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Jovan Karamata (1902-1967) JOVAN KARAMATA (1902-1967) Aleksandar M. Nikolić Faculty of Technical Sciences, University of Novi Sad Miskolc, May 2020 Karamata Family Assignat from 1948 with the signature of Athanasios Karamata Jovan Karamata – A life 1902 Born in Zagreb on the February 1st 1909-1913 Primary school in Budapest and Zemun 1913-1920 Gymnasium Zemun, Osijek, Susak (Rijeka), Cantonal Gymnase scientifique Lausanne 1920-1925 Faculty of Engineering, Faculty of Philosophy Mathematics, Belgrade 1926 PhD degree 1927-1928 Rockefeller Foundation fellow in Paris 1930-1940 The best years of his life 1931 Married Emilija Nikolajević, three children 1930-1951 Professor at the University of Belgrade 1948 Full Member of the Serbian Academy of Sciences 1951-1967 Professor at the University of Geneva 1967 Passed away in Geneva on the August 14th Parents Stevan Desanka Karamata family in 1908 With his brother Kosta and sister Smiljka Budapest, 1905 As a pupil in the primary school in Zemun Gymnasium in Lausanne In the classroom in 1919 Left: Kosta, Jovan, Ozren, Smiljka in 1922 Right: Jovan as a student in 1925 Impacts and paragons personality that touched him the most M. Petrović – the true love for science, the breadth of view, the idea liberated from the formal style, the directions in which to look for results, the monitoring of literature, theory of functions R. Kašanin – the significance of the theory of sets, measures, integrals, rigorous proofs Karamata considered himself as a self-taught man of science, acquiring his education by perusing books and research papers by masters – H. Weyl (Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann., Bd. 77 (1916)); G. Pólya and G. Szegő (he used to say that his teacher of classical analysis was their famous book Aufgaben und Lehrsätze aus der Analysis, Berlin 1925); H. Fogt (Éléments de mathématiques supérieures, Paris 1925), E. Landau (Darsellung und Begrndung einiger neurer Ergebnisse der Funktionentheorie, Berlin 1930) The first papers Karamata once said: ” I Wanted to improve my knowledge of the foundations of the theory of functions. This is the reason why I first started to study the theory of series, but when I entered into this theory, I found so many old and new results that I remained there.” Sur l’évaluation des limites se rattachant aux suites á doubles entrées, Glas SKA, 1926; On Certain Limits Similar to Definite Integrals, these, 1926. He gave the necessary and sufficient conditions for the existence and the construction of the uniform distribution function ν(x) of a double sequence generalizing Weyl’s problem of uniform distribution of numbers modulo one. Sur certaines limites rattachées aux intégrales de Stieltjes, CR de l’Acad. Des Sciences de Paris, march 1926 (F.Riesz and B.Sz.Nagy, Leçons d’analyse fonctionnelle, 1956). Jacques Hadamard realized the importance of this results. In the same CR he wrote: Clever (ingénieuse) ideas of m. Karamata … Just finished PhD studies April 1926–April 1927 Karamata during the military Service Dec. 1927-Sept. 1928, the fellow of Rockefeller Foundation in Paris. He worked in Lebesgue’s group His first conference 52. Congress AFAS La Rochelle, July 1928 The best years 1929-1940 Emilija Nikolajević (1906-1959), the judge Children: Dimitrije, Vladimir, Katarina Assistant Professor (1930), Associate Professor (1937), Full Professor (1950). Professor at University of Geneva (1951) Corresponding Member of the Yugoslav Academy of Sciences and Arts (Zagreb) (1933), Corresponding Member of RegiaSocietas Scientiarum Bohemica (Prague)(1936), Corresponding member (1939) and Full Member of the Serbian Academy of Sciences and Arts (1948) The Key papers 1930 - 1931 Über die Hardy Littlewoodsche Umkehrungen des Abelschen Stätigkeitssatzes, MZ, 1930 (Tauberian theorems) Sur une mode de croissance régulière des fonctions, Mathematica (Cluj), 1930 (regularly, slowly varying functions) Neuer beweis und verallgemeinerung einiger Tauberschen Sätze welche die Laplacesche und Stieltjessche Transformation betreffen, MZ, 1930 & Crelle, 1931 (Hardy-Littlewood-Karamata theorem) The first appearance of his proof of Hardy-Littlewood theorem and majorizability as his tauberian condition was in Serbian in the paper Théorèmes inverses de sommabilité I and II. Glas Srpske Kraljevske Akademije, 1931 Tauberian theory The proof of Littlewood theorem In spite of the efforts of many famous mathematicians of those days (Landau, Hardy, R. Schmidt) the proof of Littlewood’s theorem remained very complicated From Cesàro summability of the series ∑an and condition nan=O(1) or nan ≥-M it follows its convergence (Hardy, 1910, Landau, 1910) The proof that from A-summability of the series with the condition nan=O(1) it follows its C-summability was long, very complicated, rather difficult, not purified, far from the apparent On the contrary the proof by Jovan Karamata (1930) was extremely elegant and surprisingly simple (E.C.Titchmarsh, 1939, K.Knopp, 1951) C-summability can be omitted, as it was shown by Wielandt (1952) Karamata’s proof Karamata’s proof Still possible! Among the first 50 most important papers in 60 years of MZ ICM Zürich, 1932 Regularly varying functions Sur une mode de croissance régulière des fonctions. Mathematica (Cluj) vol. 4, 1930 croissance lente et régulière langsam und regulär wachsende slowly and regularly increasing functions The most important achievement of Jovan Karamata! The intent was to generalize the Tauberian conditions in some inverse theorems of the Tauberian type for the Laplace transform. First ideas on RVF and SVF Definitions of Karamata functions Karamata functions Applications of RVF and SVF functions Hardy-Littlewood-Karamata theorem The other results 1930-1940 The other results 1930-1940 Trigonometric and singular integrals (Konvergensatz für trigonometrische Integrale, 1937); Karamata proved a theorem on convergence of trigonometric integrals which includes the Riemann’s theorem for trigonometric series and its extension on general trigonometric series, Stieltjes integrals as well as Riez’s convergence theorem of Laplace-Stieltjes integrals. Cited by A.P. Calderón, A. Zygmund (1952, 1956) as significant for the beginning of later development of a very complicated theory of trigonometric and singular integrals, as well as its natural. Continuation in form of important theory of singular integral operators and pseudo differential operators Stirling summability as the invers of Borel procedure (1932); Cited by A.V. Lototsky (1953), R.P. Agnew (1957), A. Jakimovski (1959), V. Vučković (1959, 1965), B. Martić (1961). Their transforms are special cases of a class of a general transformations introduced by Karamata Guests of Jovan Karamata at Mathematics seminar in Belgrade Paul Montel (1876-1975) Wilhelm Blaschke (1885-1962) Paul Erdős (1913-1996) With Isaac Jacob & Charlotte Schönberg in Potsdam, Berlin 1936 Lecture on the theory of functions, Tübingen 1937/38 The World War Two Later - post war - results Tauberian theorems in the theory of prime numbers (1954- 1958) (with P. Erdős) Karamata Iteration Theoreme (1951) Cited by Nagaev (1961), Slack (1968, 1972), Bingham, Teugels (1975), Seneta (1971, 1990) Aljančić-Karamata theoreme (Frullani integrals, 1956) Complex numbers (1950) Theory and application of Stieltjes integral (1949) From 50 published papers 17 were with his students University of Geneva 1951 Left: Đuro Kurepa, de Ram (Lozana 1955) Right: Kuratowski, Aquaro (Komo 1955) Madison Mathematics Research Center 1963 M.Šaškijević, V.Marić, Č.Stanojević, R.Bojanić Lecture in Geneva 1965 Karamata school of classic analysis V. Avakumović (1910-1990) B. Bajšanski (1930-2008) P. Muzen V. Marić (1930) M. Tomić (1912-2001) D. Aranđelović (1942) S. Aljančić (1922-1993) T. Ostrogorski (1950-2005) V. Vučković (1923-2012) S. Simić (1950). R. Bojanić (1924-2017) S. Janković (1951) B. Stanković (1924-2018) Č. Stanojević (1928-2008) Werner Meyer-König D. Adamović (1928-2008) Monique Vuilleumier M. Maravić (1919-2000) Ronald Rafael Coifman M. Šteković Horst Baumann Š. Raljević (1909-1991) Karamata décédé Thank you for your attention and patience.
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