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Published online by Cambridge University Press: 24 October 2008
1. Let
be a numerical series. If for sufficiently small h > 0 the series
is convergent, we can form the upper and lower limits of J (h) as h → 0. These limits are called respectively the upper and lower sums (R, 1) of the series (1). For the purposes of the present paper it will be convenient to consider a more extended definition of these upper and lower sums. We shall suppose that for sufficiently small h the series J (h) is summable by Poisson's method. We denote the Poisson sum by PJ (h). The upper and lower limits of PJ (h) as h → 0 will be called the upper and lower sums (R, 1) of the series (1).
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† Math. Zeit. 25 (1926), 274–290 (Theorem 1, Chap. II).Google Scholar