Variants of Dirichlet's theorem on Fourier series The following is Dirichlet's theorem on Fourier series:
Theorem: If $f(t)$ is a bounded periodic function which in any one period has at most a finite number of local maxima and minima and a finite number of points of discontinuity, then the Fourier series of $f(t)$ converges to $f(t)$ at all points where $f(t)$ is continuous and converges to the average of the right- and left-hand limits of $f(t)$ at each point where $f(t)$ is discontinuous.
I would like to know about generalizations of Dirichlet's theorem. Specifically:


*

*Dirichlet's theorem gives a sufficient condition for convergence of
the Fourier series. Are necessary conditions known? i.e. Is
something like a converse known for Dirichlet's theorem?

*The step function does not satisfy Dirichlet's theorem - every value of $t$ is a local minima or maxima - yet it has a convergent
Fourier series and (I believe) the conclusions of Dirichlet's
theorem are valid in this case. Is there a generalization of
Dirichlet's theorem that accounts for the step function?
 A: The Dirichlet-Dini Theorem states that the Fourier series for a periodic integrable $f$ converges to $L$ at $\theta$ if the following improper integral exists for some $\delta > 0$.
$$
              \int_{0}^{\delta}\frac{1}{\theta'}\left|\frac{f(\theta+\theta')+f(\theta-\theta')}{2}-L\right|d\theta' < \infty.
$$
You don't need conditions elsewhere on the interval except that $f$ is integrable on $[0,2\pi]$. This is a fairly weak condition, but still not weak enough to be necessary.
For example, if $f$ has left- and right-hand derivatives at $\theta$, then the above holds. Or if $f$ has left- and right-hand limits $f(\theta\pm 0)$ and $f$ is $\alpha$-Holder continuous from the left- and the right- for some $0 < \alpha \le 1$. A variety of other conditions will work. The strangest part is that $f$ can blow up near $\theta$ and the series will converge to $0$ if $f$ is symmetric about $\theta$--that's also covered by the above condition with $L=0$. Any of these results covers the case of the step function.
The sufficiency of the Dirchlet-Dini condition can be directly verified from the Dirichlet integral formula for the truncated Fourier series. Non-trivial necessary conditions are not known.
The most impressive, difficult-to-prove result of Fourier Series convergence is Carleson's Theorem: If $f$ is square integrable on $[0,2\pi]$, then the Fourier series converges pointwise a.e. to $f$.
