的关于三角级数的研究专著——被描述如下：-

所涵盖主题的范围和深度相当广泛，她在该领域的大部分工作都包括在内。但即便在如此长篇的专著中，这一主题也无法被完全穷尽。……它已成为专攻函数论和三角级数理论的数学家的标准参考文献。

本书的十五章是：基本概念与定理；约瑟夫·傅里叶系数；一个Fourier series在一点处的收敛性；连续函数的约瑟夫·傅里叶级数；一个约瑟夫·傅里叶级数在一个集合上的收敛与发散；在一个小测度集合上“修正”一个函数；约瑟夫·傅里叶级数的可和性；共轭三角级数；约瑟夫·傅里叶级数的绝对收敛性；系数递减的正弦与余弦级数；缺项级数；一般三角级数的收敛与发散；一般三角级数的绝对收敛性；函数在三角级数中展开的唯一性问题；以及用三角级数表示函数。

Bari 还为师范学院编写了教材，如 Higher Algebra（1932年）和 The Theory of Series（1936年）。她编辑了 尼古拉·卢津 的全集，并担任两份重要数学期刊的编辑。她还将 昂利·勒贝格 关于积分的著名著作译成俄文。

她死于跌落在莫斯科地铁的一列火车前。有人声称这是由于十一年前尼古拉·卢津去世所引起的抑郁而自杀。她的一个学生P L Ul'yanov在她去世后写道[1]：-

N K Bari的过早去世是苏联数学的一大损失，也是所有认识她的人的一大不幸。Bari作为一个活泼、直率、有着无穷无尽乐观储备的人的形象，将永远留在所有认识她的人的心中。

最后让我们给出一些关于她个性和数学之外兴趣的说明。我们已经提到她对徒步旅行的热爱，这使她进入了高加索、帕米尔和天山山脉中极其困难的地形。她对音乐和艺术的热爱包括芭蕾舞、文学和诗歌。米哈伊尔·阿列克谢耶维奇·拉夫连季耶夫和L A Lyusternik写道[3]：-

……活泼的气质，直率，有着无穷无尽的青春活力……

Nina Bari's parents were Olga Eduardovna Seligson and Karl Adolfovich Bari. Karl Adolfovich was a medical doctor. Nina Karlovna was educated at a private school, L O Vyazemska's High School for Girls in Moscow, where she showed great potential in mathematics. At this time the education available for girls in Russia was of lower quality than that of boys and, as a consequence, the final examinations for girls were of a lower standard than those for boys. However, Bari wanted to show her proficiency in mathematics so she took the highly unusual step of taking the boy's leaving examinations. After the October 1917 Revolution in Russia the Bolshevik Party introduced major educational reforms, the most relevant to Bari being that for the first time universities were open to women as well as men. Before the Revolution, women who wanted further education attended colleges such as the Moscow Women's College. The year 1918 was the first in which women were allowed to enter Moscow State University and this coincided nicely with Bari achieving the entrance qualifications. In fact the University had been closed during the disruption caused by the Revolution and 1918 was the first year in which it reopened. She entered the Faculty of Mathematics and Physics at Moscow State University at a time when it was becoming an extremely vigorous research environment.

In the Moscow School of Mathematics she came under the influence of Nikolai Nikolaevich Luzin but he was only one of a number of world-class mathematicians teaching at the university at this time, including Sergei Alekseevich Chaplygin, Dimitri Fedorovich Egorov, Vyacheslaw Vassilievich Stepanov and Nikolai Egorovich Zhukovsky. Among the undergraduate students Pavel Samuilovich Urysohn was in his final year of study and remained part of the School of Mathematics undertaking research. This strong mathematical group, which the students called 'Luzitania', was joined by Pavel Sergeevich Aleksandrov in 1920 [4]:-

Nina Bari's student years coincided with the rapid growth of the Moscow Mathematical school, which was to be a very great influence in the subsequent blossoming of the whole of mathematics in the Soviet Union. This was the period of intensive development of the Moscow real variable school, headed by N N Luzin, under whose guidance Nina Bari started her own mathematical work while still an undergraduate. In fact, it was then that she first began to study the uniqueness problem for trigonometric series.

To see why 'Luzitania' was so successful, we need look no further than the comments that Aleksandrov made about this time:-

To see Luzin in those years was to see a display of what is called an inspired relationship to science. I learnt not only mathematics from him, I received also a lesson in what makes a true scholar and what a university professor can and should be. Then, too, I saw that the pursuit of science and the training of young people in it are two facets of one and the same activity - that of a scholar.

