The ebb and flow of the tides, the light from the setting sun, and the changing of the seasons are all periodic phenomena which can be described with Applied Fourier Analysis

In this video we illustrate the convergence of Fourier series to a characteristic function.

We illustrate the periodic nature of Fourier series with this video, which shows the Fourier series from above on a larger interval. Note that the Fourier series repeats itself, or is periodic.

We illustrate the Fourier Series for f(t) = t, on [-pi,pi]. Note that once again this is periodic and does not represent t outside of this interval

We illustrate the Gibbs ringing phenomena at at a discontinuity at the left. Not that the high and low peaks are both almost exactly the same, and approximately 9% above and below the level with respect to the discontinuity.

This is typical of Gibbs' ringing.

We compare the convergence of a cosine series, in red, to a sine series in yellow.

Not that the cosine series converges very quickly on the right side of zero. This is a basic idea behind compression techniques.

We illustrate the convolution of two characteristic functions at the left.

We end up with a "tent" function.