埃里克·伊瓦尔·弗雷德霍姆Erik Ivar Fredholm

埃里克·伊瓦尔·弗雷德霍姆的父母是Ludvig Oscar Fredholm和Catharina Paulina Stenberg。Ludvig Fredholm是一位商人，靠用电灯取代煤气灯发家致富。他于1861年5月2日在Arboga与Catharina结婚，她本人也是一位商人的女儿。他们是富有且受过良好教育的人，希望儿子弗雷德霍姆和弗瑞兹·约翰 Oscar（生于1875年）接受最好的教育，而且他们负担得起最好的教育。弗雷德霍姆出生于斯德哥尔摩市的Klara地区，在斯德哥尔摩的Beskowska学校证明是一名才华出众的学生，并于1885年5月16日获得中学毕业证书。

获得中学毕业证书后不久，弗雷德霍姆进入了斯德哥尔摩的皇家技术学院（称为KTH），在那里学习了一年。Bernkopf在[1]中写道：-

在这一年里，他对实用力学的技术问题产生了兴趣，这种兴趣持续了他的一生，并使他持续关注应用数学。

在皇家理工学院学习一年后，弗雷德霍姆进入乌普萨拉大学学习。他选择去乌普萨拉而不是斯德哥尔摩有一个非常明确的原因。在他大学教育的这个早期阶段，他计划最终攻读博士学位，而乌普萨拉是当时瑞典唯一授予博士学位的大学。乌普萨拉于1888年5月28日授予他理学硕士学位，但现在他面临如何继续的问题，因为只有通过乌普萨拉才能获得博士学位，而他真正想师从的人是约斯塔·米塔格-莱弗勒。八年前，约斯塔·米塔格-莱弗勒被任命为新成立的斯德哥尔摩大学（当时称为斯德哥尔摩高等学校）数学讲席的第一位持有者，因此要成为他的学生，弗雷德霍姆必须在斯德哥尔摩学习。他通过在乌普萨拉注册攻读博士学位但在斯德哥尔摩师从约斯塔·米塔格-莱弗勒学习来解决他的困难。

弗雷德霍姆的第一篇出版物出现在1890年，当时他在瑞典皇家科学院上发表了On a special class of functions。在这篇论文中，他构造了一个在单位圆盘上解析、在闭圆盘上无限可微、但在圆盘外没有解析延拓的函数。正如弗雷德霍姆产生的所有深刻数学结果一样，这个结果受到数学物理的启发，在这种情况下是热方程。这个结果比约斯塔·米塔格-莱弗勒和卡尔·魏尔斯特拉斯早期的例子更进一步。它给约斯塔·米塔格-莱弗勒留下了如此深刻的印象，以至于他将论文的副本寄给了儒勒·昂利·庞加莱。然而，我们应该注意到，弗雷德霍姆的论文中有一个错误，但幸运的是可以纠正。他使用了柯瓦列夫斯卡娅在她的教授资格论文（Habilitation）学位论文中证明的结果，但由于误解，在一个不适用的情况下应用了该结果。这篇论文在[6]中详细讨论，作者展示了如何纠正弗雷德霍姆的论证。

1893年5月30日，弗雷德霍姆被乌普萨拉大学授予博士学位，然后于1898年5月31日从同一所大学获得理学博士学位。他1898年的学位论文涉及偏微分方程的研究，其研究动机是弹性中的平衡问题。他在论文中（以及在1900年基于该论文发表的论文中）解决了在研究物理问题中出现的特定情况下的算子方程，而一般情况由弗雷德霍姆稍后解决，直到1908年才发表。

弗雷德霍姆最为人所知的是他在积分方程和谱理论方面的工作。他的思想基础在[10]中解释（另见[4]）：-

我们可能会问，在弗雷德霍姆看来，他工作的基本基础是什么。答案很直接：位势论……早在1895年，在1895年的一次研讨会演讲之后，他就曾将约翰·彼得·古斯塔夫·勒热纳·狄利克雷的问题作为消去问题来讨论。……两年后在斯德哥尔摩，一场关于Roux的“主解”及其与维多·沃尔泰拉方程的联系的演讲引发了热烈的讨论。最后，在长时间的沉默之后，弗雷德霍姆开口了，以他惯常的慢吞吞的拖腔说道：在位势论中也有这样一个方程。

