Review of Fourier Series

Part 2: Approximation of Periodic Functions with Arbitrary Period

In this part we consider the approximation of functions of period 2L by appropriate trigonometric polynomials. (Even though we are considering periods of any positive length, we will find it convenient to express the periods as multiples of 2.) Our first task will be to identify the appropriate sine and cosine functions to use in this case.

1. Experiment with values of the positive constant alpha such that the sin(x) plotted over -pi to pi has a graph of the same shape as sin((alpha/L)*x). Check several values of L.

2. Once you have determined a value for alpha, check that the graph of sin(k*x) over the interval -pi to pi has the same shape as the graph of sin(k*(alpha/L)*x) over the interval -L to L. Check several different values of k.

3. Repeat Step 2 for the cosine functions.

4. In this general setting, the approximating trigonometric polynomials are of the form
   

   where the coefficients are given by

   

   List the changes necessary to obtain these coefficient formulas from the ones for 2 pi-periodic functions.

5. Use the preceding formulas for the coefficients to approximate the step function with the graph given below
   Step Function Graph
   

   How do you know you will only need sine terms in the expansion? Compare the graphs of the approximations for n = 1 to 10 to the graph of the step function.