Fourier Series

Definition (Fourier Partial Sum)

Recall from Example 2 that we have an orthonormal list on $[-\pi ,\pi ]$. Let

$$f(x)=\sum _{k=-N}^N{e}^{ikx}$$

Then $\forall m\in [-N,N]$ we have

$$\begin{array}{rl}⟨f,e^{inx}⟩& ={\int }_{-\pi }^{\pi }f(x)\cdot \overline{e^{inx}}dx\\ & ={\int }_{-\pi }^{\pi }\left(\sum _{k=-N}^N{c}_k{e}^{ikx}\right)e^{-inx}dx\\ & =\sum _{k=-N}^N{c}_k{\int }_{-\pi }^{\pi }{e}^{i(k-m)x}dx\\ & =c_m\cdot {\int }_{-\pi }^{\pi }{e}^{0}dx\\ & =2\pi c_m\end{array}$$

To remove the summation, recognize that when $k\ne m$, we have an orthonormal basis, and so the integral becomes $0$, and when $k=m$, then the integral is $1$. Thus, we only keep $c_m$.

Thus,

$$c_m=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f\cdot e^{-imx}dx$$

Let

$$S_{N}f(x):=\sum _{k=-N}^N{c}_k{e}^{ikx}$$

This is the Fourier Partial Sum. We say that

$$\sum _{n=0}^{\infty }{c}_k{e}^{ikx}$$

is the Fourier Series of $f(x)$ and we denote $c_k$ as the Fourier Coefficients of $f(x)$.

Fourier Series Convergence

We say the Fourier Series converges at $x_0$ if

$$\underset{N\to \infty }{\lim }{S}_N\cdot f(x_0)$$

exists.

Theorem (Fourier Partial Sums are Best Approximate)

These sums are best approximations. Let $N\ge 0,f:[-\pi ,\pi ]\to \mathbb{C}$ be integrable and let

$$S_n\cdot f(x)=\sum _{k=-N}^N{c}_k{e}^{ikx}$$

be the Fourier Partial Sum and let

$$t_N(x)=\sum _{k=-N}^N{d}_k{e}^{ikx}$$

where $t_N$ is some arbitrary trigonometric polynomial of degree $N$ for $d_k\in \mathbb{C}$. Then

$${\int }_{-\pi }^{\pi }|f-S_N\cdot f{|}^{2}dx\le {\int }_{-\pi }^{\pi }|f-t_N{|}^{2}dx$$

and the equation holds $⟺c_k=d_k$ for all $k$.

We are trying to approximate to $t_N$ with $S_N$.

Proof: So,

$$\begin{array}{rl}{\int }_{-\pi }^{\pi }{t}_N\cdot \overline{f}dx& ={\int }_{-\pi }^{\pi }\left(\sum _{k=-N}^N{d}_k{e}^{ikx}\right)\overline{f}dx\\ & =\sum _{k=-N}^N{d}_k{\int }_{-\pi }^{\pi }{e}^{ikx}\overline{f}dx\\ & =\sum _{k=-N}^N{d}_k\overline{{\int }_{-\pi }^{\pi }{e}^{-ikx}fdx}\\ & =2\pi \sum _{k=-N}^N{d}_k\overline{c_k}\end{array}$$

such that

$$\begin{array}{rl}{\int }_{-\pi }^{\pi }|f-t_N{|}^{2}dx& ={\int }_{-\pi }^{\pi }f\overline{f}dx+{\int }_{-\pi }^{\pi }{t}_N\overline{f}dx+{\int }_{-\pi }^{\pi }f\overline{t_N}dx+{\int }_{-\pi }^{\pi }{t}_N\overline{t_N}dx\\ & ={\int }_{-\pi }^{\pi }|f^2|dx+\left[-2\pi \sum d_k\overline{c_k}-2\pi \sum \overline{d_k}{c}_k+2\pi \sum d_k\overline{d_k}+2\pi \sum c_k\overline{c_k}-2\pi \sum c_k\overline{c_k}\right]\\ & ={\int }_{-\pi }^{\pi }|f^2|dx+2\pi \left[\sum (c_k-d_k)\overline{(c_k-d_k)}\right]-2\pi \sum c_k\overline{c_k}\\ & ={\int }_{-\pi }^{\pi }|f{|}^2+{\int }_{-\pi }^{\pi }|S_{N}f-t_N{|}^2-{\int }_{-\pi }^{\pi }|S_{N}f{|}^{2}dx\\ & \ge {\int }_{-\pi }^{\pi }|f^2|dx-{\int }_{-\pi }^{\pi }|S_{N}f{|}^{2}dx\end{array}$$

Let $t_N=S_{N}f$. Then we have that

$${\int }_{-\pi }^{\pi }|f-S_N{|}^2={\int }_{-\pi }^{\pi }|f{|}^2-{\int }_{-\pi }^{\pi }|S_{N}f{|}^2\le {\int }_{-\pi }^{\pi }|f-t_N{|}^{2}dx$$

and thus the proof is done.

Riemann-Lesbeque Lemma

Let $f:[-\pi ,\pi ]\to \mathbb{C}$ be integrable with its associated Fourier Series:

$$\sum _{n\in \mathbb{Z}}{c}_n{e}^{inx}{c}_k=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f(x)e^{-ikx}dx$$

Recall from the previous theorem that

$$\begin{array}{rl}0\le {\int }_{-\pi }^{\pi }|f{|}^{2}dx& ={\int }_{-\pi }^{\pi }|S_{N}f{|}^2+{\int }_{-\pi }^{\pi }|f-S_{N}f{|}^2\end{array}$$

this implies that

$$0\le 2\pi \sum _{k=-N}^N{c}_k\overline{c_k}\le {\int }_{-\pi }^{\pi }|f{|}^{2}dx$$

by expanding $|S_{N}f|$ and showing that it is bounded above. By the Lemma (Convergence, Boundness) we get that RHS converges, such that

$$\sum _{k=-\infty }^{\infty }|c_k{|}^2$$

converges, such that ${\lim }_{k\to \infty }|c_k|=0$.

$$\underset{k\to \infty }{\lim }\left|{\int }_{-\pi }^{\pi }f(x)e^{-ikx}\right|=0$$

Corollary

This culminates to

$$\begin{array}{rl}\underset{N\to \infty }{\lim }\left|{\int }_{-\pi }^{\pi }f(x)\cos (Nx)\right|& =0\\ \underset{N\to \infty }{\lim }\left|{\int }_{-\pi }^{\pi }f(x)\sin (Nx)\right|& =0\end{array}$$

Intuitively, the higher the frequency, the oscillations increase rapidly, the positive and negative areas get smaller, until they approach $0$.

Lemma (Pointwise Convergence of the Fourier Sum)

Let $x_0\in [-\pi ,\pi ]$. Then

$$\underset{N\to \infty }{\lim }{S}_N(x_0)=\underset{N\to \infty }{\lim }\sum _{k=-N}^N{c}_k{e}^{ik{x}_0}=f(x)$$

if $f$ is continuous at $x_0$, then the Fourier Sum is pointwise convergent on $x_0$.

Lemma (Upgrade Fourier Sum)

Since

$$2\pi \sum _{k=-N}^N{c}_k\overline{c_k}\le {\int }_{-\pi }^{\pi }|f{|}^{2}dx$$

then

$$2\pi \sum _{k=-\infty }^{\infty }{c}_k\overline{c_k}\le {\int }_{-\pi }^{\pi }|f{|}^{2}dx$$