Winning the competition was much more than a passing event but rather a very important milestone in Szegő's career. The competition, as all mathematically inclined Hungarians know, carried a great deal of prestige. It was especially important for Szegő because it is doubtful that, as a Jew whose family had no connections, he would have been able without it to study or receive the attention that he did. His father had even tried to discourage him from entering the university since he thought that his son would have no future there as a Jew.He entered the Mathematical Society competition in his first year of study and the competition of the Philosophical Faculty of the University in his second year. He won both these competitions, having submitted the essay On the approximation of continuous functions by polynomials for the second of these competitions.
During the summers of 1913 and 1914 Szegő went to Berlin where he studied under, among others, Georg Frobenius, Hermann Schwarz, Konrad Knopp and Friedrich Schottky. He then went to Göttingen where he studied with David Hilbert, Edmund Landau and Alfréd Haar. He returned to Hungary when World War I broke out and, for a while, continued his university studies. At this time he worked under Lipót Féjer, Mihály Bauer, Manó Beke (1862-1946) who worked at the University of Budapest from 1900 to 1922, and József Kürschák. Interaction with George Pólya led to Szegő publishing the paper Ein Grenzwertsatz über die Toeplitzschen Determinanten einer reellen positiven Funktion in Mathematische Annalen in 1915. Pólya writes in :-
Our cooperation started from a conjecture which I found. It was about a determinant considered by Toeplitz and others, formed with the Fourier-coefficients of a function f (x). I had no proof, but I published the conjecture and the young Szegő found the proof [...] We have seen here a good example of the fruitful cooperation between two mathematicians. Mathematical theorems often, perhaps in most cases, are found in two steps: first the guess is found; then minutes, or hours, or days, or weeks, or months, perhaps even several years later, the proof is found. Now the two steps can be done by different mathematicians, as we have seen.However, on 15 May 1915 he enlisted in the Austro-Hungarian cavalry rather than wait until he was conscripted when he would have been made to serve in the army. As well as the cavalry, he served both in the infantry and the artillery, and spent some time in the Air force where he met Richard von Mises and Theodore von Kármán. Towards the end of the war he was in Vienna and, after the defeat of Austria-Hungary in November 1918, he remained in the military until the spring of 1919 when he returned to Budapest to continue his university studies. However, while in Vienna and still serving in the military, he was awarded a Ph.D. on 8 July 1918 by the University of Vienna for a thesis based on the paper we mentioned above, published in 1915. One of the first things that Szegő did when arriving back in Budapest was to marry Erzsébet Anna Neményi, who had just been awarded a Ph.D. in chemistry from the University of Budapest. The marriage took place on 22 May 1919 with Szegő still dressed in military uniform. Gábor and Anna Szegő had two children, Peter (born 1925) and Veronika (born 1929).
Johnny von Neumann was a pupil at the Lutheran Gymnasium in Budapest and had shown such remarkable mathematical abilities that a university teacher was sought to tutor him. This was a quite standard approach in Hungary where extremely talented pupils were supported in this way. The Lutheran Gymnasium approached József Kürschák who in turn asked Szegő, who worked as his assistant during 1919-1920, to tutor the young von Neumann. Szegő went to von Neumann's home once or twice a week and, over tea, discussed set theory, measure theory and other topics. He set Johnny von Neumann problems and his solutions are still preserved. Anna Szegő tells how her husband came home with tears in his eyes after realising the brilliance of his young pupil.
The years following World War I were difficult ones in Hungary with Mihály Károlyi's liberal government being replaced with the communist regime of Béla Kun on 21 March 1919. This was equally short-lived and was overthrown on 1 August by the military regime of Miklós Horthy. It was clear that Hungary was going to be treated harshly in a post-war settlement and conditions in the country were poor. In particular, the prospect of a permanent academic job seemed remote so when his job in Budapest came to an end in 1920, with help from Maximilian von Neumann, Johnny von Neumann's father, Szegő went to Berlin. Working towards habilitating at the University of Berlin, Szegő became a friend of Issai Schur and worked with Richard von Mises, Leon Lichtenstein and Erhard Schmidt. He was awarded the 1924 Gyula König Prize by the Eötvös Lóránd Mathematical and Physical Society on 10 April of that year. The committee making the decision was chaired by József Kürschák and had as members Gyula Farkas, Dénes König, and Frederick Riesz. Here is an extract from the report:-
This year the committee wanted to reward a member of the youngest generation and decided to recommend for the prize Gábor Szegő, Privatdozent at the University of Berlin. ... During his eight years of scientific activity Gábor Szegő has produced numerous works. Please permit me to restrict myself to those of his papers that attracted most of my attention by the novelty, beauty, and significance of their results and methods.
