Math412 Notes Select Topics in Mathematical Analysis of Functions of One or Several Variables

We now investigate the issue of convergence of the trigonometric Fourier series. To simplify the set up, we will take \(l=\pi\) from now on; the general case can be reduced to this case by a simple change of variables \(x\mapsto \pi x/l\text{.}\)

As a consequence of the above theorem, if \(g\in C[-\pi, \pi]\) and has continuous derivative on \(C[-\pi, \pi]\text{,}\) we can redefine \(g(-\pi)\) to be \(g(\pi)\) and extend this function to be a \(2\pi\) periodic function on \(\bbR\text{.}\) Then the resulting function satisfies the assumption of the above theorem at any \(x\) such that \(-\pi \lt x \lt \pi\text{,}\) so \(S_N[g]( x)\to g(x)\) as \(N\to \infty\text{.}\) At \(x=\pi\text{,}\) the left limit of the extended function is \(g(\pi)\) and the right limit of the extended function is \(g(-\pi)\text{.}\) Thus \(S_N[g]( \pi)\to \frac{g(\pi)+g(-\pi)}{2}\) as \(N\to \infty\text{.}\) Since the Fourier series expansion here is \(2\pi\) periodic, the proper interpretation of the expansion is that the series equals the \(2\pi\) periodic extension of the given function on \((-\pi, \pi]\) in the above sense.

In applying Fourier expansions, it is important to have a keen awareness of the system of orthogoal functions we are working with. For example, if we are working with the set of orthogonal functions \(\{ \sin (nx): n\in \mathbb N\}\) on \([0, \pi]\text{,}\) then the Fourier series expansion of a function \(f\in \cR[0, \pi]\) with respect to this set of orthogonal functions is \(\sum_{n=1}^{\infty} b_{n} \sin (nx)\text{,}\) where \(b_{n}=\frac{2}{\pi} \int_{0}^{\pi} f(x) \sin (nx)\, dx\text{.}\) The proper interpretation of this expansion is that the series equals the odd extension of the given function on \((-\pi, \pi)\) in the above sense. In particular, if \(f\in C[0, \pi]\) and \(f(0)\ne 0\) or \(f(\pi)\ne 0\text{,}\) then the odd extension of \(f\) is not continuous at \(0\) or at \(\pi\text{,}\) so the Fourier series expansion of \(f\) with respect to the set of orthogonal functions \(\{ \sin (nx): n\in \mathbb N\}\) at \(0\) or at \(\pi\) does not converge to \(f(0)\) or to \(f(\pi)\text{,}\) but converges to \(0\) instead. A similar phenomenon occurs if we are working with the set of orthogonal functions \(\{ \cos (nx): n\in \mathbb N\cup \{0\}\}\) on \([0, \pi]\text{,}\) then the Fourier series expansion of a function \(f\in \cR[0, \pi]\) with respect to this set of orthogonal functions is \(\frac{a_0}{2}+\sum_{n=1}^{\infty} a_{n} \cos (nx)\text{,}\) where \(a_{n}=\frac{2}{\pi} \int_{0}^{\pi} f(x) \cos (nx)\, dx\text{.}\) The proper interpretation of this expansion is that the series equals the even extension of the given function on \((-\pi, \pi)\) in the above sense.

Often, a function \(g\) is given by some analytic expression, but that expression outside of the relevant interval has no direct bearing on the Fourier series expansion of \(g\) with respect to a given set of orthogonal functions. The following figures illustrate the above phenomenon of different extensions of a given function on \([0, \pi]\) or on \([-\pi, \pi]\) and the resulting different Fourier series expansions with respect to the sets of orthogonal functions.

Fejér’s theorem implies that the span of \(\{ e^{i nx}\}\) (equivalently \(\{\cos(nx), \sin(nx)\}\)) is dense in both \(C[-\pi, \pi]\) and \(\cR[-\pi ,\pi]\) in the mean square sense, for, given any \(g\in \cR[-\pi ,\pi]\) and \(\epsilon \gt 0\text{,}\) first find a \(2\pi\) periodic and continuous function \(f\) such that \(\Vert f-g\Vert :=\left(\int_{-\pi}^{\pi}|f(t)-g(t)|^{2}\, dt\right)^{1/2}\lt \epsilon\text{.}\) Then find \(N\) such that \(\Vert f- \sigma_{N}(f)\Vert \lt \epsilon\) by the above theorem. Finally triangle inequality implies that \(\Vert g- \sigma_{N}(f)\Vert \lt 2\epsilon\text{.}\)

As a consequence we have