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Say you have a function $f(x)$, and you generate a Fourier series for it on the interval $-L \leq x \leq L$, and $f(x)$ is piecewise continuous on that interval.

Say also that $f(x)$ is a periodic function but it is not periodic with the interval $2L$, does the Fourier series then still converge to $f(x)$ for all $x$ in that interval where $f(x)$ is continuous?

Silvia Ghinassi
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Arnold Doveman
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    Consider the function $g$ which coincides with $f$ on the interval $[-L,L)$ and is $2L$-periodic. – Daniel Fischer Aug 24 '15 at 18:08
  • In fact, $f$ need not be periodic at all. Its Fourier series representation on the interval of choice, say $[-L,L]$ will be a periodic function, with period $2L$, outside $[-L,L]$. – Mark Viola Aug 24 '15 at 19:38

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