= Periodic extension
{title2=$F(x+kT)=F(x)$}
For $T>0$, values of a <function> $f$ on a half-open interval $[a,a+T)$ determine its periodic extension to the <real line>: choose the unique <integer> $k$ with $x-kT\in[a,a+T)$ and set $F(x)=f(x-kT)$. The resulting <periodic function> has <period> $T$. If the endpoint limits of $f$ differ, the extension has a <jump discontinuity> at every period boundary. Under the usual piecewise-smooth convergence hypotheses, its <Fourier series> takes the average of the two one-sided limits there; an assigned value at an isolated endpoint does not change the <Fourier coefficients>.
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