scholarly journals On (J, pn)-summability of Fourier series

1972 ◽  
Vol 18 (1) ◽  
pp. 13-17
Author(s):  
F. M. Khan
Keyword(s):  
Fourier Series ◽  
Power Series ◽  
Radius Of Convergence ◽  
Partial Sums ◽  
Summability Of Fourier Series ◽  
Image Position

Let pn>0 be such that pn diverges, and the radius of convergence of the power seriesis 1. Given any series σan with partial sums sn, we shall use the notationand

A criterion for the (e, c)-summability of Fourier series

1982 ◽  
Vol 92 (1) ◽  
pp. 121-127 ◽  
Author(s):  
Jamil A. Siddiqi
Keyword(s):  
Fourier Series ◽  
Tauberian Theorem ◽  
Partial Sums ◽  
Borel Summability ◽  
Regular Method ◽  
Summability Of Fourier Series ◽  
Image Position

1. A series with partial sums {An} is said to be summable (e, c) (c > 0) ifexists, where it is to be understood that An+k = 0 when n+k < 0. The (e, c)-sum-mability method which is a regular method of summation was introduced by Hardy and Littlewood (3) (cf. also (5)) as an auxiliary method to prove the Tauberian theorem for Borel summability, viz, if Σan is summable (B) to A and an = 0(n−½), then it converges to A.

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)

scholarly journals On absolute riesz summability of Fourier series

1975 ◽  
Vol 19 (1) ◽  
pp. 97-102
Author(s):  
G. D. Dikshit
Keyword(s):  
Fourier Series ◽  
General Theorem ◽  
Riesz Summability ◽  
Absolute Riesz Summability ◽  
Summability Of Fourier Series ◽  
Image Position

AbstractLet and .In 1951 Mohanty proved the following theorem: .In this paper a general theorem on summability |R,l (w), 1 | of Σ An(x) has been given which improves upon Mohanty's result in different ways (see Corollaries 1, 2 and 3) and it is also shown that some of the results of this note are the best possible.

A Criterion for Taylor Summability of Fourier Series

1979 ◽  
Vol 22 (3) ◽  
pp. 345-350 ◽  
Author(s):  
A. S. B. Holland ◽  
B. N. Sahney ◽  
J. Tzimbalario
Keyword(s):  
Fourier Series ◽  
Negative Integer ◽  
Summability Of Fourier Series ◽  
Image Position

Let {ank} be a matrix defined by1and n taking only non-negative integer values.Let f(x) ∈ L [0, 2π] and be periodic with period 2π outside this interval. Let the Fourier series associated with the function f(x) be given byand letwhere s is a constant.

XXII.—On Methods of Summability Based on Power Series

1957 ◽  
Vol 64 (4) ◽  
pp. 342-349
Author(s):  
D. Borwein
Keyword(s):  
Power Series ◽  
Summability Method ◽  
Radius Of Convergence ◽  
Main Concern ◽  
Image Position

SynopsisGiven a power series with real non-negative coefficients and having radius of convergence p, a summability method P is defined as follows:The main concern of this note is to establish conditions sufficient for one such method to include another.

RESTRICTED UNIVERSALITY OF POWER SERIES

2001 ◽  
Vol 33 (5) ◽  
pp. 543-552 ◽  
Author(s):  
JEAN-PIERRE KAHANE ◽  
ANTONIOS D. MELAS
Keyword(s):  
Power Series ◽  
Finite Union ◽  
Radius Of Convergence ◽  
Nonempty Interior ◽  
Partial Sums ◽  
Universal Approximation ◽  
Compact Set ◽  
Approximation Properties

We prove the existence of a power series having radius of convergence 0, whose partial sums have universal approximation properties on any compact set with connected complement that is contained in a finite union of circles centred at 0 and having rational radii, but do not have such properties on any compact set with nonempty interior. This relates to a theorem of A. I. Seleznev.

A generalization of the Hurwitz composition theorem to irregular power series

1951 ◽  
Vol 47 (3) ◽  
pp. 477-482 ◽  
Author(s):  
H. G. Eggleston
Keyword(s):  
Power Series ◽  
Radius Of Convergence ◽  
Finite Radius ◽  
Image Position ◽  
Composition Theorem

When two functions are given, each with a finite radius of convergence, a theorem due independently to Hurwitz and Pincherle (1, 2) provides information about the position of the singularities of the functionin terms of the positions of the singularities of f(z) and g(z).

scholarly journals Fatou-Riesz-theorems in general sequence spaces

1980 ◽  
Vol 23 (2) ◽  
pp. 199-205
Author(s):  
Gerhard Otto Müller ◽  
Rolf Trautner
Keyword(s):  
Power Series ◽  
Formal Series ◽  
Sequence Spaces ◽  
Partial Sums ◽  
Classical Theorem ◽  
General Sequence ◽  
Image Position

Consider a formal serieswith partial sumsand the corresponding power series. Throughout we will assume thatfis analytic for |z| <1, i.e. thatA classical theorem of Fatou-Riesz (see (1,4)) states that ifandthenis convergent to 0.

On the Cesàro summability of Fourier series and Allied series

1930 ◽  
Vol 26 (2) ◽  
pp. 173-203 ◽  
Author(s):  
R. E. A. C. Paley
Keyword(s):  
Fourier Series ◽  
Cesàro Summability ◽  
Cesàro Mean ◽  
Cesaro Summability ◽  
Cesaro Mean ◽  
Summability Of Fourier Series ◽  
Image Position

For r>-1, let denoteIfwe say that the series a0 + a1 + a2 +…+an+… is summable by Cesàro mean of order r, or more shortly summable (C, r) to sum s. If r >−1, andwe say that the series is summable by Rieszian mean of order r to the sum s. It has been shown that these two methods of summation are equivalent. Throughout this paper I shall deal with the Rieszian mean, but I shall retain the symbol (C, r). It is known† that if a series is summable (C, r), it is also summable (C, r′) to the same sum for all numbers r′ greater than r.

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