# Uniform Convergence of Fourier Series

### An Example

This is a Mathematica notebook to give some visual insight into the uniform convergence of Fourier Series (or lack thereof). The first example is from Boundary Value Problems by David Powers, p. 63. The function is f(x) = |sin(Pi*x)|.

Using Mathematica to compute the partial sums of the Fourier Series, we can plot the partial sums and the function together. From the graph, we can see that the partial sums are converging to the true function (shown in black).

### Pointwise Convergence

The following is the mathematical definition of pointwise convergence:

We can apply this definition to our example above. We choose a particular point, x = 0.05, and we compute the difference between the partial sum with n terms and the true value of the function. In order for the function to converge pointwise at this point, those differences should be less than epsilon for all n > N for some N. The figure shows that this is true for epsilon = .001 with N = 28.

### Uniform Convergence

The definition of unifrom convergence of a Fourier Series is slightly different. It says that the difference between the partial sums and the true function value must be less than epsilon for every point in the interval.

We investigate this convergence graphically by looking at the maximum difference between the partial sums and the function over the entire interval as a function of n.

Another way to visualize uniform convergence is to look at a strip of width epsilon around the function. If all of the partial sums fall within this strip eventually, then the Fourier Series converges uniformly to the function. This is shown in the two figures below for our example. As we would expect from our examination of the maximum difference between the two functions, the convergence is indeed uniform.

### Another Example

Our second example is a discontinuous step function

Here we do not expect uniform convergence of the partial sums to the function and in fact, we see that the partial sums, while they do converge pointwise, do not converge uniformly to the function. (The partial sums don't ever fall inside a strip surrounding the function.)

### Gibbs' Phenomenon

The reason that the convergence can never be uniform is the Gibbs' Phenomenon. The partial sums always over/under-shoot the true values of the function near the discontinuity. This overshoot is known as "ringing". It prevents the Fourier Series of a discontinuous function from converging uniformly.

Converted by Mathematica