Torsten Carleman - Work

Work

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

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 Carleman (1926), he introduced the Carleman inequality

valid for any sequence of non-negative real numbers a_k.

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 du⁄_dt = P(u) for u: Rk → R, where the components of P are polynomials in u, into an infinite-dimensional system of linear differential equatons.

In 1935, Torsten Carleman introduced a generalisation of Fourier transform, which foreshadowed the work of Mikio Sato on hyperfunctions; his notes were published in Carleman (1944). 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. Carleman's definition gave rise to numerous extensions.

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 (1957).

Carleman supervised the Ph.D. theses of Ulf Hellsten, Karl Persson (Dagerholm), Åke Pleijel and (jointly with Fritz Carlson) of Hans Rådström.