## Lipót Fejér by those who knew him

- Agnes P Berger (1918-2002) went to Budapest and studied for a Ph.D. advised by Lipót Fejér, and was awarded her doctorate in 1939. She was interested in research in mathematics and statistics all her life and became professor of biostatistics at Columbia University in New York. She describes Lipót Fejér, both as teacher and research supervisor, in Reuben Hersh and Vera John-Steiner, "A Visit to Hungarian Mathematics",
*The Mathematical Intelligencer***15**(2) (1993), 13-26. First she describes Fejer's teaching: - Agnes P Berger explained that she worked on interpolation advised by Fejér:
- George Pólya wrote about Lipót Fejér in a number of articles. Our first quote is from George Pólya, 'Some mathematicians I have known',
*Amer. Math. Monthly***76**(1969), 746-753: - George Pólya also wrote about Lipót Fejér in G Pólya, 'Leopold Fejér',
*J. London Math. Soc.***36**(1961), 501-506. Although this repeats some comments made in the previous quote we give a quote about his personality: - George Pólya also wrote about Lipót Fejér's mathematical style in G Pólya, 'Leopold Fejér',
*J. London Math. Soc.***36**(1961), 501-506:

Fejer gave very short, very beautiful lectures. They lasted less than an hour. You sat there for a long time before he came. When he came in, he would be in a sort of frenzy. He was very ugly-looking when you first examined him, but he had a very lively face with a lot of expression and grimaces. The lecture was thought out in very great detail, with a dramatic denouement. It was a show.

Turan was in fact my real advisor. The way a professor was expected to behave there was very different from the way it is here. I was greatly amazed when I saw that in America a professor would sit down with a graduate student. Nothing like that ever happened in Budapest. You would say to the professor, "I'm interested in this or that." And then eventually you would come back and show him what you did. There was none of the hand-holding that goes on here. I know people here who see their students every week! Have you ever heard of such a thing? Well, I did have Turan, who acted for me like an advisor. I don't think of Fejer as a college teacher. There was only one Fejer in all of Hungary. And in Szeged there was Riesz. Only two in the whole country. That is a very exalted position.

If you could see him in his rather Bohemian attire (which was, I suspect, carefully chosen) you would find him very eccentric. Yet he would not appear so in his natural habitat, in a certain section of Budapest middle-class society, many members of which had the same manners, if not quite the same mannerisms, as Fejér - there he would appear about half eccentric.

His papers are particularly well written, they are very easy to read. This is due to his style of work: When he found an idea, he tended it with loving care; he tried to perfect it, simplify it, free it from unessentials; he worked on it carefully and minutely until the idea became transparently clear. He eventually produced a work of art, not of too large dimensions, but highly finished.

He had artistic talents besides mathematics. He loved music and played the piano. He had a special gift for telling stories, he was a "raconteur." In telling his stories, he acted the part of the persons he was telling about, and underlined the points with little gestures. He liked to talk about his teacher, who was in a rather indirect way responsible for his first discovery, Hermann Amandus Schwarz. When he told about the little misadventures of this great mathematician, he was irresistible, you could not help laughing.

This variety of talents has a bearing on a question which I have often heard: Why did Hungary produce so many mathematicians? Hungary was a small country (it is even smaller today) not much industrialized, and it produced a disproportionately large number of mathematicians, several of whom were active in this country. Why was that so? There is no complete answer, I think - Hungary produced not only mathematicians, but also many musicians and some physicists. Yet, I think, as far as mathematicians are concerned a good part of the answer can be found in Fejér's personality: He attracted many people to mathematics by the success of his own work and by his personal charm. He sat in a coffee house with young people who could not help loving him and trying to imitate him as he wrote formulas on the menus and alternately spoke about mathematics and told stories about mathematicians. In fact, almost all Hungarian mathematicians who were his contemporaries or somewhat younger were personally influenced by him, and several started their mathematical career by working on his problems.

To round out the picture I must quote some of his witty remarks I heard myself.

It happened at a meeting in Germany. At that time I was a "Privatdozent." I cannot completely explain what that is: A financially shaky position, somewhat similar to, but not quite, an Assistant Professor -- thank goodness, this institution of Privatdozents has started to disappear nowadays. I was married, and my wife took photographs of the mathematicians. She also stopped Fejér in the company of three or four others, in front of the university on the street car tracks, took a picture and was about to take a second one as Fejér spoke up. "What a good wife! She puts all these full professors on the tracks of the street car so that they may be run over and then her husband will get a job!"

At another meeting (that was several years later) Fejér was very angry (and had some reason to be angry) at a Hungarian mathematician, a topologist whose name I shall not tell you. I walked up and down a long time with Fejér who could not stop talking about the target of his anger and wound up by saying: "And what he says is a topological map of the truth." You must realize how distorted a topological map may be.

Oh yes, let us not forget the question: Was Fejér eccentric? After all these stories, if you could see him in his rather Bohemian attire (which was, I suspect, carefully chosen) you would find him very eccentric. Yet he would not appear so in his natural habitat, in a certain section of Budapest middle class society, many members of which had the same manners, if not quite the same mannerisms, as Fejér - there he would appear about half eccentric.

He had artistic tastes. He deeply loved music and was a good pianist. He liked a well-turned phrase. 'As to earning a living', he said, 'a professor's salary is a necessary, but not sufficient, condition.' Once he was very angry with a colleague who happened to be a topologist, and explaining the case at length he wound up by declaring '... and what he is saying is a topological mapping of the truth'.

He had a quick eye for foibles and miseries; in seemingly dull situations he noticed points that were unexpectedly funny or unexpectedly pathetic. He carefully cultivated his talent of raconteur; when he told, with his characteristic gestures, of the little shortcomings of a certain great mathematician, he was irresistible. The hours spent in continental coffee houses with Fejér discussing mathematics and telling stories are a cherished recollection for many of us. Fejér presented his mathematical remarks with the same verve as his stories, and this may have helped him in winning the lasting interest of so many younger men in his problems.

Fejér talked about a paper he was about to write up. 'When I write a paper,' he said, 'I have to rederive for myself the rules of differentiation and sometimes even the commutative law of multiplication.' These words stuck in my memory and years later I came to think that they expressed an essential aspect of Fejér's mathematical talent; his love for the intuitively clear detail.

It was not given to him to solve very difficult problems or to build vast conceptual structures. Yet he could perceive the significance, the beauty, and the promise of a rather concrete not too large problem, foresee the possibility of a solution and work at it with intensity. And, when he had found the solution, he kept on working at it with loving care, till each detail became fully transparent.

It is due to such care spent on the elaboration of the solution that Fejér's papers are very clearly written, and easy to read and most of his proofs appear very clear and simple. Yet only the very naive may think that it is easy to write a paper that is easy to read, or that it is a simple thing to point out a significant problem that is capable of a simple solution.

JOC/EFR May 2017

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