On the Gibbs Phenomenon III: Recovering Exponential Accuracy in a Sub-Interval From a Spectral Partial Sum of a Pecewise Analytic Function

1996 ◽  
Vol 33 (1) ◽  
pp. 280-290 ◽  
Author(s):  
David Gottlieb ◽  
Chi-Wang Shu
Keyword(s):  
Analytic Function ◽  
Gibbs Phenomenon ◽  
Exponential Accuracy ◽  
Partial Sum

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.

Filters, mollifiers and the computation of the Gibbs phenomenon

Acta Numerica ◽  
2007 ◽  
Vol 16 ◽  
pp. 305-378 ◽  
Author(s):  
Eitan Tadmor
Keyword(s):  
Physical Space ◽  
Gibbs Phenomenon ◽  
Delta Function ◽  
High Accuracy ◽  
Piecewise Smooth ◽  
Spectral Information ◽  
Smooth Functions ◽  
Exponential Accuracy ◽  
Local Loss ◽  
Edge Detectors

We are concerned here with processing discontinuous functions from their spectral information. We focus on two main aspects of processing such piecewise smooth data: detecting the edges of a piecewise smooth f, namely, the location and amplitudes of its discontinuities; and recovering with high accuracy the underlying function in between those edges. If f is a smooth function, say analytic, then classical Fourier projections recover f with exponential accuracy. However, if f contains one or more discontinuities, its global Fourier projections produce spurious Gibbs oscillations which spread throughout the smooth regions, enforcing local loss of resolution and global loss of accuracy. Our aim in the computation of the Gibbs phenomenon is to detect edges and to reconstruct piecewise smooth functions, while regaining the high accuracy encoded in the spectral data.To detect edges, we utilize a general family of edge detectors based on concentration kernels. Each kernel forms an approximate derivative of the delta function, which detects edges by separation of scales. We show how such kernels can be adapted to detect edges with one- and two-dimensional discrete data, with noisy data, and with incomplete spectral information. The main feature is concentration kernels which enable us to convert global spectral moments into local information in physical space. To reconstruct f with high accuracy we discuss novel families of mollifiers and filters. The main feature here is making these mollifiers and filters adapted to the local region of smoothness while increasing their accuracy together with the dimension of the data. These mollifiers and filters form approximate delta functions which are properly parametrized to recover f with (root-) exponential accuracy.

scholarly journals Approximation of signals by general matrix summability with effects of Gibbs Phenomenon

2019 ◽  
Vol 38 (6) ◽  
pp. 141-158 ◽  
Author(s):  
B. B. Jena ◽  
Lakshmi Narayan Mishra ◽  
S. K. Paikray ◽  
U. K. Misra
Keyword(s):  
Fourier Series ◽  
Gibbs Phenomenon ◽  
Degree Of Approximation ◽  
Trigonometric Polynomials ◽  
General Matrix ◽  
Matrix Summability ◽  
Partial Sum ◽  
Approximation Of Signals

In the proposed paper the degree of approximation of signals (functions) belonging to $Lip(\alpha,p_{n})$ class has been obtained using general sub-matrix summability and a new theorem is established that generalizes the results of Mittal and Singh [10] (see [M. L. Mittal and Mradul Veer Singh, Approximation of signals (functions) by trigonometric polynomials in $L_{p}$-norm, \textit{Int. J. Math. Math. Sci.,} \textbf{2014} (2014), ArticleID 267383, 1-6 ]). Furthermore, as regards to the convergence of Fourier series of the signals, the effect of the Gibbs Phenomenon has been presented with a comparison among different means that are generated from sub-matrix summability mean together with the partial sum of Fourier series of the signals.

Investigation of discontinuous periodic signals using the measuring labview complex

2020 ◽  
pp. 72-79
Author(s):  
E.B. Solovyeva ◽  
◽  
Yu.M. Inshakov ◽  
Keyword(s):  
Fourier Series ◽  
Laboratory Experiment ◽  
Gibbs Phenomenon ◽  
Electrical Signals ◽  
Measuring Complex ◽  
Universal Measuring ◽  
Depth Study ◽  
Periodic Signals ◽  
Truncated Fourier Series

General approaches to the analysis of the Gibbs phenomenon for discontinuous periodic signals approximated by the truncated Fourier series are considered. Methods for smoothing the truncated Fourier series and improving its convergence are discussed. The software means for modeling is a universal measuring complex LabVIEW, which possesses a convenient environment for analyzing electrical signals, on the basis of this complex a laboratory experiment is carried out. The advantages of the measuring LabVIEW complex and its capabilities for in-depth study of discontinuous periodic signals are noted.

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