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 Fejer means of the series tend to zero, then all the coefficients a_n and b_n are zero.^[1]

His results on summability of trigonometric series include a generalisation of Fejér's theorem to Cesàro means of arbitrary order.^[4] He also studied the summability of power and Dirichlet series, and coauthored a book Hardy & Riesz (1915) on the latter with G.H. Hardy.^[1]

In 1916, he introduced the Riesz interpolation formula for trigonometric polynomials, which allowed him to give a new proof of Bernstein's inequality.^[5]

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 moment problem, to which he introduced the operator-theoretic approach by proving the Riesz extension theorem (which predated the closely related Hahn–Banach theorem).^[8][9]

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 potential theory and partial differential equations. He made use of "generalised potentials", generalisations of the Riemann–Liouville integral.^[2] In particular, Riesz discovered the Riesz potential, a generalisation of the Riemann–Liouville integral to dimension higher than one.^[3]

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 Harald Cramér and Einar Carl Hille.^[3] In Lund, Riesz supervised the theses of Otto Frostman, Lars Gårding, Lars Hörmander, and Olof Thorin.^[2]