Behavior of Lebesgue constants for linear methods of summing fourier series giving the best order approximation

1987 ◽  
Vol 39 (2) ◽  
pp. 154-157
Author(s):  
A. K. Kushpel'
Keyword(s):  
Fourier Series ◽  
Order Approximation ◽  
Lebesgue Constants

scholarly journals Reconstruction of Piecewise Smooth Multivariate Functions from Fourier Data

Axioms ◽  
2020 ◽  
Vol 9 (3) ◽  
pp. 88
Author(s):  
David Levin
Keyword(s):  
Fourier Series ◽  
Fourier Coefficients ◽  
Order Approximation ◽  
High Accuracy ◽  
Rectangular Domain ◽  
High Order ◽  
Piecewise Smooth ◽  
Multidimensional Case ◽  
Order Spline ◽  
The Given

In some applications, one is interested in reconstructing a function f from its Fourier series coefficients. The problem is that the Fourier series is slowly convergent if the function is non-periodic, or is non-smooth. In this paper, we suggest a method for deriving high order approximation to f using a Padé-like method. Namely, we do this by fitting some Fourier coefficients of the approximant to the given Fourier coefficients of f. Given the Fourier series coefficients of a function on a rectangular domain in Rd, assuming the function is piecewise smooth, we approximate the function by piecewise high order spline functions. First, the singularity structure of the function is identified. For example in the 2D case, we find high accuracy approximation to the curves separating between smooth segments of f. Secondly, simultaneously we find the approximations of all the different segments of f. We start by developing and demonstrating a high accuracy algorithm for the 1D case, and we use this algorithm to step up to the multidimensional case.

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

Homogenization of a second order plate model

2017 ◽  
Vol 23 (9) ◽  
pp. 1323-1332 ◽  
Author(s):  
E Pruchnicki
Keyword(s):  
Fourier Series ◽  
Series Expansion ◽  
Displacement Field ◽  
Order Approximation ◽  
Second Order ◽  
Fourier Series Expansion ◽  
Two Dimensional ◽  
Elastic Plates ◽  
Dimensional Model ◽  
Two Dimensional Model

This work is concerned with the asymptotic analysis of linearly elastic plates with periodically rapidly varying heterogeneities. For the sake of simplicity we assume that the structure of heterogeneity is homogeneous in the direction perpendicular to the mid-surface of the plate. We want to derive a homogenized two-dimensional model which is independent of the magnitude of the applied load. Consequently we have to proceed in the following manner. Firstly, we consider a two-dimensional model of the plate obtained by expanding the displacement field by Fourier-Series expansion in thickness direction of the plate with respect to a basis of scaled Legendre polynomials. We consider a second order approximation of the displacement field which gives a good compromise between the accuracy of the approximate solution and the complexity of the approximate problem. This approximation result from an approximation of the Fourier series expansion of the displacement field up to order h6 ( h denotes the thickness of the plate). By considering standard argument for this type of problem, we can rigorously formulate a two-dimensional homogenized boundary value problem for the plate.

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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