Periodic and Orthogonal Functions

Periodic functions

A function is meant to be periodic if, for every $x \in \mathbb{R}$ , there should be a $p$ such that this property holds.

f(x+ p) = f(x)

i.e. for every piece like below, you should be able to shift it to the one that goes from origin by offsetting with some $p$ .

That $p$ is called "period" of $f$ .
If from such periods, the smallest positive $p$ exists, called "Fundamental period" of $f$ .
A function having period $p$ is called a $p$ -periodic function.

A function is orthogonal in an interval if, the product of those two functions does not produce a net area under the curve.

$f$ and $g$ are orthogonal in $[a,b]$ if,

\int_{a}^{b}f(x)\cdot g(x)\,dx = 0

$sinx\cdot cosx$ is orthogonal in $[-\frac{π}{2},\frac{\pi}{2}]$

Note

A set of functions $\{f_i\}$ is said to be orthogonal on $[a,b]$ if every distinct pair of functions in the set are orthogonal on $[a,b]$

Important

Note that $[a, b]$ can be replaced with (a, b), (a, bl, or a, b).