Carleson, Lennart

Carleson, Lennart
▪ 2007

      In 2006 one of mathematics' most coveted awards, the Abel Prize—awarded by the Norwegian Academy of Science and Letters in memory of the Norwegian mathematician Niels Henrik Abel—was given to a Swede, Lennart Carleson, in recognition of his contributions to harmonic analysis and smooth dynamical systems. The $900,000 Abel Prize capped an impressive array of honours that Carleson had won, including the Leroy Steel Prize (1984), the Wolf Prize (1992), the Lomonosov Gold Medal (2002), and the Sylvester Medal (2003).

      Carleson was born on March 18, 1928, in Stockholm. He earned a doctorate (1950) from Uppsala University and conducted postdoctoral work at Harvard University (1950–51) before accepting a lectureship at Uppsala. He moved to the University of Stockholm in 1954 but returned to Uppsala the next year and remained there until he retired in 1993, although he also held various visiting appointments, including at the Massachusetts Institute of Technology, the Institute for Advanced Study, Princeton, N.J., and Stanford University. Carleson was the director of the Mittag-Leffler Institute (1968–84), editor of Acta Mathematica (1956–79), and president of the International Mathematical Union (1978–82).

      Carleson first achieved recognition with his solution in 1962 of the corona problem, which concerns structures around the edge of a disk, through the introduction of what became known as Carleson measures. His most famous result, however, was his proof in 1966 of Luzin's conjecture. In 1807 the French mathematician Joseph Fourier had shown that complex periodic phenomena, such as sound waves, could be represented by combining many simpler waves—in particular, simple sine waves. The combination of these sine functions is called a Fourier series, and the study and application of such series is the subject of harmonic analysis. Fourier believed that every function could be expressed as the sum of a (possibly infinite) number of sine functions; for example, any sound, no matter how complex, could be reproduced by an orchestra with enough diverse instruments. Early in the 20th century, however, certain mathematical functions were discovered that could not be represented by a Fourier series. This led, in 1913, to a refinement by the Russian mathematician Nikolay N. Luzin, who asserted without proof that every well-behaved function can be represented “almost everywhere” by a Fourier series. Luzin's conjecture was about the functions that are normally encountered, and Carleson's proof introduced powerful ideas that had wide applications in analysis. Among Carleson's other notable mathematical contributions was the 1991 proof, with Michael Benedicks, of the existence of a “strange attractor” (a type of stable orbit) for the Hénon map, a dynamical system first proposed by the astronomer Michel Hénon.

William L. Hosch

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▪ Swedish mathematician
born March 18, 1928, Stockholm, Swed.
 
 Swedish mathematician and winner of the 2006 Abel Prize “for his profound and seminal contributions to harmonic analysis and the theory of smooth dynamical systems.” These include his work with Swedish mathematician Michael Benedicks in 1991, which gave one of the first rigorous proofs that strange attractors exist in dynamical systems (analysis) and has important consequences for the study of chaotic (chaos theory) behaviour.

      Carleson earned a bachelor's degree (1947), master's degree (1949), and doctorate (1950) from Uppsala University. He continued postdoctoral work at Harvard University (1950–51) before accepting a lectureship at Uppsala for the following academic year. He moved to the University of Stockholm in 1954 but returned to Uppsala the next year, where he remained until he retired in 1993, although he also held various visiting appointments (such as at the Massachusetts Institute of Technology, the Institute for Advanced Study in Princeton, N.J., and Stanford University). Carleson was the director of the Mittag-Leffler Institute (1968–84), editor of Acta Mathematica (1956–79), and president of the International Mathematical Union (1978–82). During his presidency, he helped to institute the Nevanlinna Prize to recognize work in theoretical computer science.

      Carleson's most famous work clarified the relationship between a function and its Fourier series representation. These were successfully introduced into mathematics by the French mathematician Joseph Fourier (Fourier, Joseph, Baron) in 1822, when he gave a simple recipe for obtaining the Fourier series of a function and expressed the claim that every function was equal to its Fourier series. As mathematics became more rigorous, this claim seemed more and more doubtful, until in 1926 the Russian mathematician Andrey Kolmogorov (Kolmogorov, Andrey Nikolayevich) showed that there are continuous (continuity) functions for which the corresponding Fourier series fails to converge (convergence) anywhere and so is numerically meaningless. However, in 1966 Carleson showed that every function in a large class of functions that includes all continuous functions is equal to its Fourier series except on a set of measure zero. A set of measure zero is one that is negligible for the purposes of integration, and so for many purposes this result showed that, although Fourier's original claim was wrong, his hopes for the great utility of his ideas was fully justified.

 In addition to winning the Abel Prize—awarded by the Norwegian Academy of Science and Letters in memory of the Norwegian mathematician Niels Henrik Abel (Abel, Niels Henrik)—Carleson has won a Leroy Steel Prize (1984), a Wolf Prize for Mathematics (1992), a Lomonosov Gold Medal (2002), and a Sylvester Medal (2003).

Jeremy John Gray
 

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Universalium. 2010.

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