He cooperated with Pólya in bringing out a joint Problem Book: Aufgaben und Lehrsätze aus der Analysis , volumes I and II (1925) which has since gone through many editions and which has had an enormous impact on later generations of mathematicians. Pólya wrote of their collaboration (see ):-
It was a wonderful time; we worked with enthusiasm and concentration. We had similar backgrounds. We were both influenced, like all young Hungarian mathematicians of that time, by Lipót Fejér. We were both readers of the same well-directed 'Hungarian Mathematical Journal' for high school students that stressed problem solving. We were interested in the same kinds of questions, in the same topics; but one of us knew more about one topic, and the other more about some other topic. It was a fine collaboration. The book 'Aufgaben und Lehrsätze aus der Analysis', the result of our cooperation, is my best work and also the best work of Gábor Szegő.Stuart Jay Sidney, reviewing an English translation of this book which was published in 1973, writes :-
One is not often afforded the opportunity to review a book which was a classic before one's birth. The original German version of the work under review appeared in 1924 and for nearly half a century has been a major force in the education of countless mathematicians. It may well be the granddaddy of all the learn-by-doing advanced mathematics books, a species which the authors are careful to distinguish from the species of problem collection books.
In 1926 Szegő moved to Königsberg to succeed Konrad Knopp as professor; Knopp had moved to Tübingen University. The events of 1933 were devastating for Jewish academics like Szegő. After Hitler came to power the Nazi party organised Boycott Day on 1 April 1933. Jewish shops were boycotted and Jewish professors and lecturers were not allowed to enter the university. On 7 April 1933 the Nazis passed a law which, under clause three, ordered the retirement of civil servants who were not of Aryan descent, with exemptions for participants in World War I and pre-war officials. Szegő qualified for this exemption but things quickly became increasingly difficult. Pólya was worried for him and wrote from Zürich to Jacob David Tamarkin, who was in the United States, in February 1934:-
It was very difficult to write about the chief point which is the fate of Szegő. Well, I shall be brief and plain. I am terribly worried about him. I saw Mrs Szegő in December. I got a letter from Szegő in the beginning of January; although no official measure was taken against him [until the beginning of January] and no direct collision happened with the students, I cannot see how it would go on indefinitely under those circumstances. He would accept, I understand, any offer even for a short period of 1 or 2 years, he should try to get a leave of absence for that time, and see whether he can live with his family on that amount. There is no hope to get something for him in Hungary, says Fejér and also Szegő himself ... I could not do anything for him here in Switzerland ... Excuse this letter, but you see, I am worried. The whole European situation is very dark.Szegő visited Harald Bohr in Copenhagen in May 1934 and took the opportunity to write himself to Tamarkin without fear that it would be read by Nazis censors which it would have been if he had written from Germany. Reading this letter, it is clear that Szegő did not appreciate the danger that he and his family were in from the Nazis. However, Harald Bohr understood more about the danger that the Szegős were in and he also wrote to Tamarkin who found a post for Szegő at Washington University in St Louis, Missouri. Both Pólya and Harald Bohr strongly advised Szegő to accept and he arrived in the United States to take up the position in the autumn of 1934. He spent the summer of 1935 as a visiting professor at Stanford University and in 1936 he was invited to address a meeting of the American Mathematical Society. In 1938 he was offered the position of Head of Mathematics at Stanford University and he accepted, remaining at Stanford for the rest of his working life. He did undertake war work and, in 1945-46, he went to Biarritz in France where, as part of his service in the U.S. Army, he was a professor at the military university. Peter Lax writes in  about Szegő's achievements at Stanford:-
