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Published online by Cambridge University Press: 24 October 2008
1. Let {μn} be a sequence of positive numbers increasing to infinity. If the series
converges for s > 0, and if f(s) → l as s → 0, then ∑an is said to be summable (A, μn) to l; this generalization of Abel summability, first introduced by Hardy, has long been familiar. Suppose that, for x ≥ μ0, ø(x)is a positive increasing function of x, which tends to infinity as x → ∞. We write throughout λn = ø(μn), and
if this exists.
* Whenever no limits are stated, sums are to be taken to be from n = 0 to n = ∞.
† These assumptions are made throughout the paper.
‡ Cartwright, M. L., ‘The relation between the different types of Abel summation’, Proc. London Math. Soc. (2), 31 (1930), 81–96CrossRefGoogle Scholar; G. H. Hardy, ‘Theorems relating to the summability and convergence of slowly oscillating series’, ibid. (2), 8 (1910), 301–20 (319). Hardy gives the slightly weaker result in which (2) is assumed convergent for s > 0.
§ Hardy, G. H., ‘The second theorem of consistency for summable series’, Proc. London Math. Soc. (2), 15 (1916), 72–88Google Scholar; K. A. Hirst, ‘On the second theorem of consistency in the theory of summation by typical means’, ibid. (2), 33 (1932), 353–66.
* Kuttner, B., ‘On positive Riesz and Abel typical means’, Proc. London Math. Soc. (2), 49 (1947), 328–52Google Scholar. This paper will be referred to as R.A.
* Widder, D. V., The Laplace Transform (Princeton, 1941), 306–10.Google Scholar
† We suppose, as always, that ø(x) also satisfies the conditions stated in § 1. The additional restrictions now imposed involve only the values of ø(x) for x ≥ A. Thus we restrict the behaviour of ø(x) for sufficiently large x only. (It is trivial that if the conditions are satisfied for a given value of A, then they are satisfied for any greater value.)
* In the definition of R(u), c is replaced by γ.
† Wiener, N., The Fourier Integral (Cambridge, 1933), 75–98Google Scholar. The result is not explicitly stated in Wiener‘s book, but is established by the proof of Theorem 7 in the same way as Theorem 8 is established by the proof of Theorem 6. In order to obtain the result in a form convenient for our purposes, we have made some obvious changes of variable, replacing the x of Wiener‘s book by log x, K 1(x) by xψ 1(x), etc. (The function corresponding to the K 3 of Wiener‘s book is ω, not ψ 3; ψ 3 corresponds to the inner integral in (14.02).)
* See, for example, Widder, loc. cit. 160 (Theorem 12a).
* See, for example, Littlewood, J. E., Lectures on the Theory of Functions (Oxford, 1944), 113Google Scholar (Theorem 109).
