Lebesgue Constants and Best Conditions for the Norm Convergensce of Fourier Series

1985 ◽  
pp. 133-146
Author(s):  
B. Dreseler
Keyword(s):  
Fourier Series ◽  
Lebesgue Constants

Summability of multiple Fourier series. Growth of Lebesgue constants

1980 ◽  
Vol 6 (3) ◽  
pp. 255-267 ◽  
Author(s):  
Р. М. тРИгУБ
Keyword(s):  
Fourier Series ◽  
Multiple Fourier Series ◽  
Lebesgue Constants

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

The Lebesgue constants for Euler $(E,p)$ summation of Fourier series

1954 ◽  
Vol 21 (2) ◽  
pp. 309-313 ◽  
Author(s):  
Arthur E. Livingston
Keyword(s):  
Fourier Series ◽  
Lebesgue Constants

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.

scholarly journals Lebesgue constants for double Hausdorff means

1985 ◽  
Vol 31 (2) ◽  
pp. 199-214
Author(s):  
F. Ustina
Keyword(s):  
Fourier Series ◽  
Periodic Function ◽  
Lebesgue Constant ◽  
Mean Value ◽  
Lebesgue Constants ◽  
Jump Discontinuities ◽  
The Mean ◽  
Image Position ◽  
Partial Extension ◽  
Hausdorff Means

As is well known, the divergence of the set of constants known as the Lebesgue constants corresponding to a particular method of summability implies the existence of a continuous, periodic function whose Fourier series, summed by the method, diverges at a point, and of another such function the sums of whose Fourier series converge everywhere but not uniformly in the neighborhood of some point.In 1961, Lorch and Newman established that if L(n; g) is the nth Lebesgue constant for the Hausdorff summability method corresponding to the weight function g(u), thenwherewhere the summation is taken over the jump discontinuities {εk} of g(u) and M{f(u)} denotes the mean value of the almost periodic function f(u).In this paper, a partial extension of this result to the two dimensional analogue is obtained. This extension is summarized in Theorem 1.3.

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