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Localization Problem of the Absolute Riesz and Absolute Nörlund Summabilities of Fourier Series
Published online by Cambridge University Press: 20 November 2018
Extract
1.1. Let Σ an be an infinite series and sn its nth partial sum. Let (pn) be a sequence of positive numbers such that
If the sequence
(1)
is of bounded variation, that is, Σ |tn – tn –1| < ∞, then the series Σ an is said to be absolutely (R, pn, 1) summable or |R, pn, 1| summable.
Let ƒ be an integrable function with period 2π and let its Fourier series be
(2)
Dikshit [4] (cf. Bhatt [1] and Matsumoto [7]) has proved the following theorems.
THEOREM I. Suppose that (i) the sequence (pn/Pn) is monotone decreasing, (ii) mn > 0, (iii) the sequence (mnpn/Pn) decreases monotonically to zero, and (iv) the series Σ mnPn/Pn) is divergent.
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- Copyright © Canadian Mathematical Society 1970
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