I've been reading Oppenheim's "Signals and Systems" (engineering) textbook and it gives the following "Dirichlet conditions" for pointwise convergence of the Fourier series representation of a function $f$ with period $2\pi$:
- $f$ is absolutely integrable over its period
- $f$ is of bounded variation, i.e. $f$ has a finite number of minima and maxima during a single period
- $f$ has a finite number of finite discontinuities during a single period
From another question, the "i.e." part of condition 2 is not true, as there exist functions with bounded variation that have an infinite number of minima or maxima; from the Wikipedia article on Dirichlet Conditions, replacing the above condition 2 with "$f$ has bounded variation" is sufficient.
I've seen the Dirichlet-Dini Criterion on Wikipedia: the Fourier series representation of a $2\pi$-periodic, locally integrable function $f$ converges at $x_0$ to $l$ if: $$ \int_0^\pi \left| \frac{f(x_0 + t) + f(x_0 - t)}{2} - l \right| \frac{dt}{t} < \infty $$
My intuition is that this criterion is a generalization of the above Dirichlet conditions. Is it possible to prove the above Dirichlet conditions (either the finite extrema or bounded variation case) using the Dirichlet-Dini criterion? If so, how would I go about doing this?
This is related to the following, where $S_N^f(\theta)$ is the Fourier series truncated to the terms $1,\sin(n\theta),\cos(n\theta)$ for $n=1,2,3,\cdots,N$. $$ S_{N}^f(\theta)-L=\frac{1}{\pi}\int_0^{\pi}D_N(\theta')\left[\frac{f(\theta+\theta')+f(\theta-\theta')}{2}-L\right]d\theta. $$ The Dirichlet-Dini condition is formulated to make sure that the above tends to $0$ as $N\rightarrow 0$, which gives convergence of the Fourier series at $\theta$ to the mean of the left- and right-hand limits of $f$. The Dirichlet kernel is $$ \sum_{n=-N}^{N}e^{in\phi}=\frac{\sin\left[\frac{1}{2}(2N+1)\phi\right]}{\sin\left[\frac{1}{2}\phi\right]} $$