事实上，这项工作的大部分是在1899年的几个月里完成的，当时弗雷德霍姆在巴黎与儒勒·昂利·庞加莱、埃米尔·皮卡和雅克·阿达马一起研究约翰·彼得·古斯塔夫·勒热纳·狄利克雷问题。1900年，关于他的弗雷德霍姆积分方程理论的初步报告作为Sur une nouvelle méthode pour la résolution du problème de DirichletⓉ（关于解决约翰·彼得·古斯塔夫·勒热纳·狄利克雷问题的一种新方法）发表。维多·沃尔泰拉早些时候曾研究过积分方程的一些方面，但在弗雷德霍姆之前所做甚少。当然，波恩哈德·黎曼、赫尔曼·阿曼杜斯·施瓦茨、卡尔·诺伊曼和儒勒·昂利·庞加莱都解决了现在属于弗雷德霍姆积分方程一般情况的问题；这表明他的理论有多么强大。

当Holmgren于1901年在哥廷根讲授弗雷德霍姆的理论时，弗雷德霍姆的贡献迅速为数学界所熟知。大卫·希尔伯特立即看到了弗雷德霍姆理论的重要性，在20世纪的头25年里，积分方程理论成为一个主要的研究课题。弗雷德霍姆在Sur une classe d'équations fonctionelle Ⓣ（关于一类泛函方程）中发表了他的积分方程理论的更完整版本，该文于1903年出现在Acta Mathematica上。大卫·希尔伯特将弗雷德霍姆的工作扩展到包括弗雷德霍姆积分方程的完整eigenvalue理论。这项工作直接导致了大卫·希尔伯特空间的理论。

Garding在[3]中写道：-

弗雷德霍姆在积分方程方面的工作引起了极大的兴趣，并提振了此前一直处于欧洲大陆文化帝国德国和法国阴影下工作的瑞典数学家的士气和自尊。积分方程此时已成为一种新的数学工具，不再局限于对称核。它在几十年间得到发展，并被视为一种普遍工具，可以用来解决物理学中的大多数边值问题。但该理论所提供的定性洞见也可以通过更简单的方式获得。弗雷德霍姆工作的意义更多在于定性洞见，而非显式公式。

我们在讨论中向前推进了时间，以便解释弗雷德霍姆数学发现的重要性。至于他的职业生涯，在1898年获得理学博士学位后，他被任命为斯德哥尔摩大学的数学物理讲师。他的整个职业生涯都在斯德哥尔摩大学度过，并于1906年9月28日被任命为力学和数学物理讲席。1909至1910年，他在斯德哥尔摩大学担任副院长，随后担任院长。

弗雷德霍姆 还担任过各种其他职位，1899年成为公务员，1902年成为瑞典国家保险公司的部门主管。1904年至1907年，他是 Skandia 保险公司的精算师。弗雷德霍姆 还曾在许多政府委员会任职，并且还曾在国际计量委员会任职。

1911年5月31日，弗雷德霍姆 在 Sankt Olai 与 Agnes Maria Liljeblad 结婚，她是新教神职人员父母的女儿。此时 弗雷德霍姆 45岁，他的妻子33岁。他们的长子 Bengt 弗雷德霍姆 出生于1912年，后来成为瑞典军队的少校。

弗雷德霍姆 写论文时极为仔细和专注，因此他产出了高质量的工作，这很快为他赢得了整个欧洲的高度声誉。然而，他的论文需要他付出如此多的努力，以至于他只写了少数几篇，事实上他的数学 Complete Works 仅包含160页。1910年之后，除了重新审视他早期的工作外，他写得很少。

在[10]中（另见[4]），Zeilon——他是弗雷德霍姆的学生——描述了他作为讲师的风格：-

在斯德哥尔摩的讲座中，弗雷德霍姆喜欢详细谈论经典数学物理的重大问题和方法，这些曾是他科学工作的主要主题。但这并不妨碍他谈论现代物理学的所有部分。弗雷德霍姆并不是通常所谓的那种才华横溢的演讲者。他说话缓慢，声音单调，有时会在黑板上陷入计算错误。但这无关紧要。事实上，他的讲座显示出他对主题非同寻常的掌握，并且他有能力向学生传达一种对物理理论之统一性和逻辑性的感受，这种感受在他自己的著作中如此明显。

正如我们上面所指出的，弗雷德霍姆也有精算科学的职业生涯，从1902年起，他致力于研究该领域的各种问题。他通过提出一个优雅的数学公式来确定人寿保险保单的退保价值，做出了特别重要的贡献。

