Acta Mathematica

A maximal theorem with function-theoretic applications

G. H. Hardy and J. E. Littlewood

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Acta Math., Volume 54 (1930), 81-116.

First available in Project Euclid: 31 January 2017

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1930 © Almqvist & Wiksells Boktryckeri-A.-B.


Hardy, G. H.; Littlewood, J. E. A maximal theorem with function-theoretic applications. Acta Math. 54 (1930), 81--116. doi:10.1007/BF02547518.

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  • Another proof has since been found by Mr. R. E. A. C. Paley, and will be published in the Proceedings of the London Mathematical Society.
  • The arguments used in §§ 5–6 are indeed mostly of the type which are intuitive to a student of cricket averages. A batsman's average is increased by his playing an innings greater than his present average; if his average is increased by playing an innings x, it is further increased by playing next an innings y> x; and so forth.
  • If the innings to date are 82, 4, 133, 0, 43, 58, 65, 53, 86, 30, the batsman says to himself at any rate my average for my last 8 innings is 58.5′ (a not uncommon psychology).
  • Our original proof of this lemma was much less satisfactory; the present one is due in substance to Mr T. W. Chaundy.
  • In what follows the symbol ‘Max’, when it refers to an infinite aggregate of values, is always to be interpreted in the sense of upper bound.
  • We suppress the straightforward but tiresome details of the proof.
  • See for example G. H. Hardy, ‘Note on a theorem of Hilbert’, Math. Zeitschrift, 6 (1919), 314–317, and ‘Notes on some points in the integral calculus’, Messenger of Math., 54 (1925), 150–156; and E. B. Elliott, ‘A simple exposition of some recently proved facts as to convergeney’, Journal London Math. Soc., 1 (1926), 93–96. A considerable number of other proofs have been given by other writers in the Journal of the London Mathematical Society.
  • This would not necessarily be true if the interval were infinite.
  • A. Zygmund, ‘Sur les fonctions conjuguées,’Fundamenta Math., 13 (1929), 284–303.
  • This very useful inequality is due to W. H. Young, ‘On a certain series of Fourier’, Proc. London Math. Soc. (2), 11 (1913), 357–366.
  • A will not occur again in the sense of Section III. Constants B, C in future presserve their identity.
  • Sn(θ) is formed from the first n+1 terms of the Fourier series of f(θ), σn(θ) from the first n.
  • When |θ|<σ the maximum is given by r=1, and when |θ|>1/2π by r=0.
  • The usefulness of a kernel of the type of X was first pointed out by Fejér. See L. Fejér, Über die arithmetischen Mittel erster Ordnung der Fourierreihe’, Göttinger Nachrichten, 1925, 13–17.
  • E. Kogbetliantz, ‘Les séries trigonométriques et les séries sphériques’, Annales de l'Ecole Normale (3), 40 (1923), 259–323.
  • There is of course no particular point in the precise shape of Sα(θ); it is an area of fixed size and shape including all ‘Stolz-paths’ to e inside an angle 2α. The radius vector corresponds to α=0.
  • J. E. Littlewood, ‘On functions subharmonic in a circle’, Journal Lond. Math. Soc., 2 (1927), 192–196.
  • F. Riesz, ‘Über die Randwerte einer analytischen Funktion’, Math. Zeitschrift, 18 (1923), 87–95.