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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

Masako Izumi  and
Shin-Ichi Izumi
Masako Izumi
Affiliation:
The Australian National University, Canberra (ACT), Australia
Shin-Ichi Izumi
Affiliation:
The Australian National University, Canberra (ACT), Australia
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 3 , 01 June 1970 , pp. 615 - 625
Copyright
Copyright © Canadian Mathematical Society 1970

References

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