Can a fourier series generated for a function f(x) on some interval be used to calculate the value for f(x) for all x, even outside that interval?

Question

If you have fourier series for a function f(x) on some interval a < x < b. Does it series still converge to the value of f(x) even for x that is not in that interval?

Answer 1

The Fourier series over $I=[a,b]$ just depends on the values took by $f$ on $I$. You may re-define $f$ outside $I$ in any way you like, and its Fourier series over $I$ would still be the same, hence the answer is clearly negative.

Moreover, it may be wrong that the Fourier series of $f$, evaluated at $x=c$, equals $f(c)$ for every $c\in I$. If the hypothesis of Carleson's theorem hold, that just happens for almost every $c\in I$.