Frederick Riesz's Report on the 1924 Gyula König Prize
Gentlemen!
The committee entrusted with the task of making a proposal concerning the award of the second Gyula König Prize held a meeting on January 26 of this year at the Technical University with József Körschák as chairman; also present were Gyula Farkas, Dénes König, and yours truly.
The committee observed with pleasure that among the Hungarian mathematicians eligible, according to the rules of the foundation, there are more than one who are worthy of receiving the prize. This year the committee wanted to reward a member of the youngest generation and decided to recommend for the prize Gábor Szegö, Privatdozent at the University of Berlin.
The committee has charged me with the task of preparing a report and of analyzing and appraising the works of the candidate. I have the honour to present my report.
During his eight years of scientific activity Gábor Szegö 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. From among those results, if you excuse me for my perhaps excessive subjectivity, I start with a discovery that is in direct contact with my own research. It is known and easy to prove that the value of a function that is holomorphic inside a curve, say, a circle, and continuous on an arc of this circle, cannot be constant on this arc except in the trivial case when the function is constant. In 1906 the French mathematician Pierre Fatou, after showing in his famous doctoral dissertation that every function that is bounded and holomorphic inside a circle has a limiting value almost everywhere, that is, with the exception of a set of measure 0, raised the following question: since this limiting function cannot be constant on the whole arc, as was stated above, how large can the set be on which it is constant; or, and this amounts to the same, how large can the set be on which it vanishes? After showing that this set cannot fill out "almost" entirely an arc, he formulated the conjecture, which he believed was difficult to prove, that this set has measure 0. My younger brother Marcel and I proved this conjecture in a joint article, which we presented at the 1916 Stockholm Congress, not only in the bounded case but for a more general class of holomorphic functions as well.
Szegö succeeded in showing the deeper, I could say real, reason of this phenomenon in a March 1920 letter addressed to me which was published, together with my comments, in the 38th volume of Math. és Term. Értesitö in 1920 under the title "Analytikus függvény kerfileti értékeiröl." Szegö later also published his related research in the 84th volume of Math. Annalen under the title "Über die Randwerte einer analytischen Funktion." Namely, in these papers he proved that the logarithm of the absolute value of the [nontangential] boundary limit function is Lebesgue integrable. Therefore, the logarithm may be equal to negative infinity, that is, the boundary limit function itself may vanish, only on a set of measure 0. The interesting nature of this result is perhaps better shown by the following theorem which is easily seen to be equivalent to it: given a nonnegative function on the circumference of a disk, a necessary and sufficient condition for the existence of a function that is holomorphic inside the disk and is not identically vanishing inside the disk and has bounded mean value, such that the absolute value of its [nontangential] boundary value is almost everywhere equal to the given function, is that both the given function itself and its logarithm be integrable.
I note that Szegö's theorem, which he obtained in a roundabout way via the study of Toeplitz forms and the Fourier series of positive functions, can very easily be derived from a famous formula of Johan Jensen. This was pointed out not only by me in the above cited correspondence, but also by Pierre Fatou himself, who applied a similar chain of ideas to prove in just a few lines, not the theorem of Szegö, but his own conjecture.
Another, extensive group of Szegö's work belongs to the following sphere of ideas: from properties of the coefficients of power series, or from arithmetic properties of most of these coefficients, he deduces properties of the corresponding analytic functions. More specifically: (i) theorems of Jacques Hadamard and Charles Fabry about lacunary power series; (ii) the theorem conjectured by Pólya and proved by Fritz Carlson about power series with integer coefficients that are convergent inside the unit disk, which states that the function defined by such a power series either is rational or else cannot be extended beyond the unit disk; and (iii) the analogous theorem of Szegö about power series having only finitely many different coefficients ("Über Potenzreihen mit endlich vielen verschiedenen Koeffizienten," Sitzungsber. d. preuss. Akademie, 1922). In "Tschebyscheff'sche Polynome und nicht fortsetzbare Potenzreihen," which appeared in the 87th volume of Math. Annalen, Szegö shows that all these theorems follow naturally from the relationship discovered by Georg Faber that exists between the Tschebyscheff polynomials of a curve and conformal mapping, and which was used by Fritz Carlson in his proof of Pólya's conjecture. In the same article, starting from the same principle, Szegö also deduces a theorem of Alexander Markowich Ostrowski which leads us to a seemingly distant theorem of Robert Jentzsch (1890-1918), a young German mathematician who died in the war, about the distribution of zeros of the partial sums of power series. His article titled "Über die Tschebyscheff'schen Polynome," which was published in the first volume of mathematical Acta [Acta Sci. Szeged] of the Ferenc József University, and another one, a short article titled "Über die Nulstellen von Polynomen, die in einem Kreise gleichmässig konvergieren," which was recently published in the Sitzungsberichte of the Mathematical Association of Berlin belong to this area as well. In them Szegö, using completely elementary methods, throws light on the deeper causes behind the theorem of Robert Jentzsch and related phenomena.
