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?

If the function is piecewise continuous on an interval, then the Fourier series converges pointwise. — Mhenni Benghorbal, Apr 23 '13 at 05:07

score 3 · Answer 1 · answered May 27 '13 at 12:18

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.

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