Under what conditions does $f(x)$ equal its Fourier Series My question has 3 (similar) parts, and is as follows:


Under what conditions does $f(x)$ equal its Fourier Series for all $x$, $-L \leq x \leq L$?
Under what conditions does $f(x)$ equal its Fourier sine Series for all $x$, $0 \leq x \leq L$?
Under what conditions does $f(x)$ equal its Fourier cosine Series for all $x$, $0 \leq x \leq L$?
The Fourier series of a periodic continuous function of bounded variation converges pointwise to the function.  On the other hand, the Fourier series of a periodic continuous function can diverge at infinitely many points.  See e.g. R.E. Edwards, "Fourier Series - A Modern Introduction", section 10.3.1.
For your first part, any function which will be periodic with period 2L can be decomposed into fourier series and will take exact value for all the point's, given it is equal at -L,L and has no discontinuances in between. If it has discontinouities, it will take the average value between the two ends(Dirichlet theorem). 

Sine has the same conditions as above with the additional condition that the function should be odd.In cosine, the function should be even.