By Janos Horvath

A wonderful interval of Hungarian arithmetic began in 1900 while Lipót Fejér found the summability of Fourier series.This was once via the discoveries of his disciples in Fourier research and within the idea of analytic features. while Frederic (Frigyes) Riesz created useful research and Alfred Haar gave the 1st instance of wavelets. Later the subjects investigated by way of Hungarian mathematicians broadened significantly, and incorporated topology, operator thought, differential equations, likelihood, and so on. the current quantity, the 1st of 2, provides probably the most outstanding effects accomplished within the 20th century by way of Hungarians in research, geometry and stochastics. The booklet is available to an individual with a minimal wisdom of arithmetic. it's supplemented with an essay at the historical past of Hungary within the 20th century and biographies of these mathematicians who're not energetic. a listing of all people said within the chapters concludes the amount.

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1). e. e. 14) L a~ 00 ( 1)2 < 00, log , a 2 k k=l where lOg U if u log ; u:= { 1 if u ~ 2 < 2. 49 Constructive Function Th eory: I. 12) for E = O. e. for all ONS (¢k(X)). 6). 6. CESARO SUMMABILITY OF ORTHOGONAL SERIES Similarly to trigonometric Fourier series, one can expe ct better convergence behavior of orthogonal series if ordinary convergence is replaced by (C, 1)summability. In fact, this is the case , but the improvement is less than that in the case of trigonometric series. 1) with partial sum n sn(x) := L ak¢k(x) k=l is defined by 1 n n ( O"n(x) := ;; ~ Sk( X) = ~ 1 k=l k=l k~ 1) ak¢k(x) , n = 1,2 , .

He had numerous disciples in Hungary as well as all over the world. His monograph [8J on orthogonal series appeared in German in 1960, translated into English the following year , and into Russian in 1963. It has became the standard reference book for researchers in the theory of orthogonal series. Motivated by the notion of strong (C, 1)-summability introduced by G. H. Hardy and J. E. Littlewood in 1913, Alexits {2} introduced the notion of strong approximation by Fourier series in 1963, and by raising several problems in subsequent papers he laid the foundation of the so-called Constructive Function Theory: I.

E. But this requirement is too strong for a more delicate study of convergence problems. e. 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.