Experiments with Fourier Series, Part 2

Experiments with Fourier Series

Part 2: Fourier Approximations to a Triangular Wave

Our primary interest is to go in the opposite direction from Part 1, that is, to break down a periodic function into its "basic" trigonometric components. Specifically, we will express a periodic function as a linear combination of the simple trigonometric functions.

If f(x) is a periodic function of period 2 pi, the Fourier series of f(x) is the infinite series

for which the coefficients are calculated from the integral formulas

and

A partial sum of the Fourier series is called a Fourier approximation to f(x).

Our first example is the "triangular" function obtained by extending the function

to be a periodic function of period 2 pi.

First we consider the coefficients of the sine terms. What can you conclude about the coefficients b_k?
Now look at the coefficients a_k of the cosine terms. Why can we evaluate these by doubling the integral over the interval [0,pi]?
Calculate a₀, and give a geometrical explanation of this coefficient.
Now consider the general a_k for k > 0. What are the first 10 coefficients?
Compare the approximations with the original function: Plot both the function and the approximation for n = 1...11. Do the approximations appear to converge to the function? For which values of x is the convergence fastest? For which values of x is it slowest?

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