Decomposing a function into a Fourier series is possible for periodic functions. Fourier transform, on the other hand, is used for aperiodic functions. How can we use Fourier series to analyse the initial configuration of a plucked string at $t=0$?
Edit The existing answer talks about periodic extension which I am aware of. To me, the periodic extension is a redefinition of an aperiodic function in such a way so as to make it periodic. We pretend that it is periodic while in the real problem it's not. For example, in the situation I described, the configuration of the string at $t=0$, between $x=0$ to $x=L$, is not repeated in space. Here, periodic extension is something we demand by brute force.
Why is there is no difference between an actual periodic function i.e., a periodically repeated pattern in space (for example, density in a crystal lattice) and that which is repeated by brute force?