Function Orthogonality

Orthogonality

$f,g$ are orthogonal if $⟨f,g⟩=0$, orthonormal if additionally $||f||=||g||=1$. The Pythagorean Theorem for function states

$$||f\pm g|{|}^2=||f|{|}^2\pm 2⟨f,g⟩+||g|{|}^2$$

and if $f\perp g$, then

$$||f+g|{|}^2=||ƒ|{|}^2+||g|{|}^2$$

Orthogonal Set

A set of non-zero functions is mutually orthogonal on $a\le x\le b$ if

$$\int \_a^{b}f\_i(x)f\_j(x)dx=\{\begin{array}{llc}0& & i\ne j\\ c>0& & i=j\end{array}$$

Integral Values from Proving Orthogonality of Sine and Cosine

1. $$\{\begin{array}{llc}2L& & \text{if}n=m=0\\ L& & \text{if}n=m\ne 0\\ 0& & \text{if}n\ne m\end{array}$$
2. $$\{\begin{array}{llc}L& & \text{if}n=m=0\\ L/2& & \text{if}n=m\ne 0\\ 0& & \text{if}n\ne m\end{array}$$
3. $$\{\begin{array}{llc}L& & \text{if}n=m\\ 0& & \text{if}n\ne m\end{array}$$
4. $$\{\begin{array}{llc}\frac{L}{2}& & \text{if}n=m\\ 0& & \text{if}n\ne m\end{array}$$

Related

• Even and Odd Functions