# Question on fourier series convergence

• member 428835
In summary, not all Fourier series exhibit the Gibbs phenomenon, as it only occurs at jump discontinuities. For smooth functions, the convergence of the series is uniform and there is no Gibbs effect. However, for functions with jump discontinuities, the Gibbs effect can be observed but the series still converges pointwise at all points of continuity.
member 428835
hey pf!

if we have a piecewise-smooth function ##f(x)## and we create a Fourier series ##f_n(x)## for it, will our Fourier series always have the 9% overshoot (gibbs phenomenon), and thus ##\lim_{n \rightarrow \infty} f_n(x) \neq f(x)##?

thanks!

First, not every Fourier series exhibits the overshoot/Gibbs phenomenon. If the function is smooth (a special case of piecewise smooth), then the convergence of the Fourier series is uniform, and there is no Gibbs effect.

The Gibbs effect occurs at jump discontinuities. There are many examples of functions with jump discontinuities where the Fourier series converges pointwise at all points of continuity. For example, a "square wave" exhibits the Gibbs effect but the series converges pointwise everywhere. At the points of continuity, it converges to the original function, and at the points of discontinuity, it converges to the midpoint between the upper and lower values of the "square wave." Any partial sum will show the Gibbs effect, but as you sum more and more terms, the main part of the "ripple" is limited to smaller and smaller neighborhoods around the discontinuities. In the limit, it goes away entirely.

1 person

## 1. What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function as a sum of sine and cosine functions. It is named after the French mathematician Joseph Fourier and is commonly used in signal processing, engineering, and physics.

## 2. How do you determine if a Fourier series converges?

There are several tests that can be used to determine the convergence of a Fourier series, such as the Dirichlet test, the Abel-Poisson test, and the Dini test. These tests evaluate the behavior of the coefficients and the periodic function to determine if the series converges or not.

## 3. What is the difference between pointwise and uniform convergence of a Fourier series?

Pointwise convergence means that the Fourier series converges at each individual point of the function, while uniform convergence means that the series converges at every point simultaneously. Uniform convergence is generally considered stronger than pointwise convergence.

## 4. Can a Fourier series converge to a non-periodic function?

No, a Fourier series can only converge to a periodic function. This is because the series is made up of periodic functions, and the sum of periodic functions can only result in another periodic function.

## 5. What is the relationship between the convergence of a Fourier series and the smoothness of the function?

The smoother the function, the faster the Fourier series will converge. This is because the coefficients of the series are related to the derivatives of the function, and a smoother function will have smaller derivatives, resulting in faster convergence.

Replies
1
Views
237
Replies
11
Views
1K
Replies
17
Views
3K
Replies
5
Views
624
Replies
8
Views
4K
Replies
11
Views
1K
Replies
1
Views
2K
Replies
3
Views
838
Replies
9
Views
2K
Replies
6
Views
2K