The dissertation of Carleman under
Erik Albert Holmgren, as well as his work in the early 1920s, was devoted to
singular integral equations. He developed the
spectral theory of
integral operators with
Carleman kernels, that is,
kernels K(
x,
y) such that
K(
y,
x) =
K(
x,
y) for
almost every (
x,
y), and : \int | K(x, y) |^2 dy for almost every
x. In the mid-1920s, Carleman developed the theory of
quasi-analytic functions. He proved the necessary and sufficient condition for quasi-analyticity, now called the
Denjoy–Carleman theorem. As a corollary, he obtained a
sufficient condition for the determinacy of the
moment problem. As one of the steps in the proof of the Denjoy–Carleman theorem in , he introduced the
Carleman inequality : \sum_{n=1}^\infty \left(a_1 a_2 \cdots a_n\right)^{1/n} \le e \sum_{n=1}^\infty a_n, valid for any sequence of non-negative real numbers
ak. At about the same time, he established the
Carleman formulae in
complex analysis, which reconstruct an
analytic function in a domain from its values on a subset of the boundary. He also proved a generalisation of
Jensen's formula, now called the
Jensen–Carleman formula. In the 1930s, independently of
John von Neumann, he discovered the
mean ergodic theorem. Later, he worked in the theory of
partial differential equations, where he introduced the
Carleman estimates, and found a way to study the spectral asymptotics of
Schrödinger operators. In 1932, following the work of
Henri Poincaré,
Erik Ivar Fredholm, and
Bernard Koopman, he devised the
Carleman embedding (also called
Carleman linearization), a way to embed a finite-dimensional system of nonlinear differential equations =
P(
u) for
u:
Rk →
R, where the components of
P are polynomials in
u, into an infinite-dimensional system of linear differential equations. In 1933 Carleman published a short proof of what is now called the
Denjoy–Carleman–Ahlfors theorem. This theorem states that the number of asymptotic values attained by an
entire function of order ρ along curves in the
complex plane going outwards toward infinite absolute value is less than or equal to 2ρ. In 1935, Torsten Carleman introduced a generalisation of
Fourier transform, which foreshadowed the work of
Mikio Sato on
hyperfunctions; his notes were published in . He considered the functions
f of at most polynomial growth, and showed that every such function can be decomposed as
f =
f+ +
f−, where
f+ and
f− are analytic in the upper and lower half planes, respectively, and that this representation is essentially unique. Then he defined the Fourier transform of (
f+,
f−) as another such pair (
g+,
g−). Though conceptually different, the definition coincides with the one given later by
Laurent Schwartz for
tempered distributions. Returning to
mathematical physics in the 1930s, Carleman gave the first proof of global existence for
Boltzmann's equation in the
kinetic theory of gases (his result applies to the space-homogeneous case). The results were published posthumously in . Carleman supervised the Ph.D. theses of Ulf Hellsten, Karl Persson (Dagerholm),
Åke Pleijel and (jointly with
Fritz Carlson) of
Hans Rådström. ==Life==