Rating:

Date added: 29.1.2015

156 816

FB2PDFEPUB

Henri Léon Lebesgue ForMemRS (French: [ɑ̃ʁi leɔ̃ ləbɛɡ]- June 28, 1875 – July 26, 1941) was a French mathematician most famous for his theory of integration, which was a generalization of the 17th century concept of integration—summing the area between an axis and the curve of a function defined for that axis. His theory was published originally in his dissertation Intégrale, longueur, aire (Integral, length, area) at the University of Nancy during 1902.Lebesgues first paper was published in 1898 and was titled Sur lapproximation des fonctions. It dealt with Weierstrass theorem on approximation to continuous functions by polynomials. Between March 1899 and April 1901 Lebesgue published six notes in Comptes Rendus. The first of these, unrelated to his development of Lebesgue integration, dealt with the extension of Baires theorem to functions of two variables. The next five dealt with surfaces applicable to a plane, the area of skew polygons, surface integrals of minimum area with a given bound, and the final note gave the definition of Lebesgue integration for some function f(x). Lebesgues great thesis, Intégrale, longueur, aire, with the full account of this work, appeared in the Annali di Matematica in 1902. The first chapter develops the theory of measure (see Borel measure). In the second chapter he defines the integral both geometrically and analytically. The next chapters expand the Comptes Rendus notes dealing with length, area and applicable surfaces. The final chapter deals mainly with Plateaus problem. This dissertation is considered to be one of the finest ever written by a mathematician.His lectures from 1902 to 1903 were collected into a Borel tract Leçons sur lintégration et la recherche des fonctions primitives. The problem of integration regarded as the search for a primitive function is the keynote of the book. Lebesgue presents the problem of integration in its historical context, addressing Augustin-Louis Cauchy, Peter Gustav Lejeune Dirichlet, and Bernhard Riemann. Lebesgue presents six conditions which it is desirable that the integral should satisfy, the last of which is If the sequence fn(x) increases to the limit f(x), the integral of fn(x) tends to the integral of f(x). Lebesgue shows that his conditions lead to the theory of measure and measurable functions and the analytical and geometrical definitions of the integral.He turned next to trigonometric functions with his 1903 paper Sur les séries trigonométriques. He presented three major theorems in this work: that a trigonometrical series representing a bounded function is a Fourier series, that the nth Fourier coefficient tends to zero (the Riemann–Lebesgue lemma), and that a Fourier series is integrable term by term. In 1904-1905 Lebesgue lectured once again at the Collège de France, this time on trigonometrical series and he went on to publish his lectures in another of the Borel tracts. In this tract he once again treats the subject in its historical context. He expounds on Fourier series, Cantor-Riemann theory, the Poisson integral and the Dirichlet problem.In a 1910 paper, Représentation trigonométrique approchée des fonctions satisfaisant a une condition de Lipschitz deals with the Fourier series of functions satisfying a Lipschitz condition, with an evaluation of the order of magnitude of the remainder term. He also proves that the Riemann–Lebesgue lemma is a best possible result for continuous functions, and gives some treatment to Lebesgue constants.In measure-theoretic analysis and related branches of mathematics, the Lebesgue–Stieltjes integral generalizes Riemann–Stieltjes and Lebesgue integration, preserving the many advantages of the latter in a more general measure-theoretic framework. Lecons Sur LIntegration Et La Recherche Des Fonctions Primitives by Henri Lebesgue