Description - Studies on Divergent Series and Summability Volume 2 by Walter Burton Ford
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: ...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 Holdek sense2 ( 36) according to which a given series (convergent or divergent) (1) Jlun n=0 has its sum s defined by the equation (2) s = lim - (r); r = fixed integer = 0, n=oo where - 0 = - = -o + -i + + -n, - (1)--4-; ((0) + i(0) +. + (0)), n-f i n-f-l Moreover, if the terms un are functions of the (real) variable x (as will now always be the case) when considered throughout an interval (a, b), the series (1) will be termed uniformly summable throughout (a, b) in accordance with the definition II of 43--i. e., provided that the limit (2) is approached uniformly for the same values of x. In view of the fact that any discussion of the summability of Fourier series and other allied developments is intimately connected with the corresponding discussion of convergence, the latter being in fact but the case of summability in which r = 0 (cf. (2)), we shall as a matter of course elaborate both aspects of the subject.3 No attempts will be made however to obtain theorems containing the minimum restrictions for a given function f(x) in order that it be 1 See explanatory remarks in the Preface. 2 The results obtained will therefore ( 38) be convertible at any point into those for summability in the Cesaro sense. 'Since all convergent series are summable but...
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