The Dini criterion for the convergence of the partial sums states that if $~\exists ~ \delta>0$ for some point $x$ such that $$ \int_{|t|<\delta}\left| \frac{f(x+t)-f(x)}{t} \right|dt<\infty, $$ then we have the pointwise convergence of the partial sums $$ \lim_{R\to\infty}S_Rf(x)=f(x). $$ Differentiation from first principles states that we can write the first derivative of a differentiable function $f$ can be calculated via the formula $$ f'(x)=\lim_{t\to 0}\frac{f(x+t)-f(x)}{t} $$ My question is regarding whether there is a relationship between the two concepts, given that the limit and the integrand are remarkably similar. Is there some deep hidden relationship between the two theorems based on this similarity? If there is, is it because the conclusion of the two theorems is that they are making approximations at a particular point?
Thank you in advance for any comments and help.
Differentiability at $x$ implies the Dini condition. But the Dini condition is much weaker. For example, if you have the following estimate for $t$ near $0$ and for some $\alpha > 0$, then the Dini condition holds: $$ |f(x+t)-f(x)| \le C|t|^{\alpha} $$ So it doesn't take much more than continuity to get the Fourier series to converge. Differentiability is much stronger than what you need. Unfortunately, there's no perfect characterization of the Dini condition in terms of a simple growth restriction. But that's okay, because even the Dini condition is stronger than it needs to be to get convergence of the Fourier series. For example, if $f \in L^{2}$, then the Fourier series for $f$ converges pointwise almost everywhere to $f$; that's a famous theorem of Lenart Carleson.