Marcel Riesz

Marcel Riesz (born November 16, 1886 in Győr , Hungary ; † September 4, 1969 in Lund , Sweden ) was a Hungarian mathematician .

Life

Marcel Riesz was the son of a doctor and the younger brother of Frigyes Riesz . As a high school student he won the Lorand Eötvös competition in 1904. He studied mathematics at the University of Budapest and was influenced by Leopold Fejér . In 1907 he received his doctorate (Summable trigonometric series and power series). In his dissertation he generalized Georg Cantor's proof of the uniqueness of the representation of a function by Fourier series from convergent Fourier series to Cesàro-summable . He regularly attended the University of Göttingen and Paris, where he was in 1910/11.

In 1908 he was invited by Magnus Gösta Mittag-Leffler (whom he knew from the International Congress of Mathematicians in Rome in 1908) to give lectures at Stockholm University . Riesz accepted this invitation and initially stayed permanently in Sweden. In 1926 Riesz accepted a professorship at Lund University . In 1923, Torsten Carleman was preferred to him there , but in 1926 he went to Stockholm, where Riesz had also applied to be the successor of Helge von Koch in vain. In 1952 he retired in Lund and went to the USA, where he worked at the University of Chicago (where he was visiting professor in 1947/48), the Courant Institute , Stanford University , the University of Washington , Indiana University and the University of Maryland was. He returned to Lund in 1960 because of health problems.

His fields of work included analytical functions , harmonic analysis , functional analysis , potential theory and wave equations , the Kolmogorow-Riesz theorem and the Riesz mean are associated with his name. The Riesz remedy in particular made him known internationally at the time. He presented the related theory in a book with Godfrey Harold Hardy . In the 1930s he generalized the theory of the Liouville-Riemann integral (see fractional calculus , here Riesz potentials are named after him), which he applied to the Cauchy, among other things -Problem of wave equation applied. In 1949 he published a large paper on this in Acta Mathematica. Later in his career he studied spinors and Clifford algebras .

In 1916 he proved the equivalence of the Riemann hypothesis to a conjecture about the asymptotic behavior of the Riesz function:

${\displaystyle {\rm {Riesz}}(x)=-\sum _{k=1}^{\infty }{\frac {(-x)^{k}}{(k-1)!\zeta (2k)}}.}$

where the zeta function is in even places in the denominator . The Riemann hypothesis is equivalent to

${\displaystyle \operatorname {Riesz} (x)=O(x^{{\frac {1}{4}}+\epsilon })\qquad ({\text{für }}x\to \infty )}$

for each (using Landau symbols ). ${\displaystyle \epsilon >0}$

He is also known for his theorem on conjugate functions and his convexity theorem (often also named after his student Olof Thorin (1912-2004), who found a simple proof), both from 1927. He published with his brother Frigyes Riesz only one work from 1916 that contained an important result about the boundary behavior of an analytic function. In 1916 he proved an interpolation formula for trigonometric polynomials, with which he provided a new proof of the Bernstein inequality.

His students include Lars Gårding , Einar Hille , Harald Cramér , Otto Frostman (1907–1977, professor at Stockholm University) and Lars Hörmander .

He was a member of the Royal Swedish Academy of Sciences (1936), the Royal Danish Academy of Sciences and the Physiographical Society in Lund and an honorary member of the Swedish Mathematical Society. Riesz was an honorary doctor from the universities of Lund and Copenhagen.

Fonts

• Collected papers . - Berlin: Springer, 1988. - ISBN 3-540-18115-6
• with Godfrey Harold Hardy General Theory of Dirichlet Series , Cambridge Tracts, 1915, Reprint 1952
• with Emil Hilb Recent studies on trigonometric series , Encyclopedia of Mathematical Sciences , 1922
• Clifford numbers and spinors , Chapters I through IV, The Institute for Fluid Dynamics and Applied Mathematics, Lecture Series N ° 38, University of Maryland, College Park, Maryland, 1957-1958

literature

• Lars Gårding, Marcel Riesz in Memoriam, in Acta Mathematica, 124 (1970), i – xi
• John Horvath: Riesz, Marcel . In: Frederic Lawrence Holmes (Ed.): Dictionary of Scientific Biography . tape 18 , Supplement II: Aleksander Nikolaevich Lebedev - Fritz Zwicky . Charles Scribner's Sons, New York 1981, p. 743-745 (on- line ).

Web links

• Literature by and about Marcel Riesz in the catalog of the German National Library
• John J. O'Connor, Edmund F. Robertson :  Marcel Riesz. In: MacTutor History of Mathematics archive .
• Jean Horvath L'oeuvre mathématique de Marcel Riesz I , Cahiers du séminaire d'histoire des mathématiques, Vol. 3, 1982, pp. 83-121 , part 2

Individual evidence

1. ↑ Published in expanded form as Über summbaren trigonometrische Serien , Mathematische Annalen, Volume 71, 1912, pp. 54–75, online
2. ^ Riesz L'integrale de Riemann – Liouville et le probleme de Cauchy, Acta Mathematica, Volume 81, 1949, pp. 1-223
3. ↑ Riesz Sur l'hypothèse de Riemann , Acta Mathematica, Volume 40, 1916, pp. 185-190
4. ↑ Riesz Sur les fonctions conjuguées , Mathematische Zeitschrift, Volume 27, 1927, pp. 218–244
5. ↑ Riesz Sur les maxima des formes bilinéaires et sur les fonctionnelles linéaires , Acta Mathematica, Volume 49, 1927, pp. 465-497
6. ^ M. and F. Riesz on the boundary values ​​of an analytic function , Quatrième Congrès des Mathématiciens Scandinaves, Stockholm 1916, Uppsala, Almquist & Wiksells, 1920, pp. 27–44
7. ↑ Riesz on a sentence by Mr. Serge Bernstein, from two letters to Mr. G. Mittag-Leffler , Acta Mathematica, Volume 40, 1916, pp. 337–347
8. ^ Mathematics Genealogy Project