Fourier Series on Fractals

Kealey Dias

The idea of Fourier Series is to write an arbitrary function as a linear combination of basis functions. Suppose is an orthonormal basis of eigenfunctions of the form with . We can write where . This is exact if we take the infinite sum. In practice, we use the partial sum as an approximation.

Working on the unit interval may give us some insight about what to do on the Sierpinski Gasket.

Fourier Series on the Interval

Say we want to represent this function as a Fourier Sine Series:

Perfect Wave Propogator

Then we want to approximate this function using the partial sums of the Fourier Sine Series

Here are 2 partial sum approximations of this graph of 50 terms and 250 terms. (in both cases, y=.5 and t=.25)

When you take partial sums of a higher number of terms, you get a better approximation.

Notice that there is an under and overshoot (Gibbs Phenomena) where the discontinuities occur. Also notice that the height of the phenomena don't seem to decrease with closer approximations.

If one looks closely, he might notice also that the number of oscillations increases with the number of terms of the series.

Due to the work from previous summers on the subject, it is expected that the gibbs phenomena will be better behaved in this respect on fractals.

Fourier Series on the Sierpinski Gasket

We want to study the analagous situation on the Sierpinski Gasket (SG). Similarly, we want to write an arbitrary function as a linear combination of eigenfunctions of the form where Again, is an orthonormal basis of eigenfunctions of the form with where with and for and is the dense set of all vertices in SG.

We wanted to represent a discontinuous, constant function on the SG defined as being 1 everywhere except the boundary (where it is 0). This is the analog of the Fourier Sine Series on the interval. This will be a Fourier Series of the form (since the function is identically 1, the inner product is just there's level 2

You can see that this approximation isn't great so we take more terms to get a better approximation.

You can definitely see the gibbs phenomena in this slide the height of the phenomena hovers around 1.3 and stays there for higher partial sums.

Here is the level 4 approximation.

Notice that the phenomena does appear to be better behaved here. You see one large over shoot and an undershoot and then it looks almost perfectly flat.

One interesting aspect of the Gibbs Phenomena on the gasket is in the self similarity of where the maximum occurs.

The red dot is the place in which the maximum occurs for level 2 in the top triangle segment in the gasket (this is symmetrical with respect to the other triangle parts). The green dot is representative of where the maximum for level 3 occurs, and the yellow dot for level 4.

I wanted to take a closer look at the phenomena, so I zoomed in on one vertex of the gasket (namely, vertex [0]). So here is a side-by-side comparison of levels 4, 3, and 2 (from left to right). Zooming in once means you are looking at the triangle with vertices [0], [0,1], and [0,2]; zooming in twice means that you are looking at the triangle with vertices [0], [0,0,1], and [0,0,2], etc. Each of these pictures is a side view of these triangles.

zoom 1, level 4

zoom 2, level 4zoom 1, level 3

zoom 3, level 4zoom 2, level 3zoom 1, level 2

zoom 4, level 4zoom 3, level 3zoom 2, level 2

You can see that these pictures are the same when you move up a level and zoom one more time. In fact, they are almost identical. This is another way in which the Gibbs phenomena is self-similar from level to level. It is like the Gibbs Phenomena are being smashed toward the vertex as you move up a level.

I wanted to see if the Fourier Series is actually better behaved on the SG (wrt the oscillations). From the pictures of the constant discontinuous function on the whole gasket, it looks totally flat. I looked closer to see if there were actually any oscillations at all. There are. In particular, there is the same number of oscillations as there are terms of the partial series. Keep in mind that lvl2: the series has 2 terms lvl3: the series has 4 terms lvl4: the series has 8 terms Their height just decays very rapidly. With each subsequent hump (moving closer to the center of the gasket), the magnitude decreases by a factor of 10.

Here is a list of programs used for this project:

  • 2serpartiallineplots.mws.gz
  • aveFS.mws.gz
  • borderline.mws.gz
  • bordervert.mws.gz
  • compareconst.mws.gz
  • confirmgibbs.mws.gz
  • masterabridged.mws.gz
  • masterscriptremakeall.mws.gz
  • masterscriptremakeall2.mws.gz
  • masterscriptremakeall3.mws.gz
  • masterscriptremakeall.mws.gz
  • maxargmax.mws.gz
  • oscillate.mws.gz
  • removeouter.mws.gz
  • sums2d.mws.gz
  • zoom2series.mws.gz
  • zoomgibbs.mws.gz
  • zoomgibbs2.mws.gz
  • biblambda.mws.gz

    You can download the programs that these call by visiting the Sampling website linked below.

    Programs and other information used in this project were downloaded from this website: Sampling on the Sierpinski Gasket

    For more websites done for REUs, click here

    This project was supported by the NSF through an REU grant at Cornell University under the advisement of Robert Strichartz