Fourier series, XV. Gibbs' phenomenon

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...

Two-Dimensional Gibbs Phenomenon for Fractional Fourier Series and Its Resolution

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 polyno...

Computer Graphics and a New Gibbs Phenomenon for Fourier - Bessel Series

The Resolution of the Gibbs Phenomenon for Fourier Spectral Methods

Fourier spectral methods have emerged as powerful computational techniques for the simulation of complex smooth physical phenomena. Their exponential convergence rate depends on the smoothness and periodicity of the function in the domain of interest. If the function has a discontinuity at even one point, the convergence rate deteriorates to first order and spurious oscillations develop near th...

Gibbs’ Phenomenon and Surface Area

If a function f is of bounded variation on TN (N ≥ 1) and {φn} is a positive approximate identity, we prove that the area of the graph of f ∗φn converges from below to the relaxed area of the graph of f . Moreover we give asymptotic estimates for the area of the graph of the square partial sums of multiple Fourier series of functions with suitable discontinuities.