Two-Dimensional Gibbs Phenomenon for Fractional Fourier Series and Its Resolution
نویسندگان
چکیده
The truncated Fourier series exhibits oscillation that does not disappear as the number of terms in the truncation is increased. This paper introduces 2-D fractional Fourier series (FrFS) according to the 1-D fractional Fourier series, and finds such a Gibbs oscillation also occurs in the partial sums of FrFS for bivariate functions at a jump discontinuity. In this study, the 2-D inverse polynomial reconstruction method (IPRM) which is a method based on the inversion of the transformation matrix that represents the fraction Fourier space has been used to remove the Gibbs effect. The aim of this study is to verify the 2-D IPRM has the similar effection for removing the Gibbs oscillation for partial fractional Fourier sums of bivariate functions. Numerical experiments verify the efficiency and accuracy of IPRM. Keywords-fractional Fourier series; Gibbs phenomenon; inverse polynomial reconstruction method; Gegenbauer polynomials
منابع مشابه
Fourier series and the Gibbs phenomenon
An understanding of Fourier series and their generalizations is important for physics and engineering students, as much for mathematical and physical insight as for applications. Students are usually confused by the so-called Gibbs phenomenon-the persistent discrepancy, an "overshoot," between a discontinuous function and its approximation by a Fourier series as the number of terms in the serie...
متن کاملOn the resolution power of Fourier extensions for oscillatory functions
Functions that are smooth but non-periodic on a certain interval have only slowly converging Fourier series, due to the Gibbs phenomenon. However, they can be represented accurately by a Fourier series that is periodic on a larger interval. This is commonly called a Fourier extension. Fourier extensions have been mostly used to solve PDE’s on complicated domains, by embedding the domain into a ...
متن کاملOn the Gibbs Phenomenon and Its Resolution
The nonuniform convergence of the Fourier series for discontinuous functions, and in particular the oscillatory behavior of the finite sum, was already analyzed by Wilbraham in 1848. This was later named the Gibbs phenomenon. This article is a review of the Gibbs phenomenon from a different perspective. The Gibbs phenomenon, as we view it, deals with the issue of recovering point values of a fu...
متن کاملFilters, mollifiers and the computation of the Gibbs phenomenon
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, th...
متن کاملRemoval of the Gibbs phenomenon and its application to fast-Fourier-transform-based mode solvers.
A simple strategy for accurately recovering discontinuous functions from their Fourier series coefficients is presented. The aim of the proposed approach, named spectrum splitting (SS), is to remove the Gibbs phenomenon by making use of signal-filtering-based concepts and some properties of the Fourier series. While the technique can be used in a vast range of situations, it is particularly sui...
متن کامل