Fourier Series - 2

Inner Product

Definition. The inner product of two vectors $\underline{u}$ and $\underline{v}$ is defined such that

$$( \underline{u}, \underline{v} ) = u_1\overline{v_1} + u_2\overline{v_2} + \cdots + u_n\overline{v_n}. \tag{1}$$

 

Definition. The inner product of two functions $f$ and $g$ that are continuous on an interval $[a,b]$ is defined by the formula analogous to Eq. (1)

$$ (f,g) = \int_a^b f(t)\overline{g(t)}dt.$$

 

Orthogonality and Norm

Definition. Two elements are said to be orthogonal if their inner product is zero.

Definition. The nonnegative number $\| x \| $ defined by the equation

$$ \| x \| = (x, x)^{1/2} $$ 

is called the norm.

 

The complex exponentials $e^{2\pi int}, n=0, \pm1, \pm2, \cdots$, are orthonormal.

Writing $e_n(t) = e^{2\pi int}, $

when $n \neq m,$ we have

$$\begin{align*} (e_n, e_m) &= \int_0^1 e^{2\pi int}\overline{e^{2\pi imt}}dt = \int_0^1 e^{2\pi int}e^{-2\pi imt} = \int_0^1 e^{2 \pi i(n-m)t}dt \\ &= \frac{1}{2\pi i(n-m)} e^{2\pi i(n-m)t} \\ &= \frac{1}{2\pi i(n-m)} \left( e^{2\pi i(n-m)} - e^0 \right) \\ &= 0. \hspace{2cm} \text{(orthogonal)} \end{align*} $$

When $n=m$,

$$ \begin{align*} \| e_n \|^2 &= (e_n, e_n) = \int_0^1 e^{2\pi int} \overline{e^{2\pi int}}dt = \int_0^1 (1)dt \\ &= 1. \hspace{2cm}

Therefore, we have

$$ (e_n, e_m) = \delta_{nm} = \begin{cases}1, & {n=m} \\ 0, & {n \ne m} \end{cases} $$

The functions $e_n(t)$ are orthonormal.

 

The inner product of $f$ and $e_n$ is given by

$$\begin{align*} (f, e_n) &= \int_0^1 f(t)\overline{e^{2\pi int}}dt = \int_0^1 f(t)e^{-2\pi int}dt \\ &= \hat{f}(n) \end{align*}$$

The nth Fourier coefficient $\hat{f}(n)$ is the projection of $f$ onto the direction of $e_n(t).$

$$\begin{align*} \therefore f(t) &= \sum_{n=-\infty}^{\infty} \hat{f}(n)e^{2\pi int} \\ &= \sum_{n=-\infty}^{\infty} (f, e_n)e_n \end{align*}$$

 

The Fourier series expansion can be interpreted as the decomposition in terms of an orthonormal basis and associated inner product.