Question

How to proof that the Fourier series of a function exist? vice versa, if the Fourier series of a function doesn't exist, how do i proof it?

Answer 1

Any periodic function of period 2pi which satisfies certain conditions known as Dirchlet's conditions can be expressed in the form of a Fourier Series in the interval

The Dirchlets conditions are,

1. f(x) and its integrals are finite and single values
2. f(x) has discontinuities , finite in number
3. f(x) has finite number of maxima and minima

Hence, by checking if the given function satisfies the above conditions or not, we can determine if its Fourier series exist.