037S-6218/85/020194-03$01.50 + 0.20/0 @ 1985 Birkhiiuser Verlag, Basel
Results in Mathematics, Vol. 8 (1985)
Alfred Haar (1885-1933) Alfred Haar was born in Budapest one hundred years ago, on October 11th, 1885. During his relatively short life he became a professor of mathematics of world wide fame, member of the Hungarian Academy of Sciences, one of the founders of the internationally renowned Szeged school of mathematics and its journal Acta Scientiarum Mathematicarum. 194
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He was the son of the land-owner Igmk Haar and his wife nee Emma Fuchs. He studied in the Evangelical School of his native town, where his teacher of mathematics was Laszlo Ratz, the excellent editor of the Mathematical Journal for Secondary Schools (»Kozepiskolai Matematikai Lapok«). During his grammar and high school studies Haar was a diligent contributor to this journal. Shortly after he had received his certificate of final examination at this school, he won, in the Autumn of 1903, the first prize of the »Lorand Eotvos Mathematical Competition«, annually organized for students who just finished their secondary school studies. It was this success that compelled the young Haar, already enrolled as a student of chemical engineering, to change and devote his life entirely to mathematics. Thus, from Easter 1904 on, he attended courses and seminars of mathematics, physics, and astronomy at the University of Budapest, with Professors Beke, Eotvos, Kiirschak, Rados, and others. In Autumn 1905 he entered the University of Gottingen, where excellent scholars as Caratheodory, Hilbert, Felix Klein, Minkowski, Prandtl, Runge, Schwarzschild, Voigt, and Zermelo taught at that time. He received the Ph. D. degree in June 1909 at the Philosophical Faculty of the University of Gottingen; Hilbert was the consultant of his Thesis. Just a few months after his Ph. D., Haar was also qualified Privatdozent of the University of Gottingen. Haar's Thesis dealt with, among other problems, the asymptotic behaviour of orthogonal function systems of Sturm-Liouville type, and spherical harmonics, including some studies analogous to those of Lebesgue concerning divergence properties of Fourier series. It was also in this paper that Haar introduced the orthogonal system which since then bears his name. After being qualified Privatdozent at the University of Gottingen, Haar first remained an assistent of Hilbert for some time, and then worked temporarily at the Technical University in Ziirich. In 1912, at the age of 27, he was appointed associate professor at the University of Kolozsvar, to succeed Professor Lipot Fejer, who had been invited to the University of Budapest. He became full professor in 1917. Among his colleagues were Frigyes Riesz and Gyula Farkas, first rate mathematicians who had initiated and successfully supported Haar's invitation to the University. After the First World War, when Transylvania was ceded to Romania, professors of the University of Kolozsvar had to leave this city. After a short provisional period in Budapest they settled down in Szeged. There, in a relatively short time, Alfred Haar and Frigyes Riesz had developed the mathematical department, later named »Bolyai Institute«, of the new university, to a center of international recognition. The periodical »Acta Scientiarum Mathematicarum«, started by Haar and Riesz in 1922 as the mathematical section of the »Acta« of the University, played an important part in this development.
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In all these changing circumstances, Haar untiringly and successfully continued his scientific research work. He extended the classical Du Bois-Reymond lemma concerning onedimensional variation problems to the multidimensional case. This result, later called »Haar lemma«, enabled him to successful researches on the solution of the classical Plateau problem relating to minimal surfaces, and in the treatment of the regular and adjoint variation problems. In 1929, at the invitation of Hamburg University, he summarized his own results in this field and those of others on the same problem in three lectures; he also discussed the future possibilities of these researches. His lectures were published in the mathematical periodical of Hamburg University. He also continued his studies on orthogonal function systems, this time from a novel, algebraic point of view. Namely, he characterized the structural properties of multiplicative systems generated by these function systems. Sursprisingly enough, he used to this effect some facts from the spectral theory of operators on Hilbert space, as developed shortly before by Riesz and J. von Neumann. Meanwhile he took deep interest on one of the famous problems raised by Hilbert at the International Mathematical Congress in Paris, 1900. This was Problem 5 which asked whether continuous groups can be parametrized by analytical parameters. The problem is of basic importance in geometry, theoretical physics and other fields of mathematics. It is intimately connected with the problem of whether a notion of measure can be defined on each continuous group such that the measure be invariant with respect to the group operation.
With a masterful display of deep mathematical reasoning, Alfred Haar eventually succeeded in finding a positive answer to this question. The discovery of the existence of invariant measure on groups, now generally called »Haar measure«, opened the way to the solution of the Hilbert problem. Moreover, it led to rapid development in several other branches of mathematics, as well, such as harmonic analysis, topology, etc. This was Haar's greatest achievement in mathematics. Upon his election as a corresponding member of the Hungarian Academy of Sciences in 1931, he chose this discovery as the subject of his inaugural address at the Academy. Most tragically, the epoch-making discovery of invariant measures on groups was Haar's last work. He did not live to see the extensions and many applications of his results. On the 16th of March, 1933, he died of cancer in Szeged, at the very peak of his splendid scientific career. Bela Szokefalvi-Nagy (Szeged)
