Marcel Riesz ( hu, Riesz Marcell ; 16 November 1886 – 4 September 1969) was a Hungarian

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

, known for work on summation methods, potential theory, and other parts of

analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...

, as well as

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...

partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...

, 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 Győr ( , ; german: Raab, links=no; names in other languages) is the main city of northwest Hungary, the capital of Győr-Moson-Sopron County and Western Transdanubia region, and – halfway between Budapest and Vienna – situated on one of ...

Austria-Hungary Austria-Hungary, often referred to as the Austro-Hungarian Empire,, the Dual Monarchy, or Austria, was a constitutional monarchy and great power in Central Europe between 1867 and 1918. It was formed with the Austro-Hungarian Compromise of ...

; 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 , motto = Ad utrumque , mottoeng = Prepared for both , established = , type = Public research university , budget = SEK 9 billion 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 In mathematics, a trigonometric series is a infinite series of the form : \frac+\displaystyle\sum_^(A_ \cos + B_ \sin), an infinite version of a trigonometric polynomial. It is called the Fourier series of the integrable function f if the term ...

: :

\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

summability In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must ...

of trigonometric series include a generalisation of

Fejér's theorem In mathematics, Fejér's theorem,Leopold FejérUntersuchungen über Fouriersche Reihen ''Mathematische Annalen''vol. 58 1904, 51-69. named after Hungarian mathematician Lipót Fejér, states the following: Explanation of Fejér's Theorem's Ex ...

to Cesàro means of arbitrary order. He also studied the summability of

power Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may a ...

and

Dirichlet series In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in analy ...

, 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. 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(z) = 0, n=1,2,3\cdots,\,

then the variation , ''μ'', of ''μ'' and the Lebesgue measure on the circle are mutually

absolutely continuous In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central ope ...

Functional-analytic methods

Part of the analytic work of Riesz in the 1920s used methods of

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...

. 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 In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, of a real variable and produces another function of a real variable . This linear operator is given by convolution with the functi ...

is a bounded operator in ''L''_''p'' 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 Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...

, what is now called the ''Kolmogorov–Riesz compactness criterion'' in ''L''_''p'': a subset ''K'' ⊂''L''_''p''(R^''n'') is precompact if and only if the following three conditions hold: (a) ''K'' is bounded; (b) for every there exists so that :

\int_ , f(x), ^p dx < \epsilon^p\,

for every (c) for every there exists so that :

\int_ , f(x+y)-f(x), ^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 and

. He made use of "generalised potentials", generalisations of the

Riemann–Liouville integral In mathematics, the Riemann–Liouville integral associates with a real function f: \mathbb \rightarrow \mathbb another function of the same kind for each value of the parameter . The integral is a manner of generalization of the repeated antide ...

. In particular, Riesz discovered the

Riesz potential In mathematics, the Riesz potential is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz. In a sense, the Riesz potential defines an inverse for a power of the Laplace operator on Euclidean space. They generalize to ...

, 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 David Orlin Hestenes (born May 21, 1933) is a theoretical physicist and science educator. He is best known as chief architect of geometric algebra as a unified language for mathematics and physics, and as founder of Modelling Instructio ...

"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 Olov (or Olof) is a Swedish form of Olav/Olaf, meaning "ancestor's descendant". A common short form of the name is ''Olle''. The name may refer to: *Per-Olov Ahrén (1926–2004), Swedish clergyman, bishop of Lund from 1980 to 1992 * Per-Olov Br ...

Publications

* * *

References

External links

* * {{DEFAULTSORT:Riesz, Marcel 1886 births 1969 deaths 20th-century Hungarian mathematicians 20th-century Hungarian people 20th-century Swedish people Swedish mathematicians Mathematical analysts Functional analysts Measure theorists People connected to Lund University People from Lund Members of the Royal Swedish Academy of Sciences Emigrants from the Austro-Hungarian Empire to Sweden People from Győr Swedish Jews Austro-Hungarian mathematicians