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The Symmetric Derivative and its application to the Theory of Trigonometric Series

Published online by Cambridge University Press:  24 October 2008

S. Verblunsky
Affiliation:
Magdalene College

Extract

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).

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1931

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References

Khintchine, , Fund. Math. 9 (1927), 212279 (§ 1).CrossRefGoogle Scholar

* Proc. Lond. Math. Soc. 31 (1930), 373.Google Scholar

Math. Zeit. 25 (1926), 274290 (Theorem 1, Chap. II).Google Scholar