Download Seminaire de Theorie du Potentiel Paris No 4 by F. Hirsch, G. Mokobodzki, M. Brelot, G. Choquet, J. Deny PDF

By F. Hirsch, G. Mokobodzki, M. Brelot, G. Choquet, J. Deny

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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 .

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