人们很容易认为，由于两条数学职业生涯并行发展，即应用于物理应用数学和应用于精算科学，弗雷德霍姆 几乎没有时间顾及其他兴趣。然而，事实并非如此，因为他也是一位音乐家，对他来说这不仅仅是一种休闲活动。小时候他吹长笛，但后来开始拉小提琴。他特别喜欢演奏巴赫。他再次将自己的才能结合起来，将他的机械技能应用于音乐以及数学。听起来不太可能，但他用半个椰子壳制作了他的第一把小提琴，同时他还利用自己制造机器的才能制造了一台解微分方程的机器。他对机器和机械的兴趣使他成为瑞典工程师学会的会员，他经常为该学会提供科学建议。在他去世时，他正在研究小提琴声学的数学，但他留下的关于这个主题的未完成工作已被证明无法理解。

弗雷德霍姆因其数学贡献获得了许多荣誉，包括1903年因微分方程理论获得的V A Wallmarks奖、1908年法国科学院颁发的让-维克托·彭赛列奖，以及1909年莱比锡大学授予的名誉博士学位。

Ivar Fredholm's parents were Ludvig Oscar Fredholm and Catharina Paulina Stenberg. Ludvig Fredholm was a merchant who made his fortune replacing gas lamps with electric lamps. He married Catharina, who was herself the daughter of a merchant, on 2 May 1861 in Arboga. They were wealthy, well-educated people who wanted the very best education for their sons Ivar and John Oscar (born 1875), and they were able to afford the very best. Ivar, who was born in the Klara area of Stockholm City, proved a brilliant school pupil at the Beskowska School in Stockholm and he was awarded his baccalaureate on 16 May 1885.

Soon after the award of his baccalaureate, Fredholm entered the Royal Technological Institute in Stockholm (known as KTH), studying there for a year. Bernkopf writes in [1]:-

During this single year he developed an interest in the technical problems of practical mechanics that was to last all his life and that accounted for his continued interest in applied mathematics.

After the year at the Royal Technological Institute, Fredholm enrolled as a student at the University of Uppsala. There was a very definite reason why he chose to go to Uppsala rather than Stockholm. At this early stage of his university education he had plans to eventually undertake doctoral studies and Uppsala was the only university in Sweden at that time which granted doctorates. Uppsala awarded him a Master of Science degree on 28 May 1888 but now he had some problems as how to proceed for it was only through Uppsala that he could be awarded a doctorate yet the person whom he really wanted to work under was Mittag-Leffler. It was eight years earlier that Mittag-Leffler had been appointed as the first holder of the chair of mathematics at the newly founded University of Stockholm (called Stockholms Högskola at this time) so to be his student Fredholm had to study at Stockholm. He solved his difficulties by remaining registered for his doctorate at Uppsala but studying at Stockholm under Mittag-Leffler.

Fredholm's first publication came in 1890 when he published On a special class of functions in the Royal Swedish Academy of Sciences. In this paper he constructed a function which is analytic on the unit disk, is infinitely differentiable on the closed disk, but has no analytic continuation outside the disk. As was always the case with all the deep mathematical results which Fredholm produced, this result was inspired by mathematical physics, in this case by the heat equation. The result went further than earlier examples by Mittag-Leffler and Weierstrass. It so impressed Mittag-Leffler that he sent a copy of the paper to Poincaré. We should note, however, that there is an error in Fredholm's paper, but fortunately one which can be corrected. He used a result proved by Kovalevskaya in her Habilitation thesis but, through a misinterpretation, applied the result in a case where it was not valid. This paper is discussed in detail in [6] where the authors show how Fredholm's argument can be corrected.

On 30 May 1893 Fredholm was awarded his Ph.D. from the University of Uppsala and then on 31 May 1898 he received the degree of Doctor of Science from the same university. His 1898 doctoral dissertation involved a study of partial differential equations, the study of which was motivated by an equilibrium problem in elasticity. He solved his operator equation in the particular cases which arise in the study of the physical problem in his thesis (and in the paper which appeared in 1900 based on that thesis) while the general case was solved by Fredholm somewhat later and not published until 1908.

Fredholm is best remembered for his work on integral equations and spectral theory. The basis for his thinking is explained in [10] (see also [4]):-

We may ask what in Fredholm's eyes was the essential basis of his work. The answer is immediate: potential theory ... Already in 1895 after a seminar lecture in 1895 he had talked about Dirichlet's problem as one of elimination. ... Two years later in Stockholm a lecture about the 'principal solutions' of Roux and their connections with Volterra's equation led to a vivid discussion Finally, after a long silence Fredholm spoke and remarked in his usual slow drawl: in potential theory there is also such an equation.

