Torsten Carleman

Days Gills Torsten Carleman (born July 8, 1892 in Visseltofta , Osby municipality , † January 11, 1949 in Stockholm ) was one of the leading Swedish mathematicians of the 20th century.

Life

Carleman studied mathematics at Uppsala University , where he also received his doctorate under Erik Holmgren in 1917 ( on the Neumann-Poincaré problem for a region with corners ) and then became a lecturer. After several stays abroad, he received a call to the 1923 University of Lund , but one year later as a successor of Helge von Koch at the University of Stockholm . In 1927, after the death of Magnus Gösta, Mittag-Leffler was appointed the first director of the newly founded Mittag-Leffler Institute . Carleman was considered the leading Swedish mathematician at the time, but could not help the institute to shine, so that it mainly consisted of an excellently equipped library that was still compiled by Mittag-Leffler.

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Carleman proved important statements about singular integral equations . In particular, he investigated integral operators whose kernel satisfies the conditions for almost all and for almost all . Such cores are now called Carleman cores. Building on previous results from Arnaud Denjoy , he gave a characterization of quasi-analytic functions that is now known as the Denjoy and Carleman Theorem. In the proof, he used an inequality now known as the Carleman inequality. The Denjoy-Carleman-Ahlfors theorem deals with an entirely different subject than the Denjoy-Carleman theorem: ${\displaystyle L^{2}(I)}$ ${\displaystyle K}$ ${\displaystyle K(x,y)={\overline {K(y,x)}}}$ ${\displaystyle x,y\in I}$ ${\displaystyle \int _{I}|K(x,y)|^{2}dy<\infty }$ ${\displaystyle x\in I}$

It says that an entire function of finite order has at most asymptotic values, which Joy had proved for a special case and assumed that this is generally true. Carleman was able to show this with instead of before Ahlfors then fully proved Denjoy's guess. Shortly afterwards, Carleman gave another evidence. Another function-theoretical result is the Carleman-Jensen formula, which can be viewed as an analogue of Jensen's formula for the semicircle. Carleman used this formula to prove an analogue of Müntz's theorem on approximation by powers for analytic functions. Further results from Carleman deal with, among other things, ergodic theory , partial differential equations and mathematical physics , where he proved an existence theorem for the Boltzmann equation . ${\displaystyle \rho }$ ${\displaystyle 2\rho }$ ${\displaystyle 5\rho /2}$ ${\displaystyle 2\rho }$

In 1932 he gave a plenary lecture at the International Congress of Mathematicians in Zurich (on the theory of linear integral equations and their applications, held in French). In 1934 he was elected a corresponding member of the Saxon Academy of Sciences and in 1946 of the Académie des Sciences .

Åke Pleijel is one of his PhD students .

Web links

• Literature by and about Torsten Carleman in the catalog of the German National Library
• John J. O'Connor, Edmund F. Robertson :  Torsten Carleman. In: MacTutor History of Mathematics archive .

literature

• Lars Gårding , Mathematics and Mathematicians. Mathematics in Sweden Before 1950. American Mathematical Society, History of Mathematics, Volume 13, 1997.

Individual evidence

1. ^ Mathematics Genealogy Project
2. ^ A. Denjoy, Sur les fonctions entiéres de genre fini, Comptes Rendus 145, 106-108 (1907)
3. ^ T. Carleman, Sur les fonctions inverses des fonctions entières d'ordre fini, Arkiv för Matematik, Astronomi och Fysik 15, No. 10, 7 pp. (1921).
4. ↑ L. Ahlfors, On the asymptotic values ​​of the entire functions of finite order, Annales Academiae Scientiarum Fennicae 32 (Lindelöf Festschrift), No. 6, 15 pp. (1929).
5. ↑ T. Carleman, Sur une inégalité différentielle dans la théorie des fonctions analytiques, Comptes Rendus 196, 995-997 (1933).
6. ^ List of members since 1666: Letter C. Académie des sciences, accessed on October 25, 2019 (French).