L2 convergence of Fourier Series

Let us approximate a function $f(x)$ by a linear combination of $\{e^{ikx}|k=-n,\cdots ,0,\cdots ,n\}$. We show that the partial Fourier sum is the ``best'' of such approximators. Then, we prove that the Fourier series of the function $f$ converges to $f$ in the $L^2$ norm.

Lemma (``best'' approximator)

Let any $n\in \mathbb{N}$, consider the set of $2n+1$ functions of the form

$${\phi }_k(x)=e^{ikx}(|k|\le n).$$

Let $v_n$ be an arbitrary linear combination of these functions:

$$v_n(x)=\sum _{|k|\le n}{\gamma }_k{\phi }_k(x)$$

where ${\gamma }_k\in \mathbb{C}$ are constants. Suppose $f\in C_{2\pi }^0$. Then,

$$\Vert f-S_n\Vert \le \Vert f-v_n\Vert$$

where $S_n=S_n[f]$ is the partial sum of the Fourier series $S[f]$. In other words, $S_n[f]$ is the best approximator of $f$ among all the linear combinations of $\{{\phi }_k{\}}_{|k|\le n}$ with respect to the $L^2$ norm.

Proof. 

$$\begin{array}{rcl}\Vert f-v_n{\Vert }^2& =& \Vert (f-S_n)+(S_n-v_n){\Vert }^2\\ & =& \Vert f-S_n{\Vert }^2+2\mathrm{\Re }(f-S_n,S_n-v_n)+\Vert S_n-v_n{\Vert }^2.\end{array}$$

But the inner product $(f-S_n,S_n-v_n)=0$ because $S_n-v_n$ is a linear combination of ${\phi }_k$ and

$$\begin{array}{rcl}(f-S_n,{\phi }_k)& =& (f,{\phi }_k)-(S_n,{\phi }_k)\\ & =& 2\pi c_k-2\pi c_k=0.\end{array}$$

Therefore,

$$\Vert f-v_n{\Vert }^2=\Vert f-S_n{\Vert }^2+\Vert S_n-v_n{\Vert }^2\ge \Vert f-S_n{\Vert }^2$$

where the equality holds if and only if $v_n=S_n$. ■

Definition ($L^2$ convergence)

For the sequence of functions $\{f_n\}$ and some function $f_0$, $\{f_n\}$ is said to converges to $f_0$ in the $L^2$ norm if

$$\Vert f_n-f_0\Vert \to 0(n\to \mathrm{\infty }).$$

Example. Consider the functions $u_n(x)=\sin (\pi x^n),n=1,2,\cdots$ on the interval $(0,1)$. For each $x\in (0,1)$, $\underset{n\to \mathrm{\infty }}{lim}{u}_n(x)=0$. But $\{u_n\}$ does not converge uniformly to $u_0(x)\equiv 0$. In fact, for $x_n={\left(\frac{1}{2}\right)}^{\frac{1}{n}}$, $u_n(x_n)=1$. On the other hand, with the change of variables $t=x^n$,

$$\begin{array}{rcl}\Vert u_n{\Vert }^2& =& {\int }_0^1|u_n(x){|}^{2}dx={\int }_0^1{\sin }^2(\pi x^n)dx={\int }_0^1{\sin }^2(\pi t)\cdot \frac{t^{\frac{1}{n}-1}}{n}dt\\ & \le & \frac{1}{n}{\int }_0^1(\pi t{)}^2{t}^{\frac{1}{n}-1}=\frac{{\pi }^2}{n}{\int }_0^1{t}^{\frac{1}{n}+1}dt=\frac{{\pi }^2}{n}\frac{1}{\frac{1}{n}+2}\\ & =& \frac{{\pi }^2}{2n+1}\to 0.\end{array}$$

Thus, $u_n(x)$ converges to $u_0(x)\equiv 0$ in $L^2$ (on the interval $(0,1)$). □

Remark. We can define a distance function based on a norm. That is, the distance $d(f,g)$ between two functions $f$ and $g$ can be defined by

$$d(f,g)=\Vert f-g\Vert .$$

Thus, the convergence in the $L^2$ norm is a generalization of the convergence of a point sequence in ${\mathbb{R}}^n$ (or ${\mathbb{C}}^n$).

Theorem ($S[f]$ converges to $f$ in the $L^2$ norm)

If $f\in C_{2\pi }^0$, then the Fourier series $S[f]$ converges to $f$ in the $L^2$ norm.

Proof. For the partial sum $S_n$ and Fejér partial sum ${\sigma }_n$ of the Fourier series of $f$ (see Kernels arising from the Fourier series), by the above Lemma (``best'' approximator) , we have

$$\Vert f-S_n\Vert \le \Vert f-{\sigma }_n\Vert .$$

Note that ${\sigma }_n$ is also a linear combination of $\{e^{ikx}\}$. Furthermore, ${\sigma }_n$ uniformly converges to $f$ (Lemma (Uniform convergence of ${\sigma }_n[f]$)) so that $\Vert f-{\sigma }_n\Vert \to 0$. Hence $\Vert f-S_n\Vert \to 0$. ■

This theorem can be further extended to include discontinuous functions. We list the theorem without proof.

Theorem

If $f\in {\mathcal{R}}_{2\pi }^2$, then $S[f]$ converges to $f$ in the $L^2$ norm.