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.

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 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 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 On the Absolute Summability (A) of Infinite Series

1932 ◽  
Vol 3 (2) ◽  
pp. 132-134 ◽  
Author(s):  
M Fekete
Keyword(s):  
Bounded Variation ◽  
Infinite Series ◽  
Absolute Summability ◽  
The Absolute ◽  
Image Position

§1. A serieshas been defined by J. M. Whittaker to be absolutely summable (A), ifis convergent in (0 ≤ x < 1) and f (x) is of bounded variation in (0, 1), i.e.for all subdivisions 0 = x0 < x1 < x2 < . … < xm < 1.

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

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

On the Absolute Nörlund Summability of a Fourier series (II)

1969 ◽  
Vol 9 (1-2) ◽  
pp. 161-166 ◽  
Author(s):  
Fu Cheng Hsiang
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Partial Sums ◽  
Nörlund Means ◽  
The Absolute ◽  
Image Position ◽  
Absolute Nörlund Summability

Let be a given series with its partial sums {Sn} and {Pn} a sequence of real or complex parameters. Write. The transformation given by defines the Nörlund means of {Sn} generated by {Pn}. The series Σann is said to be absolutely summable (N, pn) or summable ∣N, pn∣, if {tn} is of bounded variation, i.e., Σ|tn—tn−1| converges.

On the absolute summability and convergence of Fourier series and associated series

1971 ◽  
Vol 70 (3) ◽  
pp. 421-433
Author(s):  
M. K. Nayak
Keyword(s):  
Fourier Series ◽  
General Result ◽  
Absolute Summability ◽  
Partial Sum ◽  
The Absolute ◽  
Image Position

1. We suppose that f(t) is integrable L and periodic with period 2π and we denote the Fourier series of f(t) at the point t = x byWe denote the nth partial sum of the Fourier series by sn. We will prove here a general result (i.e Theorem 1) and some interesting results in connexion with associated series of a Fourier series of the typeWe writeThe object of the present paper is to prove the following theorems. (In all the following cases k is any constant > π if nothing else is mentioned.)

scholarly journals A Note on the Absolute Nörlund Summability of Conjugate Fourier Series

1971 ◽  
Vol 12 (1) ◽  
pp. 86-90 ◽  
Author(s):  
G. D. Dikshit
Keyword(s):  
Fourier Series ◽  
Infinite Series ◽  
Partial Sums ◽  
Conjugate Fourier Series ◽  
Nörlund Means ◽  
Sequence Transformation ◽  
The Absolute ◽  
Image Position ◽  
Absolute Nörlund Summability

Let σan be an infinite series, with sequence of partial sums {un}. Let {pn} be a sequence of constants, real or complex, and write Pn = po+p1+ … +pn The sequence-to-sequence transformation defines the sequence {tn} of Nörlund means of the sequence {u}, generated by the sequence {pn}. The series σan is said to be surnmable (N, pn), to sum s, if limn→∞ tn = s. It is said to be absolutely sum.mable (N, pn), or summable |N, pn|, if {tn} ∈BV.

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

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