Orthogonal and Orthonormal Systems of Functions

Orthogonal and Orthonormal Systems of Functions

We will soon begin to look at a special type of series called a Fourier series but we will first need to get some concepts out of the way first. We will begin by defining two types of "systems" of functions called orthogonal systems and orthonormal systems.

Definition: Let $I$ be any interval in $\mathbb{R}$ and let $L^2(I)$ denote the space of all complex-valued square Lebesgue integrable functions on $I$ with inner product $( \cdot, \cdot )$ defined for all $f, g \in L^2(I)$ by $\displaystyle{(f, g) = \int_I f(x) \overline{g(x)} \: dx}$ and with norm $\| f \| = (f, f)^{1/2}$. A collection of functions $\{ \varphi_0, \varphi_1, \varphi_2, ... \}$ defined on $I$ is said to be an Orthogonal System of Functions (or simply an Orthogonal System) on $I$ in $L^2(I)$ if $(\varphi_i, \varphi_j) = 0$ for all $i \neq j$. $\{ \varphi_0, \varphi_1, \varphi_2, ... \}$ is said to be an Orthonormal System of Functions (or simply an Orthonormal System) on $I$ in $L^2(I)$ if $(\varphi_i, \varphi_j) = 0$ for all $i \neq j$ and additionally $(\varphi_i, \varphi_i) = 1$ for all $i$.

If $\{ \varphi_0, \varphi_1, \varphi_2, ... \}$ is an orthogonal system of nonzero functions on $I$ in $L^2(I)$ and $\varphi_i \neq 0$ then we can obtain an orthonormal system of functions on $I$ in $L^2(I)$ to be:

\begin{align} \quad \left \{ \frac{\varphi_0}{\| \varphi_0 \|}, \frac{\varphi_1}{\| \varphi_1 \|}, \frac{\varphi_2}{\| \varphi_2 \|}, ... \right \} \end{align}

We will now look at a very well-known example of an orthogonal system of functions $S = \{ \varphi_0, \varphi_1, \varphi_2, ... \}$ called the trigonometric system. For any interval $I$ of length $2\pi$ and for each $n \in \{0, 1, 2, ... \}$ let:

\begin{align} \quad \varphi_0(x) = \frac{1}{\sqrt{2\pi}} \quad , \quad \varphi_{2n-1}(x) = \frac{\cos nx}{\sqrt{\pi}} \quad , \quad \varphi_{2n}(x) = \frac{\sin nx}{\sqrt{\pi}} \end{align}

The trigonometric system can also be written explicitly as:

\begin{align} \quad \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}}, \frac{\cos 3x}{\sqrt{\pi}}, \frac{\sin 3x}{\sqrt{\pi}}, ... \right \} \end{align}

Another very important complex-function orthonormal system is the system given for each $n \in \{0, 1, 2, ... \}$ by:

\begin{align} \quad \varphi_n(x) = \frac{e^{inx}}{\sqrt{2\pi}} = \frac{\cos nx + i \sin nx}{\sqrt{2\pi}} \end{align}

This system can be written explicitly in terms of sine and cosine functions as:

\begin{align} \quad \left \{ \frac{1}{\sqrt{2\pi}}, \frac{\cos x + i \sin x}{\sqrt{2\pi}}, \frac{\cos 2x + i \sin 2x}{\sqrt{2\pi}}, ... \right \} \end{align}
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