Pointwise convergence of sine series of $x^{-2}$

Question

I was wondering if the sine series of $x^{-2}$ converges pointwise on the open interval $(0,1)$.

What is the most general criterion to ensure pointwise convergence of a Fourier series?

Answer 1

Some results about pointwise convergence: list in Wikipedia, notes on the Dini test. There is no such thing as the most general criterion.

The function $x^{-2}$ on the interval $(0,1)$ does not have a Fourier series, since it is not integrable. So the question of its convergence is moot.