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.,

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.

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.

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 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

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.

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 summability |N, pn| of a Fourier series at a point

1969 ◽  
Vol 65 (2) ◽  
pp. 495-506 ◽  
Author(s):  
H. P. Dikshit
Keyword(s):  
Fourier Series ◽  
Infinite Series ◽  
Partial Sums ◽  
Image Position

Let ∑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 write

On the non-absolute summability of a Fourier series and the conjugate of a Fourier series by a Nörlund method

1967 ◽  
Vol 63 (2) ◽  
pp. 407-411 ◽  
Author(s):  
R. Mohanty ◽  
B. K. Ray
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Absolute Summability ◽  
Partial Sums ◽  
Nörlund Means ◽  
Sequence Transformation ◽  
Image Position

Let {Sn} be the sequence of partial sums of the infinite seriesΣαn. Let {pn} be a sequence of constants real or complex and let us setThe sequence {tn} of Nörlund means (5) or simply (N, pn) means of the sequence {Sn} generated by the sequence of coefficients {pn} is defined by the following sequence -to-sequence transformationThe series ∑αn or the sequence {Sn} is said to be summable (N, pn) to the sum S, ifand is said to be absolutely summable (N, pn) or summable |N, pn|, if the sequence {tn} is of bounded variation, that is, the series ∑|tn − tn−1| is convergent (2).

scholarly journals Metric Fourier Approximation of Set-Valued Functions of Bounded Variation

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Elena E. Berdysheva ◽  
Nira Dyn ◽  
Elza Farkhi ◽  
Alona Mokhov
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Error Bounds ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Dirichlet Kernel ◽  
Partial Sums ◽  
Functions Of Bounded Variation ◽  
Moduli Of Continuity ◽  
Fourier Approximation

AbstractWe introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.

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