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Torsten Carleman (8 July 1892, Visseltofta, Osby Municipality – 11 January 1949, Stockholm), born Tage Gillis Torsten Carleman, was a Swedish mathematician, known for his results in classical analysis and its applications. As the director of the Mittag-Leffler Institute for more than two decades, Carleman was the most influential mathematician in Sweden. ==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 , 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 = P(u) for u: R''k'' → 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.〔 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 supervised the Ph.D. theses of Ulf Hellsten, Karl Persson (Dagerholm), Åke Pleijel and (jointly with Fritz Carlson) of Hans Rådström. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Torsten Carleman」の詳細全文を読む スポンサード リンク
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