Marcel Riesz

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Marcel Riesz ( hu, Riesz Marcell ; 16 November 1886 – 4 September 1969) was a
HungarianHungarian may refer to: * Hungary, a country in Central Europe * Kingdom of Hungary, state of Hungary, existing between 1000 and 1946 * Hungarians, ethnic groups in Hungary * Hungarian algorithm, a polynomial time algorithm for solving the assignmen ...
mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

, known for work on summation methods,
potential theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
, and other parts of
analysis Analysis is the process of breaking a complex topic or substance Substance may refer to: * Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes * Chemical substance, a material with a definite chemical composit ...
, as well as
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

,
partial differential equations In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, and
Clifford algebras In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a Unital algebra, unital associative algebra. As algebra over a field, ''K''-algebras, they generalize the real numbers, complex numbers, ...
. 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. He obtained his PhD at Eötvös Loránd University under the supervision of Lipót Fejér. In 1911, he moved to Sweden upon the invitation of Gösta Mittag-Leffler. From 1911 to 1925 he taught at ''Stockholms högskola'' (now Stockholm University). From 1926 to 1952 he was professor at Lund University. After retiring, he spent 10 years at universities in the United States. He returned to Lund in 1962, and died there in 1969. Riesz was elected a member of the Royal Swedish Academy of Sciences in 1936.

Mathematical work

Classical analysis

The work of Riesz as a student of Fejér in Budapest was devoted to trigonometric series: :$\frac + \sum_^\infty \left\.\,$ One of his results states that, if :$\sum_^\infty \frac < \infty,\,$ and if the Fejer means of the series tend to zero, then all the coefficients ''a''''n'' and ''b''''n'' are zero. His results on divergent series, summability of trigonometric series include a generalisation of Fejér's theorem to Cesàro means of arbitrary order. He also studied the summability of power series, power and Dirichlet series, and coauthored a book on the latter with G.H. Hardy. In 1916, he introduced the Riesz interpolation formula for trigonometric polynomials, which allowed him to give a new proof of Bernstein's inequality (mathematical analysis), Bernstein's inequality. 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 :$\int z^n d\mu\left(z\right) = 0, n=1,2,3\cdots,\,$ then the complex measure#Variation of a complex measure and polar decomposition, variation , ''μ'', of ''μ'' and the Lebesgue measure on the circle are mutually Absolute continuity#Generalizations 2, 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 theory, 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 space, ''L''''p'' The generalisation of the interpolation theorem by his student Olof Thorin, 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 ''L''''p'': a subset ''K'' ⊂''L''''p''(R''n'') is totally bounded space, precompact if and only if the following three conditions hold: (a) ''K'' is bounded; (b) for every there exists so that :$\int_ , f\left(x\right), ^p dx < \epsilon^p\,$ for every (c) for every there exists so that :$\int_ , f\left(x+y\right)-f\left(x\right), ^p dx < \epsilon^p\,$ for every with , ''y'',  < ''ρ'', and every .

Potential theory, PDE, and Clifford algebras

After 1930, the interests of Riesz shifted to
potential theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
and
partial differential equations In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. He made use of "generalised potentials", generalisations of the Riemann–Liouville integral. In particular, Riesz discovered the Riesz potential, a generalisation of the Riemann–Liouville integral to dimension higher than one. In the 1940s and 1950s, Riesz worked on Clifford algebras. His 1958 lecture notes, the complete version of which was only published in 1993 (), were dubbed by the physicist David Hestenes "the midwife of the rebirth" of Clifford algebras.

Students

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

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