The Fourier Series of Functions Relative to an Orthonormal System

Recall from The Best Approximation of a Function from an Orthonormal System page that if $\mathcal S = \{ \varphi_0(x), \varphi_1(x), ... \}$ is an orthonormal system of functions on $I$ and $f \in L^2(I)$ then for $b_0, b_1, ..., b_n \in \mathbb{C}$ and $c_k = (f, \varphi_m)$ for all $k \in \{0, 1, ..., n \}$, if $\displaystyle{s_n(x) = \sum_{k=0}^{n} c_k \varphi_k(x)}$ and $\displaystyle{t_n(x) = \sum_{k=0}^{n} b_k \varphi_k(x)}$ then for all $n \in \mathbb{N}$:

(1)

\begin{align} \quad \| f(x) - s_n(x) \| \leq \| f(x) - t_n(x) \| \end{align}

Furthermore, equality above holds if and only if $b_k = c_k$ for all $k \in \{0, 1, ..., n \}$. In other words, if the $L^2$-norm is used as a measure of error for how closely a linear combination of functions in the orthonormal system approximates $f$ then this error is minimized when the coefficients of such a linear combination are chosen to be the values of $c_k$ as above.

We are now ready to define the Fourier series of a function relative to an orthonormal system of functions.

Definition: Let $\mathcal S = \{ \varphi_0(x), \varphi_1(x), ... \}$ be an orthonormal system of functions on $I$ and let $f \in L^2(I)$. The Fourier Coefficients of $f$ are the numbers $c_n = (f, \varphi_n)$ where $n \in \{ 0, 1, ... \}$, and the Fourier Series of $f$ Relative to $\mathcal S$ is the series $\displaystyle{\sum_{n=0}^{\infty} c_n \varphi_n(x)}$.

The notation $\displaystyle{f(x) \sim \sum_{n=0}^{\infty} c_n \varphi_n(x)}$ means that $\displaystyle{\sum_{n=0}^{\infty} c_n \varphi_n(x)}$ is the Fourier series of $f$ relative to $\mathcal S$, i.e., the coefficients $c_n$ are determined as above where of course, $\displaystyle{c_n = (f, \varphi_n) = \int_I f(x) \overline{\varphi_n(x)} \: dx}$.

The trigonometric system on the interval $I = [0, 2\pi]$ is so special that we give Fourier series relative to this system a special name.

Definition: Let $f \in L^2([0, 2\pi])$ and consider the trigonometric system $\left \{ \frac{1}{\sqrt{2\pi}}, \frac{\cos x}{\sqrt{\pi}}, \frac{\sin x}{\sqrt{\pi}}, \frac{\cos 2x}{\sqrt{\pi}}, \frac{\sin 2x}{\sqrt{\pi}}, ... \right \}$ on $[0, 2\pi]$. Then with this orthonormal system we call the Fourier series of $f$ relative to the trigonometric system more generally the Fourier Series Generated by $f$.

Let $f \in L^2([0, 2\pi])$ and consider the trigonometric system. Then the first few Fourier coefficients of $f$ for the Fourier series generated by $f$ are:

(2)

\begin{align} \quad c_0 &= \left ( f(x), \frac{1}{\sqrt{2\pi}} \right ) = \int_0^{2\pi} f(t) \frac{1}{\sqrt{2\pi}} \: dt = \frac{1}{\sqrt{2\pi}} \int_0^{2\pi} f(t) \cos(0x) \: dt \\ \\ \quad c_1 &= \left ( f(x), \frac{\cos x}{\sqrt{\pi}} \right ) = \int_0^{2\pi} f(t) \frac{\cos t}{\sqrt{\pi}} \: dt = \frac{1}{\sqrt{\pi}} \int_0^{2\pi} f(t) \cos t \: dt \\ \\ \quad c_2 &= \left ( f(x), \frac{\sin x}{\sqrt{\pi}} \right ) = \int_0^{2\pi} f(t) \frac{\sin t}{\sqrt{\pi}} \: dt = \frac{1}{\sqrt{\pi}} \int_0^{2\pi} f(t) \sin t \: dt \\ \\ \quad c_3 &= \left ( f(x), \frac{\cos 2x}{\sqrt{\pi}} \right ) = \int_0^{2\pi} f(t) \frac{\cos 2t}{\sqrt{\pi}} \: dt = \frac{1}{\sqrt{\pi}} \int_0^{2\pi} f(t) \cos 2t \: dt \\ \\ \quad c_4 &= \left ( f(x), \frac{\sin 2x}{\sqrt{\pi}} \right ) = \int_0^{2\pi} f(t) \frac{\sin 2t}{\sqrt{\pi}} \: dt = \frac{1}{\sqrt{\pi}} \int_0^{2\pi} f(t) \sin 2t \: dt \\ & \vdots \end{align}

Hence, by letting $\displaystyle{a_n = \frac{1}{\pi} \int_0^{2\pi} f(t) \cos nt \: dt}$ and $\displaystyle{b_n = \frac{1}{\pi} \int_0^{2\pi} f(t) \sin nt \: dt}$ for all $n \in \{ 0, 1, ... \}$, the Fourier series generated by $f$ is given as:

(3)

\begin{align} \quad f(x) & \sim \sum_{n=0}^{\infty} c_n \varphi_n(x) \\ & \sim \frac{1}{\sqrt{2\pi}} \int_0^{2\pi} f(t) \cos(0x) \: dt \frac{1}{\sqrt{2\pi}} + \frac{1}{\sqrt{\pi}} \int_0^{2\pi} f(t) \cos t \: dt \frac{\cos x}{\sqrt{\pi}} + \frac{1}{\sqrt{\pi}} \int_0^{2\pi} f(t) \sin t \: dt \frac{\sin x}{\sqrt{\pi}} + \frac{1}{\sqrt{\pi}} \int_0^{2\pi} f(t) \cos 2t \: dt \frac{\cos 2x}{\sqrt{\pi}} + ... \\ & \sim \frac{1}{2} \frac{1}{\pi} \int_0^{2\pi} f(t) \cos (0x) \: dt + \frac{1}{\pi} \int_0^{2\pi} f(t) \cos t \: dt \cos x + \frac{1}{\pi} \int_0^{2\pi} f(t) \sin t \: dt \sin x + \frac{1}{\pi} \int_0^{2\pi} f(t) \cos 2t \: dt \cos 2x + ... \\ & \sim \frac{a_0}{2} + a_1 \cos x + b_1 \sin x + a_2 \cos 2x + b_2 \sin 2x + ... \\ & \sim \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos nx + b_n \sin nx) \end{align}