When is fourier series non-differentiable?

[jostpuur]

I know that if

$$\sum_{n=-\infty}^{\infty} |n| |\hat{f}(n)| < \infty$$

then

$$\sum_{n=-\infty}^{\infty} \hat{f}(n) e^{2\pi i nx}$$

is continuously differentiable as a function of $x$.

Now I'm interested to know what kind of conditions exist for Fourier coefficients such that they guarantee the non-differentiability.

It is a fact that just because some abstract integral $\int d\mu(x)\psi(x)$ diverges, it doesn't mean that a limit of other integrals $\lim_{n\to\infty} \int d\mu(x) \psi_n(x)$ would diverge too, even when $\psi_n\to\psi$ point wisely. So this means that even if I know that

$$\sum_{n=-\infty}^{\infty} |n| |\hat{f}(n)| = \infty$$

this will not obviously imply that

$$\lim_{\delta\to 0} \sum_{n=-\infty}^{\infty} \hat{f}(n) e^{2\pi inx} \frac{e^{2\pi in\delta} - 1}{\delta}$$

would diverge too.

What condition will suffice to prove that the Fourier series is not differentiable?

 

[fresh_42]

I think that Hardy (1916) is the original work you're looking for:
https://www.ams.org/journals/tran/1...-1916-1501044-1/S0002-9947-1916-1501044-1.pdf
(see e.g. Theorem 4.31)
And here are other examples:
https://www.sciencedirect.com/science/article/pii/0021904586900535