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

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|.

On the convergence factor of a Fourier series and a differentiated Fourier series

1969 ◽  
Vol 65 (1) ◽  
pp. 75-85 ◽  
Author(s):  
R. Mohanty ◽  
B. K. Ray
Keyword(s):  
Fourier Series ◽  
Convergence Factor ◽  
Partial Sum ◽  
Logarithmic Means ◽  
Image Position

Definition A. The serieswith partial sumUn (or the sequence {Un}) is said to be summable by logarithmic means to the sum U, if

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

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.

scholarly journals Absolute Riesz summability factors for Fourier series

1970 ◽  
Vol 17 (1) ◽  
pp. 65-70
Author(s):  
Prem Chandra
Keyword(s):  
Fourier Series ◽  
Infinite Series ◽  
Summability Factors ◽  
Monotonic Sequence ◽  
Riesz Mean ◽  
Riesz Summability ◽  
Absolute Riesz Summability ◽  
Image Position

Let ∑an be a given infinite series and {λn} a non-negative, strictly increasing, monotonic sequence, tending to infinity with n. We write, for w > λ0,and, for r>0, we write is known as the Riesz sum of “ type ” λn and “ order ” r, andis called the Riesz mean of type λn and order r.

On the absolute Hausdorff summability factors of a Fourier series

1969 ◽  
Vol 66 (2) ◽  
pp. 355-363
Author(s):  
N. Tripathy
Keyword(s):  
Fourier Series ◽  
Infinite Series ◽  
Partial Sums ◽  
Summability Factors ◽  
The Absolute ◽  
Image Position

Let be a given infinite series with the sequence of partial sums {sn}. Then the sequence-to-sequence Hausdorff transformation of the sequence {sn} is given by

On the Absolute Nörlund Summability of a Fourier Series

1973 ◽  
Vol 16 (4) ◽  
pp. 599-602
Author(s):  
D. S. Goel ◽  
B. N. Sahney
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Infinite Series ◽  
Partial Sums ◽  
The Absolute ◽  
Image Position ◽  
Absolute Nörlund Summability

Let be a given infinite series and {sn} the sequence of its partial sums. Let {pn} be a sequence of constants, real or complex, and let us write(1.1)If(1.2)as n→∞, we say that the series is summable by the Nörlund method (N,pn) to σ. The series is said to be absolutely summable (N,pn) or summable |N,pn| if σn is of bounded variation, i.e.,(1.3)

On the absolute Nörlund summability of a Fourier series

1967 ◽  
Vol 7 (2) ◽  
pp. 252-256 ◽  
Author(s):  
Fu Cheng Hsiang
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Infinite Series ◽  
Partial Sums ◽  
The Absolute ◽  
Image Position ◽  
Absolute Nörlund Summability

Let Σn−0∞an, be a given infinite series and {sn} the sequence of its partial sums. Let {pn} be a sequence of constants, real or complex, and let us writeIfas n → ∞, then we say that the series is summable by the Nörlund method (N, pn) to σ And the series a,Σan, is said to be absolutely summable (N, pn) or summable |N, Pn| if {σn} is of bounded variation, i.e.,

scholarly journals Absolute weighted arithmetic mean summability factors of infinite series and trigonometric fourier series

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4963-4968 ◽  
Author(s):  
Hüseyin Bor
Keyword(s):  
Fourier Series ◽  
Infinite Series ◽  
Arithmetic Mean ◽  
Trigonometric Fourier Series ◽  
Summability Factors ◽  
Weighted Arithmetic ◽  
Quasi Power Increasing Sequence

In this paper, we generalized a known theorem dealing with absolute weighted arithmetic mean summability of infinite series by using a quasi-f-power increasing sequence instead of a quasi-?-power increasing sequence. And we applied it to the trigonometric Fourier series

Sign in / Sign up

Export Citation Format

Share Document