University of Michigan Studies Volume 2; Scientific Series by University of Michigan
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Author: University of Michigan
Page Count: 54 pages
Published Date: 01 May 2012
Publisher: Rarebooksclub.com
Publication Country: Miami Fl, United States
Language: English
Type: eBook
ISBN: 9781236260505
File size: 37 Mb
File Name: University.of.Michigan.Studies.Volume.2;.Scientific.Series.pdf
Download Link: University of Michigan Studies Volume 2; Scientific Series
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1916 Excerpt: ... Cf. Hardy, Quarterly Journ. of Math., Vol. 35 (1904), pp. 25, 28. M Cf. Hardy, I. c, pp. 43-44. - Cf. Hardy, I. c, pp. 44.45 Cf. Bromwich, I. c, 24. This result evidently becomes of especial significance for all series (11) of the form ao--Oi + 02--at-f-...; am "positive for which the successive differences between the quantities a0, ai, a2, ... all eventually vanish--e. g., the series 1-2 + 3-4 + 5 wherein the quantities E0, E, E2, etc., become Eo--1, Ei =--1, Ei = E3 =... = En = 0, and hence s = 5--=. The proof of statement (8) may be readily supplied when we make use of the Lemma of 37. Thus, in the notation there employed, let us take in the present instance and condition (4) of the lemma is at once seen to be satisfied (cf. (16)). Application of the lemma thus gives 00 (0Z7) I = lim e-2" --p -n = lim e-2as(2a) = lim ---(-). p--00 n=0 Til a--to a--n where s(a) is defined by (13). Moreover, this result may be written (cf. (25)) in the form CHAPTER V THE SUMMABILITY AND CONVERGENCE OF FOURIER SERIES AND ALLIED DEVELOPMENTS 45. In the present chapter it is proposed to derive the principal known results concerning the summability of Fourier series and other allied developments for functions of one real variable (developments in terms of Bessel functions, Legendre functions, etc.).1 We shall take the word " sum " in the Holder sense2 ( 36) according to which a given series (convergent or divergent) (1) Z-n has its sum s defined by the equation (2) s = lim snM; r = fixed integer = 0, n=oo where n(0) = sn = -0 + -i +... + -, = (o(0) + i(0) +... + n(0)), n-f-l n-+-l Moreover, if the terms un are functions of the (real) variable x (as will now always be the case) wh...
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