"Suppose we have a function defined on an interval [0,K], then we extend it as an even or odd function of period K so as to produce a Fourier cosine or sine series."
(1): What exactly is extending a function?
(2): How do you extend a function to become odd or even?
Best Answer
The idea is to force the function to be even or odd on the interval $[-K, K]$. E.g. if you want to extend it as an odd function define $g$ on $[-K, K]$ by $g(x) = -f(-x)$ for $-K \leq x < 0$ and $g(x) = f(x)$ for $0 \leq x \leq K$.
This function is then odd as $g(-x) = -g(x)$.
Similarly you can extend it to an even function, i.e. $g(-x) = g(x)$ for $x\in [-K, K]$.
edit: this is of period $2K$ which is what I assume you meant.