Contributions To Fourier Analysis by A. Zygmund

By A. Zygmund

Within the conception of convergence and summability even if for usual Fourier sequence or different expansions emphasis is put on the phenomenon of localization every time such happens, and within the current paper a undeniable element of this phenomenon can be studied for the matter of most sensible approximation to boot.

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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

Part (ill) follows very directly from (y) if we compare the expressions (3*0 and (68), and we are still left with proving the more comprehensive part (Iv). Since P - H has a boundary function zero, it suffices to prove that in general we have (69) (F,H) - o DIRICHLET PROBLEM II, for any function P with boundary function H(x) (70) hn r ^2^j^ fV 0, 37 and any harmonic function (1) Pn ^ ^ (6) for which 2Li (71) n ^ n h njs (1) , < oo Now (P,H) - lim r (72) and, due to AH = o, .. s the integral on the right is flr ' e) (73) dv e ' and this has the value (r) However (75) (H lnf (r) (i)|) s ^ E nf (D 2 T> 2 (i f ) and (69) follows now from (66) and (7OIf B is the outer spherical boundary of S, and if Other Domains ^.

16) ^ P IJ,' TT P < ^ J I + y^j 1 JL> ky+ J |CTX pc,TT-yl Si A xjl LO , ky J \v -*y Tj > 9 } fcy+yl JL9* ky I ( I I J ( y (u,v) = r>, W*>V(u,v) = * esc - sin (11^) x Cu> We shall refer to the relation (u) sin - esc esc (|) sin co y (v) esc (TLl) sin TSJL ^ (v ) , III.

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