Marcel Riesz

Marcel Riesz (

Marcel is the younger brother of Frigyes Riesz, who was also an important mathematician and at times they worked together (see F. and M. Riesz theorem).

Biography

Marcel Riesz was born in Győr, Austria-Hungary. He was the younger brother of the mathematician Frigyes Riesz. In 1904, he won the Loránd Eötvös competition.^[1] Upon entering the Budapest University, he also studied in Göttingen, and the academic year 1910-11 he spent in Paris. Earlier, in 1908, he attended the 1908 International Congress of Mathematicians in Rome. There he met Gösta Mittag-Leffler, in three years, Mittag-Leffler would offer Riesz to come to Sweden.^[2]

Riesz obtained his PhD at

From 1926 to 1952, he was a professor at Lund University. According to Lars Gårding, Riesz arrived in Lund as a renowned star of mathematics, and for a time his appointment may have seemed like an exile. Indeed, there was no established school of mathematics in Lund at the time. However, Riesz managed to turn the tide and make the academic atmosphere more active.^[3][2]

Retired from the Lund University, he spent 10 years at universities in the United States. As a visiting research professor, he worked in Maryland, Chicago, etc.^[3][2]

After ten years of intense work with little rest, he suffered a breakdown. Riesz returned to Lund in 1962. After a long illness, he died there in 1969.^[3][2]

Riesz was elected a member of the Royal Swedish Academy of Sciences in 1936.^[3]

Mathematical work

Classical analysis

The work of Riesz as a student of Fejér in Budapest was devoted to trigonometric series:

${\displaystyle {\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }\left\{a_{n}\cos(nx)+b_{n}\sin(nx)\right\}.\,}$

One of his results states that if

${\displaystyle \sum _{n=1}^{\infty }{\frac {|a_{n}|+|b_{n}|}{n^{2}}}<\infty ,\,}$

and if the

He also introduced the Riesz function Riesz(x), and showed that the Riemann hypothesis is equivalent to the bound {{{1}}} as x → ∞, for any ε > 0.^[6]

Together with his brother Frigyes Riesz, he proved the F. and M. Riesz theorem, which implies, in particular, that if μ is a complex measure on the unit circle such that

${\displaystyle \int z^{n}d\mu (z)=0,n=1,2,3\cdots ,\,}$

then the variation |μ| of μ and the Lebesgue measure on the circle are mutually absolutely continuous.^[5][7]

Functional-analytic methods

Part of the analytic work of Riesz in the 1920s used methods of functional analysis.

In the early 1920s, he worked on the

Later, he devised an interpolation theorem to show that the Hilbert transform is a bounded operator in L_p (1 < p < ∞). The generalisation of the interpolation theorem by his student Olaf Thorin is now known as the Riesz–Thorin theorem.^[2][10]

Riesz also established, independently of Andrey Kolmogorov, what is now called the Kolmogorov–Riesz compactness criterion in L_p: a subset K ⊂L_p(Rⁿ) is precompact if and only if the following three conditions hold: (a) K is bounded;

(b) for every ε > 0 there exists R > 0 so that

${\displaystyle \int _{|x|>R}|f(x)|^{p}dx<\epsilon ^{p}\,}$

for every f ∈ K;

(c) for every ε > 0 there exists ρ > 0 so that

${\displaystyle \int _{\mathbb {R} ^{n}}|f(x+y)-f(x)|^{p}dx<\epsilon ^{p}\,}$

for every y ∈ Rⁿ with |y| < ρ, and every f ∈ K.^[11]

Potential theory, PDE, and Clifford algebras

After 1930, the interests of Riesz shifted to

In the 1940s and 1950s, Riesz worked on Clifford algebras. His 1958 lecture notes, the complete version of which was only published in 1993 (Riesz (1993)), were dubbed by the physicist David Hestenes "the midwife of the rebirth" of Clifford algebras.^[12]

Students

Riesz's doctoral students in Stockholm include

• JFM 45.0387.03
  .
• Riesz, Marcel (1988). Collected papers. Berlin, New York:
  MR 0962287
  .
• Riesz, Marcel (1993) [1958]. Clifford numbers and spinors. Fundamental Theories of Physics. Vol. 54. Dordrecht: Kluwer Academic Publishers Group.
  MR 1247961
  .