Originally Posted by
Opalg Can you provide a bit of context for this? It doesn't make much sense on its own.
The only way in which I can think of this happening would be if you were given an odd function defined on half of a fundamental interval, and then you make it into an even function by defining it symmetrically on the other half of the interval.
For example, if you are given the odd function $\displaystyle f(x) = x^3$, defined on the interval $\displaystyle [0,\pi]$, then you can make it into an even function on the interval $\displaystyle [-\pi,\pi]$ by defining $\displaystyle f(-x) = f(x)$. That gives you the even function $\displaystyle |x^3|$, which will have an even Fourier expansion.
But if a function is odd on the whole fundamental interval then it can only have an odd Fourier expansion.