Finally, I turn to the area belonging both to complex and real analysis to which Szegö devoted the largest part of his work: the theory of orthogonal systems and the corresponding series expansions. To start with a smaller, very interesting work, which also shows his great ability in formal calculations, let me mention the article "Über die Lebesgue'schen Konstanten bei den Fourier'schen Reihen" which appeared in the 9th volume of the Math. Zeitschrifl. Here he gives very simple numerical expressions for Lebesgue constants whose properties were previously studied by Lipót Fejér and Thomas Gronwall. Then he proves in a straightforward fashion properties of these constants some of which were proved by the above-mentioned authors in a much more complicated way and some of which were conjectured by them. In more extensive work which appeared in the same volume under the title "Über orthogonale Polynome, die zu einer gegebenen Kurve der komplexen Ebene gehören," he examines Fourier expansions in polynomials which are orthogonal on a closed curve. This contains, as a special case, both Legendre and power series. These expansions, even in the general case, behave very much like power series, and provide a new and most natural solution to the following problem of Georg Faber: given a domain, find a system of polynomials in which every function that is holomorphic in this domain has an expansion. The expansions studied by Szegö have an interesting and very simple relationship with the conformal mapping of the finite and infinite domains bounded by the given curve onto the unit disk. For example, the conformal mapping between the exterior of the curve and the exterior of the unit disk which maps infinity onto itself is the limit of the ratios P_{n+1}(z) / P_{n}(z) formed from consecutive polynomials.
Among the papers of Szegö related to the problems just discussed there are two that deserve the greatest acclaim. In these two papers he examines the so-called "inner" asymptotics for orthogonal systems and the corresponding series expansions. In other words, he discusses questions concerning asymptotic behaviour on those curves and intervals on which the polynomials are orthogonalised with respect to some weight function p(x). In this area, where the first classical results are linked with the names of Laplace and Darboux, Szegö not only obtains very general results, far overshadowing anything known previously, but he obtains these results exactly because he examines these questions, considered very difficult, using a simple, one can say elementary, method. The main point of his method is that he squeezes the weight function P(x) between two functions of a very simple structure that have the form √(1 - x^{2})/P(x) where P(x) is a polynomial. He shows that these functions may be viewed as majorants and minorants, respectively, from the point of view of the problems which are studied. After reducing the problems to the case of weight functions of this special type, he evaluates explicitly the corresponding expressions, using a theorem of Lipót Fejér on positive trigonometric polynomials. Using this method that we have just sketched, in his article, "Über den asymptotischen Ausdruck von Polynomen, die durch eine Orthogonalitätseigenschaft definiert sind" which appeared in the 86th volume of Math. Annalen, he gives [inner] asymptotic expressions of orthogonal polynomials for every point x where p"(x) exists. It is known, especially after Alfréd Haar's dissertation, that with the help of these asymptotic expressions one can reduce questions of convergence and summability of series expansions to certain special cases, e.g., Fourier series. In addition, in another article titled "Über die Entwicklung einer willkürlichen Funktion nach den Polynomen eines Orthogonalsystems" published in the 12th volume of the Math. Zeitschrift, Szegö also shows that the same elementary method, without the use of asymptotic expressions of the polynomials, directly gives asymptotics for the partial sums [of orthogonal series] and in this way reduces convergence problems to analogous questions for Fourier series.
I wish to mention another merit of Szegö of a different nature. Namely, the many careful, precise, to-the-point, and, if needed, critical reviews which he wrote for the last two volumes of Jahrbuch über die Fortschritte der Mathematik, which dealt with the literature from 1914 to 1918. In my judgement, with these reviews, together with those of other collaborators, he has considerably contributed to raising the quality of the yearbook. This is of permanent value while contacts between scientists of different nations is made difficult by financial and other considerations. Based on these observations I recommend that the board approve the committee's recommendation.
Szeged, 7 March 1924
Frederick Riesz
JOC/EFR July 2014
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