Marcel Riesz

Marcel Riesz ( ; 16 November 1886 – 4 September 1969) was a Hungarian mathematician, known for work on summation methods, potential theory, and other parts of analysis, as well as number theory, partial differential equations, and Clifford algebras. He spent most of his career in Lund, Sweden.

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.

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.

Riesz was elected a member of the Royal Swedish Academy of Sciences in 1936.

His results on summability of trigonometric series include a generalisation of Fejér's theorem to Cesàro means of arbitrary order. He also studied the summability of power and Dirichlet series, and coauthored a book on the latter with G.H. Hardy.

He also introduced the Riesz function Riesz(x), and showed that the Riemann hypothesis is equivalent to the bound as for any

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

then the variation |μ| of μ and the Lebesgue measure on the circle are mutually absolutely continuous.

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 moment problem, to which he introduced the operator-theoretic approach by proving the Riesz extension theorem (which predated the closely related Hahn–Banach theorem).

Later, he devised an interpolation theorem to show that the Hilbert transform is a bounded operator in Lp The generalisation of the interpolation theorem by his student Olaf Thorin is now known as the Riesz–Thorin theorem.

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

(b) for every there exists so that

for every

(c) for every there exists so that

for every with |y| < ρ, and every .

Potential theory, PDE, and Clifford algebras

After 1930, the interests of Riesz shifted to potential theory and partial differential equations. He made use of "generalised potentials", generalisations of the Riemann–Liouville integral.

Students Riesz's doctoral students in Stockholm include Harald Cramér and Einar Carl Hille. In Lund, Riesz supervised the theses of Otto Frostman, Lars Gårding, Lars Hörmander, and Olof Thorin.

Publications