D.8 Completeness of Fourier modes

The purpose of this note is to show completeness of the
Fourier modes

for describing functions that are periodic of period

allthese functions can be written as combinations of the Fourier modes above. Assume that

or

for short. Such a representation of a periodic function is called a “Fourier series.” The coefficients

Because of the Euler formula, the set of exponential Fourier modes
above is completely equivalent to the set of real Fourier modes

so that

The extension to functions that are periodic of some other period than

and similarly the real version of them becomes

See [41, p. 141] for detailed formulae.

Often, the functions of interest are not periodic, but are required to
be zero at the ends of the interval on which they are defined. Those
functions can be handled too, by extending them to a periodic
function. For example, if the functions

If the half period

The basic completeness proof is a rather messy mathematical
derivation, so read the rest of this note at your own risk. The fact
that the Fourier modes are orthogonal and normalized was the subject
of various exercises in chapter 2.6 and will be taken for
granted here. See the solution manual for the details. What this
note wants to show is that any arbitrary periodic function

in other words, as a combination of the set of Fourier modes.

First an expression for the values of the Fourier coefficients

a requirement that was already noted by Fourier. Note that

Now the question is: suppose you compute the Fourier coefficients

a valid approximation to the true function

To find out, the trick is to substitute the integral for the
coefficients

The sum in the square brackets can be evaluated, because it is a geometric series with starting value

This expression is called the

Dirichlet kernel. You now have

The second trick is to split the function

Now if you backtrack what happens in the trivial case that

To manipulate this error and show that it is indeed small for large

Using l’Hôpital's rule twice, it is seen that since by assumption

And since the integrand of the final integral is continuous, it is bounded. That makes the error inversely proportional to

It may be noted that under the stated conditions, the convergence is
uniform; there is a guaranteed minimum rate of convergence regardless
of the value of

The condition for