By Janos Horvath

An excellent interval of Hungarian arithmetic began in 1900 whilst Lipót Fejér came across the summability of Fourier series.This was once by way of the discoveries of his disciples in Fourier research and within the thought of analytic features. whilst Frederic (Frigyes) Riesz created practical research and Alfred Haar gave the 1st instance of wavelets. Later the themes investigated through Hungarian mathematicians broadened significantly, and integrated topology, operator thought, differential equations, likelihood, and so on. the current quantity, the 1st of 2, offers probably the most awesome effects accomplished within the 20th century through Hungarians in research, geometry and stochastics. The publication is obtainable to somebody with a minimal wisdom of arithmetic. it really is supplemented with an essay at the heritage of Hungary within the 20th century and biographies of these mathematicians who're now not energetic. a listing of all individuals pointed out within the chapters concludes the amount.

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**Example text**

Lakl ~ ... be a decreasing rearrangement of the sequence (ak) of numbers. e. if and only if L L 00 p=O { lakf(log k)2 }1/2 < 00 . kElp The idea of studying unconditional convergence is due to Wladislaw Orlicz {12}, who proved the first basic result in this direction. His result , which is actually a consequence of the above theorem of Tandori, is formulated in terms of the so-called Weyl multiplier. 52 F. M6ricz Corollary. Let (A(k) : k = 1,2 , ... ) be an increasing sequence of positive numbers.

Orthogonal Series A 21f-periodic function I is said to satisfy the uniform Lipschitz condition of order a > 0, in symbol: I E Lip21r a, if Only the case 0 < a :S 1 is interesting: if a > 1, then w(f,o)/o tends to zero with o. Consequently, in this case f'(x) exists and is zero everywhere, and I is constant. The function w(f, 0), 0 :S 0 < 21f, is called the modulus of continuity of I. It is clear that a function I is uniformly continuous if and only if w(f, 0) tends to zero with o. On the other hand, I belongs to LiP21r 1 if and only if I is the antiderivative of a bounded function .

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 . Math. (Szeged) , 23 (1962), 185-22l. {21} M. Zamansky, Classes de saturation de certains precedes d'approximation des series de Fourier, Annales de l'Ecole Normale Superieure, 66 (1949), 19-93 .