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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 $f(0)=0$.2012-06-02

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