scholarly journals On the absolute Cesaro summability factors of trigonometric series

1968 ◽  
Vol 16 (1) ◽  
pp. 71-75 ◽  
Author(s):  
Niranjan Singh
Keyword(s):  
Trigonometric Series ◽  
Infinite Series ◽  
Summability Factors ◽  
Cesàro Summability ◽  
Partial Sum ◽  
Cesaro Summability ◽  
The Absolute ◽  
Image Position

Let be any given infinite series with sn as its n-th partial sum.We writeandwhere

scholarly journals The Absolute Cesaro Summability of the Successively Derived Allied Series of a Fourier Series

1950 ◽  
Vol 8 (4) ◽  
pp. 163-176
Author(s):  
R. Mohanty
Keyword(s):  
Fourier Series ◽  
Cesàro Summability ◽  
Cesaro Summability ◽  
The Absolute ◽  
Image Position

We suppose that f(t) is integrable in the Lebesgue sense m (π, π) and is periodic with period 2π. We denote its Fourier series byThen the allied series is

scholarly journals Some sufficient conditions for the absolute Cesàro summability of series

1939 ◽  
Vol 6 (1) ◽  
pp. 51-56 ◽  
Author(s):  
J. M. Hyslop
Keyword(s):  
Sufficient Conditions ◽  
Cesàro Summability ◽  
Cesàro Mean ◽  
Cesaro Summability ◽  
Cesaro Mean ◽  
The Absolute ◽  
Image Position

denote the n-th Cesàro mean of order k for the series aΣan, that is,whereand let

On absolute summability factors of infinite series and their application to Fourier series

1967 ◽  
Vol 63 (1) ◽  
pp. 107-118 ◽  
Author(s):  
R. N. Mohapatra ◽  
G. Das ◽  
V. P. Srivastava
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Infinite Series ◽  
Absolute Summability ◽  
Summability Factors ◽  
Partial Sum ◽  
Absolute Summability Factors ◽  
Image Position

Definition. Let {sn} be the n-th partial sum of a given infinite series. If the transformationwhereis a sequence of bounded variation, we say that εanis summable |C, α|.

Summability factors for Riesz loǵarithmic means of order one for a Fourier series

1970 ◽  
Vol 67 (2) ◽  
pp. 307-320
Author(s):  
R. N. Mohapatra
Keyword(s):  
Fourier Series ◽  
Infinite Series ◽  
Summability Factors ◽  
Partial Sum ◽  
Logarithmic Means ◽  
Image Position

Let 0 < λ1 < λ2 < … < λn → ∞ (n→∞). We writeLet ∑an be a given infinite series with the sequence {sn} for its nth partial sum. The (R, λ, 1) mean of the sequence {sn} is given by

Localization Problem of the Absolute Riesz and Absolute Nörlund Summabilities of Fourier Series

1970 ◽  
Vol 22 (3) ◽  
pp. 615-625 ◽  
Author(s):  
Masako Izumi ◽  
Shin-Ichi Izumi
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Infinite Series ◽  
Integrable Function ◽  
Localization Problem ◽  
Partial Sum ◽  
The Absolute ◽  
Image Position

1.1. Let Σ an be an infinite series and sn its nth partial sum. Let (pn) be a sequence of positive numbers such thatIf 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.

A new note on generalized absolute Cesàro summability factors

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3093-3096
Author(s):  
Hüseyin Bor
Keyword(s):  
Infinite Series ◽  
Summability Method ◽  
Summability Factors ◽  
Cesàro Summability ◽  
Monotone Sequences ◽  
Cesaro Summability ◽  
Power Increasing Sequences ◽  
Increasing Sequences ◽  
General Summability

Quite recently, in [10], we have proved a theorem dealing with the generalized absolute Ces?ro summability factors of infinite series by using quasi monotone sequences and quasi power increasing sequences. In this paper, we generalize this theorem for the more general summability method. This new theorem also includes some new and known results.

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