Bari graduated in 1921, and having completed the four-year course in only three years, she became the first woman to graduate from Moscow State University. She began teaching mathematics at a number of institutions, the Moscow Forestry Institute, the Moscow Polytechnic Institute, and the Sverdlov Communist Institute. However, her aim was to become a university teacher and having sampled the excitement of 'Luzitania' while an undergraduate she applied for a research fellowship. She won one of a small number of available scholarships and joined the Research Institute of Mechanics and Mathematics at Moscow State University. Bari began research at the Institute, with Luzin as her thesis advisor, but continued to hold her teaching posts. She undertook research on the theory of trigonometrical series and her major results were announced in her first paper Sur l'unicite du developpement trigonometrique Ⓣ published by the Académie des Sciences in Paris in 1923. In the same year she presented the results in a lecture to the Moscow Mathematical Society (becoming the first woman to address the Society). In addition to support from Luzin, she was strongly influenced by Dmitrii Evgenevich Menshov who had undertaken research with Luzin but returned to Moscow State University as a lecturer in 1922. Bari's thesis On the uniqueness of trigonometric expansions was submitted in 1925 and the degree awarded after she defended her thesis in January 1926. The exceptional quality of this thesis led to her being awarded the Glavnauk Prize. After this Bari became a research assistant at the Institute of Mathematics and Mechanics in Moscow. She published further papers in 1926, namely Sur la representation analytique d'une classe de fonctions continues Ⓣ and (with D E Menshov) Sur l'integrale de Lebesgue-Stieltjes et les functions absolument continues de fonctions absolument continues Ⓣ both of which had the distinction of also appearing in the Académie des Sciences publications. In 1927 she published detailed proofs of the results of her thesis which she had announced in her first paper [4]:-

Already this first piece of work by Nina Bari testified to her great mathematical talent, since it included the solution of several very difficult problems in the theory of trigonometric series that had lately been engaging the attention of many outstanding mathematicians.

During 1927-29 she spent time at the Sorbonne and the Collège de France in Paris, attending lectures by Jacques Hadamard. After attending the Polish Mathematical Congress in Lwów, she also attended the International Congress of Mathematicians in Bologna in 1928 at which she gave the invited lecture Sur la structure analytique d'une fonction continue arbitraire Ⓣ. Following this, she was awarded a Rockefeller fellowship which funded a second year-long visit to Paris. In 1929 she returned to Moscow State University where she became a full professor in 1932. Three years later she was awarded the equivalent of a D.Sc. in Physical-Mathematical Sciences. This degree, normally conferred after submitting a thesis, was awarded to Bari without the thesis requirement [4]:-

... since she was already recognised as one of the leading mathematicians specializing in the theory of functions of a real variable.

The year Bari graduated from Moscow State University, Viktor Vladmirovich Nemytski entered the university to read mathematics. They became close friends sharing not only mathematical interests but also a love of hiking in the mountains. They were eventually married. The vigorous school at Moscow State University headed by Luzin began to run out of steam in the last few years of the 1920s as he concentrated on writing monographs. He left the University in the early 1930s to work at the USSR Academy of Sciences and from 1935 on he led the Department of the Theory of Functions of Real Variables at the Steklov Institute. This left Bari and Menshov to take over his role of leading the School of Function Theory which they did in the 1940s.

Bari was an outstanding research mathematician who wrote over fifty research articles. The authors of [4] write:-

Nina Bari was one of the acknowledged leaders of the Soviet real variable school, and a worthy representative of this discipline at the Moscow University with which it is so closely associated. Her numerous papers exerted a great influence on the development of such fundamental branches of the theory of functions as the theory of trigonometric series, orthogonal series and bases, etc. Several of her investigations are rightly regarded as classics, for example her papers on the theory of uniqueness for trigonometric series and on the superpositions of functions. ... a number of her papers, for example those on the theory of uniqueness and on bases, have served as starting-points for new lines of research in the theory of functions, afterwards intensively developed ...

Let us mention in particular her paper The uniqueness problem of the representation of functions by trigonometric series (Russian) which she published in 1949. Antoni Zygmund describes it as:-

... an exhaustive review, in many cases accompanied by complete proofs, of the existing results in the theory of uniqueness of representation of functions by trigonometric series.

This paper was translated into English and was published as a 90-page book by the American Mathematical Society in 1951. In [1] her final publication, a 936-page research monograph on trigonometric series, is described as follows:-

The range and depth of topics covered is quite extensive, and most of her work in the field is included. But even within so long a monograph, the subject could not be completely exhausted. ... It has become a standard reference for mathematicians specializing in the theory of functions and the theory of trigonometric series.

The fifteen chapters of the book are: Basic concepts and theorems; Fourier coefficients; Convergence of a Fourier series at a point; Fourier series of continuous functions; Convergence and divergence of a Fourier series on a set; "Correcting" a function on a set of small measure; Summability of Fourier series; Conjugate trigonometric series; Absolute convergence of Fourier series; Sine and cosine series with decreasing coefficients; Lacunary series; Convergence and divergence of general trigonometric series; Absolute convergence of general trigonometric series; The problem of uniqueness of the expansion of a function in a trigonometric series; and Representation of functions by trigonometric series.

Bari also wrote textbooks for use in teaching training institutes such as Higher Algebra (1932) and The Theory of Series (1936). She edited the complete works of Luzin and was the editor of two important mathematics journals. She also translated Lebesgue's famous book on integration into Russian.

She died by falling in front of a train on the Moscow Metro. It has been claimed that this was suicide due to depression caused by Luzin's death eleven years earlier. One of her students, P L Ul'yanov, wrote after her death [1]:-

The untimely death of N K Bari is a great loss for soviet mathematics and a great misfortune for all who knew her. The image of Bari as a lively, straightforward person with an inexhaustible reserve of cheerfulness will remain forever in the hearths of all who knew her.

Finally let us give some indication of her personality and interests outside mathematics. We have already mentioned her love of hiking which took her into seriously difficult terrains in the Caucausus, Pamir and Tian Shan mountains. Her love of music and the arts included ballet, literature and poetry. Mikhail Lavrent'ev and L A Lyusternik write of her [3]:-

... lively temperament, directness, with an inexhaustible supply of young vigour ...