Notes on Fourier Series

A function or a real variable f is said to be periodic with period P if f(x+ P ) = f(x) holds for all x. Hence, if we know the values of f on an interval of length P , we know its values everywhere. If f is a function defined on an interval [a, b), we can extend f to a function defined for all x which is periodic of period b− a. We simply define f(x) to be f(x+ n(b− a)), where n is the integer such that a ≤ x+ n(b− a) < b. The graph of a periodic function f has the same shape in any two consecutive intervals corresponding to a period. Thus, if P is the period of f , ∫ P−a