q-Analogues of Guillera’s two series for π±2 with convergence rate 27 64

2020 ◽  
pp. 1-20
Author(s):  
Xiaojing Chen ◽  
Wenchang Chu
Keyword(s):  
Convergence Rate ◽  
Infinite Series ◽  
Classical Formula ◽  
Partial Sum

Three transformation formulae are established for the partial sum of Bailey’s well-poised [Formula: see text]-series. Their particular cases provide [Formula: see text]-analogues of Guillera’s two series for [Formula: see text] with convergence rate [Formula: see text], and for other classical [Formula: see text]-related infinite series.

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

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.

Extension of a theorem of Sunouchi

1969 ◽  
Vol 65 (2) ◽  
pp. 489-494 ◽  
Author(s):  
V. P. Srivastava
Keyword(s):  
Infinite Series ◽  
Cesàro Means ◽  
Cesaro Means ◽  
Partial Sum ◽  
Cesáro Means

1·1. Definitions and notations. Let ∑an be a given infinite series, and let sn be its nth partial sum. We denote by and the nth Cesàro means of order α(α > − 1) of the sequences {sn} and {n. an} respectively.

Toroidal Helical Fields

1987 ◽  
Vol 42 (10) ◽  
pp. 1124-1132 ◽  
Author(s):  
M. Y. Kucinski ◽  
I. L. Caldas
Keyword(s):  
Coordinate System ◽  
Infinite Series ◽  
Plasma Column ◽  
Scalar Potential ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Small Curvature ◽  
Partial Sum ◽  
Magnetic Surfaces ◽  
Series Of Functions

Using the conventional toroidal coordinate system Laplace’s equation for the magnetic scalar potential due to toroidal helical currents is solved. The potential is written as a sum of an infinite series of functions. Each partial sum represents the potential within some accuracy. The effect of the winding law is analysed in the case of small curvature. Approximate magnetic surfaces formed by toroidal helical currents flowing around a standard tokamak chamber are determined. Stability of the plasma column in this system against displacements is discussed.

Convergence of Finite Fifference Method for Parabolic Problem

2003 ◽  
Vol 3 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Dejan Bojović
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Parabolic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

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