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"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?

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What you write is not possible. You need period $2K$ (at least) in order to have sufficient freedom to make the function even or odd as well as periodic. And even then, making it odd only works if already $f(0)=0$. –  Marc van Leeuwen Jun 2 '12 at 15:50
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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.

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