In the forties and fifties I spent many summers at Stanford University at the invitation of the Head of the Mathematics Department, Gábor Szegő, who was my uncle by marriage. The Head of a department was in those days a much more powerful figure than a mere chairperson today; he made all decisions, including hiring and firing. Szegő used his powers to turn the provincial mathematics department that Stanford had been under Hans Frederick Blichfeldt and James Victor Uspensky - both remarkable mathematicians - into one of the leading departments of the country that Stanford is today. ... As a young man, Szegő was very shy; by the time he came to the United States, he was a self-assured man with old world courtly manners. Underneath his somewhat aristocratic appearance he was a warm-hearted person, ever willing to help others. He was aware of the absurdities of life and savoured them. He had high standards, but did not expect everyone to live up to them. ... it would be impossible today, as Szegő did in the summer of 1946, to invite all graduate students to his home for a supper of stuffed cabbage and plum dumplings, cooked expertly by his wife. Cooking wasn't Mrs Szegő's only expertise; she was a chemist, and supported the family while Szegő served in the prestigious but barely remunerated position of Privatdozent in Berlin. She was a voracious reader, in four languages, and provided intellectual companionship and stimulation to her husband, and their children. It was a happy household, and I was privileged to be part of it for three summers.Szegő worked mainly in function theory (of one complex variable), classical orthogonal polynomials, isoperimetric inequalities, and Toeplitz form. His most important work was in the area of extremal problems and Toeplitz matrices. This work led him to introduce the notion of the Szegő reproducing kernel. From these beginnings he moved to prove a number of limit theorems, now known as the Szegő limit theorem, the strong Szegő limit theorem and Szegő's orthogonal polynomials and on the unit circle. He produced over 130 research articles as well as several influential books. In addition to the books he wrote with Pólya, described above, Szegő wrote research monographs on his own work. Orthogonal Polynomials appeared in 1939 and was published by the American Mathematical Society. It has proved highly successful, running to four editions and many reprints over the years. Edward Copson writes :-
The whole work is characterised by the clarity of exposition and the sustaining of interest which one expects of one of the authors of "Pólya-Szegő".James Alexander Shohat writes :-
It is a remarkable source of powerful methods and far-reaching results in an important field, which will inspire and stimulate further research.Dunham Jackson writes :-
... no one man has made more significant contributions than the author of the book under review. With creative mastery in particular domains he combines an unusually extensive and penetrating acquaintance with the whole background of mathematical analysis, and almost unique experience in presenting the essentials of complicated theories with the greatest possible compactness.
In a collaboration with Ulf Grenander, Szegő wrote Toeplitz forms and their applications which was published in 1958. Mark Kac praises the book highly, writing :-
I recommend the book as an outstanding example of the power and beauty of analysis.
Szegő retired in 1960 and was made Professor Emeritus by Stanford. The health of both Szegő and his wife began to deteriorate and Anna died in 1966. Szegő was diagnosed with Parkinson's disease in 1970. Between 1973 and 1980 he spent part of the year in Budapest and part in Palo Alto. It was a long difficult illness and he died at the age of ninety.
We have already given some quotes which tell us about Szegő's character. Here is another by Halsey Royden :-
Szegő was very distinguished and autocratic; wore elegant tailor-made suits, and was always regarded with awe by the students and most of the faculty.Among his Ph.D. students at Stanford, we mention Paul Rosenbloom (the author of  and ) and Joseph Ullman. In these papers by Rosenbloom says that, in addition to his mathematical needs, Szegő also was concerned for his cultural education, giving him a ticket to a concert given by Bela Bartók at Stanford in the early 1940s. Szegő's colleagues at Stanford wrote these words in the tribute :-
He was an outstanding human being. His leadership at our department created a spirit of relaxed but very active cooperation and stimulating exchange of ideas. At the same time his hospitality created a warm social atmosphere and his cultured personality produced in all of us a feeling of respect and admiration. His work remains and will not be forgotten.Szegő received a number of honours in addition to the 1924 Gyula König Prize mentioned above. Among these was his election to the Austrian Academy of Sciences (1960) and to the Hungarian Academy of Sciences (1965). In addition we note that his bust has been erected in Kunhegyes with copies in St Louis and in Stanford. In 2010 the Gábor Szegő prize was established to be awarded to an early-career researcher for outstanding research contributions in the area of orthogonal polynomials and special functions.
Article by: J J O'Connor and E F Robertson