Contributions To Fourier Analysis by A. Zygmund

By A. Zygmund

Within the idea of convergence and summability even if for traditional Fourier sequence or different expansions emphasis is put on the phenomenon of localization each time such happens, and within the current paper a undeniable point of this phenomenon could be studied for the matter of most sensible approximation to boot.

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Also make an anomalous use of g/u, g/v, g/uv, g uv etc. as designating functions with values , uv g(u,v) etc. \ respectively. Y 5. ^ ! P LO' oj The interval \uv 9 , y 1 LO+,, + J \ A xy (0 < x TT) is the Cartesian product (0,x] Co < y X(0,y]. IT) ' III. THE PRECHET VARIATION 55 -o,,, fx ,, L, P , L2 P , L' J iv y -x pT A TT -yl f 1 , "TT LX f -yl ,y _\ J pTT -x - o(x); ^y^\ P y\ f 1 * oj \u , *xfj > |_* - o(y) A \^xy I^J O (1) T(? ; -x ,TT -y] i , Lp P Lkx P B;^ , 1"xy J ley -x, If f L. -jF] P Lo>^ Jiv^y^j , i 1 /, m\ p [Tx- ; S] {** (f )] .

J, j , . , k, But now take a sequence of such gradient fields say. ) in norm, and the functions 'i converge in |f. U J F^ (x) themselves also converge towards some continuous function P(x), uniformly in every S', say. (x) with continuous derivatives in all of S, but vanishing outside some neighborhood in S each, we have the DuBois Reymond relations Inclenr-n^ent of towards + f .

5. Soc. son, "Uniformly convex spaces," Trans Compare J. A. Clark - IKD (1936), pp. Amer. Math. II. , x- then we again have h = h throughout. Finally we note that these /V -statements also hold for the modified 1 norm <$ s ^ 2 f 2)P/2 * which is structurally closer to 7 ) than (95) is. 2 it was shown by a factor the value from thet the Dirichlet Integral (20) has apart Remarks on the Problem 1. Douglas (91 2TT For k Plateau. of. 2TT - (95 \ ) ^ sin 2 ded

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