Minimum condition for convergence of Fourier series

Question

I do study about Fourier Analysis with Stein and Shakarchi’s text book. Recently, I get the striking result that is, if $f$ is an integrable function on the circle, then The partial sum of Fourier series of function $f$ is converging to $f$ at a point $\theta_0$ on which $f$ is differentiable.

Indeed, this statement still holds if we only assume that $f$ satisfies a Lipschitz condition at $\theta_0$.

My question is follow. What is the minimum condition of (uniformly/pointwisely) converge of partial sum of Fourier series of function $f$ ??

Answer 1

You can write the truncated Fourier series in terms of the Dirichlet kernel $$ D_N(\theta)=\frac{\sin((n+1/2)\theta)}{\sin(\theta/2)}. $$ The truncated Fourier series $S_N^f(\theta)$ that involves $1,\cos(n\theta),\sin(n\theta)$ for $1 \le n \le N$ becomes $$ S_N^f(\theta)-L = \frac{1}{\pi}\int_{0}^{\pi}D_N(\theta')\left[\frac{f(\theta+\theta')+f(\theta-\theta')}{2}-L\right]d\theta'. $$ As $N\rightarrow\infty$, the integral over $[\delta,\pi]$ of the right side always tends to $0$ because of the Riemann-Lebesgue lemma, assuming only that $f$ is absolutely integrable on $[0,2\pi]$. So the Fourier series at $\theta$ converges to $L$ if, for every $\epsilon > 0$ there exists $\delta > 0$ and $N_0$ such that $$ \left|\frac{1}{\pi}\int_{0}^{\delta}D_N(\theta')\left[\frac{f(\theta+\theta')+f(\theta-\theta')}{2}-L\right]d\theta'\right| < \epsilon,\;\;\; N > N_0. $$ This is about the best you can say in general. The above does hold if, for example, the Dirichlet-Dini condition holds, meaning that the term in square brackets is absolutely integrable on $[0,\delta]$ for some $\delta$. A Holder condition on $f$ at $\theta$ for any small positive Holder exponent will certainly imply this absolute integrability, but such a condition is also too strong, meaning that it is not necessary for convergence. No one has found a "best" condition. If the graph of $f$ is nearly a straight line at $\theta$, then you get convergence; the bracketed quantity measures the asymmetry of the graph of $f$ near $\theta'=\theta$. This is why conditions of differentiability easily imply convergence.

Answer 2

Your question can be reformulated as follows. Let $U$ be the space of uniformly converging Fourier series. That space admits a norm given by $$ \|f\|_U = \sup_{N \geq 1} \| S_N(f) \|_\infty, $$ where $S_n$ is $N^{th}$-term Fourier series. Your question is what is the largest subspace $U_0 \subset U$ that can be described in terms of "smoothness/continuity conditions".

The case of pointwise convergence can be defined similarly. Fix $\theta_0 \in [0,2 \pi)$. Let $V_{\theta_0}$ the the space of functions for which $S_{N}(f)(\theta_0) \to f(\theta_0)$. What is the largest $V_0 \subset V_{\theta_0}$ that can be described in terms of continuity conditions?

I highly doubt that such minimal conditions exist. In the summability of series there is a well-known heuristic stating that there is no "boundary" between convergent and divergent series, i.e: any criteria for convergence can be relaxed and any criteria for divergence can be made stiffer[*]. In the case of pointwise convergence that heuristic can be turn into a concrete statement. Let $\omega:\mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0} $ be a continuous and increasing function satisfying that $\omega(0) = 0$. Let $\Lambda_\omega$ be the functions satisfying that $$ |f(\theta) - f(\theta_0)| \leq C \, \omega \big(|\theta - \theta_0| \big), $$ for all $\theta$ close enough to $\theta_0$.

Observation: There if no minimal $\omega$ for which: $f \in \Lambda_\omega$ imply that $S_N(f)(\theta_0) \to f(\theta_0)$.

[*] See How did Rudin conclude his argument there is no "boundary" between convergent and divergent series?.