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Another proof has since been found by Mr. R. E. A. C. Paley, and will be published in theProceedings 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 inningsx, it is further increased by playing next an inningsy>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 theJournal 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. ConstantsB, C in future presserve their identity.
Sn(θ) is formed from the firstn+1 terms of the Fourier series off(θ), σn(θ) from the firstn.
When |θ|<σ the maximum is given byr=1, and when |θ|>1/2π byr=0.
The usefulness of a kernel of the type ofX 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 ofSα(θ); it is an area of fixed size and shape including all ‘Stolz-paths’ toe iθ 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.
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Hardy, G.H., Littlewood, J.E. A maximal theorem with function-theoretic applications. Acta Math. 54, 81–116 (1930). https://doi.org/10.1007/BF02547518
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DOI: https://doi.org/10.1007/BF02547518