In fact much of this work was accomplished during the months of 1899 which Fredholm spent in Paris studying the Dirichlet problem with Poincaré, Émile Picard, and Hadamard. In 1900 a preliminary report on his theory of Fredholm integral equations was published as Sur une nouvelle méthode pour la résolution du problème de Dirichlet Ⓣ. Volterra had earlier studied some aspects of integral equations but before Fredholm little had been done. Of course Riemann, Schwarz, Carl Neumann, and Poincaré had all solved problems which now came under Fredholm's general case of an integral equation; this was an indication of how powerful his theory was.

Fredholm's contributions quickly became well known to the world of mathematics when Holmgren lectured on Fredholm's theory at Göttingen in 1901. Hilbert immediately saw the he importance of Fredholm's theory, and during the first quarter of the 20th century the theory of integral equations became a major research topic. Fredholm published a fuller version of his theory of integral equations in Sur une classe d'équations fonctionelle Ⓣ which appeared in Acta Mathematica in 1903. Hilbert extended Fredholm's work to include a complete eigenvalue theory for the Fredholm integral equation. This work led directly to the theory of Hilbert spaces.

Garding writes in [3]:-

Fredholm's work on integral equations was met with great interest and boosted the morale and self-respect of Swedish mathematicians who so far had been working under the shadow of the continental cultural empires Germany and France. Integral equations had now become a new mathematical tool not confined to symmetrical kernels. It was developed during several decades and was seen as a universal tool with which it was possible to solve the majority of boundary value problems of physics. But the qualitative insight that the theory gave could also be achieved in a simpler way. The significance of Fredholm's work was more the qualitative insight than the explicit formulas.

We have moved forward in time in our discussions in order to explain something of the importance of Fredholm's mathematical discoveries. As to his career, after the award of his Doctor of Science degree in 1898 he was appointed as a lecturer in mathematical physics at the University of Stockholm. He spent his whole career at the University of Stockholm being appointed to a chair in mechanics and mathematical physics on 28 September 1906. In 1909-1910 he was Pro-Dean and then Dean in Stockholm University.

However, Fredholm also held various other positions, becoming a civil servant in 1899 and then a Head of Department for the Swedish State Insurance Company in 1902. He was an actuary in the Skandia Insurance Company from 1904 to 1907. Fredholm also served on many government committees and he also served on the International Committee for Weights and Measures.

On 31 May 1911 Fredholm married Agnes Maria Liljeblad, the daughter of Protestant clergy parents, in Sankt Olai. At this time Fredholm was 45 years old and his wife was 33. Their eldest son, Bengt Ivar Fredholm, was born in 1912 and he became a major in the Swedish army.

Fredholm wrote papers with great care and attention so he produced work of high quality which quickly gained him a high reputation throughout Europe. However his papers required so much effort from him that he wrote only a few and in fact his Complete Works in mathematics comprises of only 160 pages. After 1910 he wrote little beyond revisiting his earlier work.

In [10] (see also [4]), Zeilon, who was a student of Fredholm's, described his style as a lecturer:-

In his Stockholm lectures Fredholm loved to talk at length about the great problems and methods of classical mathematical physics which had been the main theme of his scientific work. But this did not prevent him from talking about all parts of modern physics. Fredholm was not what is usually called a brilliant speaker He talked slowly in a monotone voice and it could happen that he got embroiled in computational mistakes at the blackboard But this had little importance In fact, his lectures revealed an unusual mastery of his subject and he had the ability of communicating to his students a feeling for the unity and logic of physical theory which is so apparent in his own written work.

Fredholm also, as we indicated above, had a career in actuarial science and from 1902 onwards he occupied himself with studying various questions in this area. He made a particularly important contribution by proposing an elegant mathematical formula to determine the surrender value of a life insurance policy.

It is tempting to think that with two mathematical careers running in parallel, namely applications to physical applied mathematics and applications to actuarial science, Fredholm would have had little time for other interests. This, however, was not so for he was also a musician and for him this was more than just a leisure activity. As a young boy he played the flute, but later took up playing the violin. He particularly loved to play Bach. Again he combined his talents, applying his mechanical skills to music as well as to mathematics. Unlikely as it sounds, he built his first violin from half a coconut, while he also used his talents at building machines to make one to solve differential equations. His interest in machines and mechanics led to his membership of the Swedish Society of Engineers and he frequently provided scientific advice to that Society. At the time of his death he was working on the mathematics of the acoustics of the violin, but the unfinished work he left on this topic has proved impossible to understand.

Fredholm received many honours for his mathematical contributions, including the V A Wallmarks Prize for the theory of differential equations in 1903, the Poncelet Prize from the French Academy of Sciences in 1908, and an honorary doctorate from the University of Leipzig in 1909.