# Contributions To Fourier Analysis by A. Zygmund

By A. Zygmund

Within the conception 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 top approximation besides.

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Over Q. See In general a function gePL(Q) is not in L(Q). ' See Th. 7-1 NfP (7). we If g is in L over ) k. The continuous extension of an PL-integral g over Q will again be denoted by g. THE PRECHET VARIATION III. 12) 53 v \ fTT g(3,t) dS dt - g(U,v) , TT! 13) - s

S. Bochner, "Linear partial differential equations with constant coefficients," Annals of Math, ki (191*6), pp. 202-212. Compare also Bochner and W. T. Martin, Several complex variables, Princeton 19^8, p. 158 ff. kk S. BOCHNER Returning to the Principle itself, we now define a Banach space by using the norm (95) I If I ! p C ( (IfjP + ... + |f P p k ) l /\. dv/ The subspace iP of gradient fields of harmonic instead of (91 ), for p ^ functions is again a closed subspace, and we claim that i . \ f - infh XI o I If - hi I P again Iff (n) h such that h| I for some Mf h.

If A and B are two such conditions A + B shall denote the set of conditions A and B. 17)' A.. 17) g(u,v) g(u,v) (x > o) g(u,v) (y > o) - - g(u+x,v) g(u,v+y) + g(u,v) . 18) A g A g g In addition to coordinates (u,v), coordinates (s,t) will be used. 17) replacing x and y by s and t. The object io to this somewhat imperfect notation seem outweighed by its convenience. 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. ^ !