I read in my text book that the Dirichlet conditions are sufficient conditions for a real-valued, periodic function $f(x)$ to be equal to the sum of its Fourier series at each point where $f$ is continuous.
However, it further stated that although the conditions are sufficient but they are NOT necessary.
Why are the Dirichlet conditions "not necessary" ?
Example : One of the Dirichlet conditions state that the function can not have infinite discontinuities. Hence we can not express, a function like $ tan x $ in terms of a Fourier series since (as it appears) violates one of the conditions. So, why is that they are 'not necessary'?
P.S.: The Wikipedia Link to the Dirichlet conditions.
Best Answer
The conditions are "not necessary" because no one proved a theorem that if the Fourier series of a function $f(x)$ converge pointwise then the function satisfies the Dirichlet conditions.
Some general information: the issues of the pointwise converges of Fourier series are very delicate matter. As far as I understand we still do not have the exact condition to fully identify the class of functions whose Fourier series converge. It was proved, as @chaohuang pointed in another answer, that if $f\in L^2$, then the Fourier series converge. On the other hand, A. Kolmogorov provided an example of a function $f\in L^1$ whose Fourier series do not converge at any point. I am sure you can find much more information on these issues in any serious text on Fourier analysis.