The Gibbs Phenomenon and Lebesgue Constants for Regular Sonnenschein Matrices

1962 ◽  
Vol 14 ◽  
pp. 723-728 ◽  
Author(s):  
W. T. Sledd
Keyword(s):  
Fourier Series ◽  
Gibbs Phenomenon ◽  
Summability Method ◽  
Jump Discontinuity ◽  
Partial Sums ◽  
Lebesgue Constants ◽  
Dirichlet Conditions ◽  
Image Position

Ifψ(x)is a real-valued function which has a jump discontinuity atx= ε and otherwise satisfies the Dirichlet conditions in a neighbourhood ofx= ε then{sn(x)}the sequence of partial sums of the Fourier series forψ(x)cannot converge uniformly atx =ε. Moreover, it can be shown that given τ in [ — π, π] then there is a sequence {tn} such thattn→ ε andThis behaviour of{sn(x)}is called the Gibbs phenomenon. If {σn(x)} is the transform of{sn(x)}by a summability methodT, and if {σn(x)} also has the property described then we say thatTpreserves the Gibbs phenomenon.

The Gibbs Phenomenon for Generalized Taylor and Euler Transforms

1975 ◽  
Vol 27 (2) ◽  
pp. 384-395
Author(s):  
Robert E. Powell ◽  
Richard A. Shoop
Keyword(s):  
Fourier Series ◽  
Gibbs Phenomenon ◽  
Jump Discontinuity ◽  
Partial Sums ◽  
Dirichlet Conditions

Let f be a real-valued function satisfying the Dirichlet conditions in a neighborhood of x = x0, at which point f has a jump discontinuity. If {Sn(x)} is the sequence of partial sums of the Fourier series of f at x, then ﹛Sn(x)﹜ cannot converge uniformly at x — x0. Moreover, for any number , there exists a sequence ﹛tn﹜, where tn → x0 and

A Remark on the Gibbs Phenomenon and Lebesgue Constants for a Summability Method of Melikov

1968 ◽  
Vol 11 (2) ◽  
pp. 301-303
Author(s):  
Fred Ustina
Keyword(s):  
Gibbs Phenomenon ◽  
Summability Method ◽  
Lebesgue Constants ◽  
Image Position

Let u = ∑ukbe a given series and letMelikov [4] has defined the n-th σ- transform of u bywhere ε and θ are assumed to be non-negative. This is easily shown to be equivalent toThe method is a generalization of a method used by Kaufman [l], and of another one used by Melikov [5]. It reduces to the (n-l)th (C;l) mean when θ = 0 and ε = 0, and to the n-th (C;l) mean when θ = 1 and ε = 0.

Gibbs phenomenon for the Hausdorff means of double sequences

1970 ◽  
Vol 11 (2) ◽  
pp. 169-185 ◽  
Author(s):  
Fred Ustina
Keyword(s):  
Fourier Series ◽  
Weight Function ◽  
Gibbs Phenomenon ◽  
Partial Sums ◽  
Double Sequences ◽  
Image Position ◽  
Hausdorff Means

let g (u) be a regular Hausdorff weight function, and let hm (Ψ x m) denote the mthe corresponding Hausdorff transform, evaluated at x m, of the sequence of partial sums of the Fourier series of Ψ (x), where . In [3], Szász investigated the Gibbs phenomenon for Ψ(x) for these means. His main results are contained in the following two theorems: . (3) THEOREM 2. Taking the limit superior as. If this maximum is attained for τ = τ′ then.

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 (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

scholarly journals Improving Fourier Partial Sum Approximation for Discontinuous Functions Using a Weight Function

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Beong In Yun
Keyword(s):  
Target Function ◽  
Gibbs Phenomenon ◽  
Weighted Averaging ◽  
Jump Discontinuity ◽  
Partial Sums ◽  
Left Hand ◽  
Partial Sum ◽  
Hand Part ◽  
Approximate Function ◽  
The Right

We introduce a generalized sigmoidal transformation wm(r;x) on a given interval [a,b] with a threshold at x=r∈(a,b). Using wm(r;x), we develop a weighted averaging method in order to improve Fourier partial sum approximation for a function having a jump-discontinuity. The method is based on the decomposition of the target function into the left-hand and the right-hand part extensions. The resultant approximate function is composed of the Fourier partial sums of each part extension. The pointwise convergence of the presented method and its availability for resolving Gibbs phenomenon are proved. The efficiency of the method is shown by some numerical examples.

The Lebesgue Constants for (γ, r) Summation of Fourier Series

1963 ◽  
Vol 6 (2) ◽  
pp. 179-182 ◽  
Author(s):  
Lee Lorch ◽  
Donald J. Newman
Keyword(s):  
Fourier Series ◽  
Imaginary Part ◽  
Complex Number ◽  
Circle Method ◽  
Summation Method ◽  
Lebesgue Constants ◽  
Taylor Method ◽  
Image Position

The (γ, r) summation method, 0 < r < 1, is the "circle method" employed by G. H. Hardy and J. E. Littlewood. It is also known as the Taylor method. Its Lebesgue constants, say L(Tr, n), n = 1, 2, …, were studied by K. Ishiguro [1] in the notation L*(n;1-r). He noted that1where Im{z} denotes the imaginary part of the complex number z, and proved that2Here3

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 dual summability methods

1972 ◽  
Vol 71 (1) ◽  
pp. 67-73 ◽  
Author(s):  
B. Kuttner
Keyword(s):  
Summability Method ◽  
Partial Sums ◽  
Summability Methods ◽  
Sequence Transformation ◽  
Image Position

1. Let A be a summability method given by the sequence-to-sequence transformationWe suppose throughout that, for each nconverges; this is a much weaker assumption than the regularity of A. Then we defineWe also suppose throughout that the sequence {sk} is formed by taking the partial sums of the series Σak; that is to say thatLet A' denote the summability method given by the series-to-sequence transformationFollowing Lorent and Zeller (4), (5), we describe A, A' as dual summability methods. We recall that formally,

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