Download A Panorama of Hungarian Mathematics in the Twentieth by Janos Horvath PDF

By Janos Horvath

A wonderful interval of Hungarian arithmetic begun in 1900 while Lipót Fejér came upon the summability of Fourier series.This was once via the discoveries of his disciples in Fourier research and within the concept of analytic services. whilst Frederic (Frigyes) Riesz created useful research and Alfred Haar gave the 1st instance of wavelets. Later the subjects investigated via Hungarian mathematicians broadened significantly, and integrated topology, operator concept, differential equations, likelihood, and so forth. the current quantity, the 1st of 2, offers essentially the most notable effects completed within the 20th century by way of Hungarians in research, geometry and stochastics. The booklet is obtainable to somebody with a minimal wisdom of arithmetic. it truly is supplemented with an essay at the heritage of Hungary within the 20th century and biographies of these mathematicians who're not energetic. an inventory of all folks said within the chapters concludes the amount.

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3) is satisfied . 3). (i) Sufficient condition: Rademacher (1922), Menshov (1923). The most important convergence test was discovered nearly simultaneously by Hans Rademacher {13} and D. E. Menshov {8}. 44 F . M6ricz Rademacher--Menshov theorem. e, Here and in the sequel the logarithm is to the base 2, but any base greater than 1 would be appropriate. It was D. E. 4) is the best possible in general. Theorem of Menshov. If ( w(k) : k = 1,2, ... 1) diverges everywhere. (ii) Necessary condition: Tandori (1957).

The majority of these results are collecte d in t he monograph by G. Alexits [8], which serves as a by G. Alexit s [8], which serves as a reference book even nowadays . The divergenc e theorem of Tandori has several consequ ences. We cite here only one of them. It follows from the Rademacher-Menshov theorem and the familiar Kronecker lemma that if ()"(k) : k = 1,2 , .. 9) Now, from his divergence theorem K. 9). Corollary. 2(k) n )\t 0, bu t it is no longer t rue for e = O.

Tandori, Einfacher Beweis eines Satzes von B. S. Kasin , Acta Sci . Math . (Szeged) , 39 (1977), 175-178 . {17} K. Tandori, tiber beschrankte orthonormierte Systeme , Acta Math. Acad. Sci. , 31 (1978), 279-285 . {18} K. Tandori , tiber die Divergenz der Orthogonalreihen, Publicationes Math . Debrecen, 8 (1961), 291-307. {19} K. Tandori, tiber die orthogonalen Funktionen. II (Summation) , Acta Sci . Math . (Szeged), 18 (1957), 149-168 . {20} K. Tandori, tiber die orthogonalen Funktionen. X (Unbedingte Konvergenz) , Acta Sci .

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