Question: let $f:\Bbb R \to \Bbb R, f(0)=1 \forall x\in[-\pi,\pi] \setminus \lbrace0\rbrace , f(x)=1+\sin \frac {\pi^2}x$ Does the fourier series of this function converge at zero? If it does what is the sum of the series in x=0?
What we tried Using the formula to look for the series(and formulas for product of sin and cos), we got stuck with an integral of a sin with a non-linear inner function. Any other directions?
Write $f(x)=1+g(x)$. The function $g$ is odd. It's Fourier series has only sine terms. Then the Fourier series of $f$ is the Fourier series of the constant function $1$ plus a series of sine terms, all of which vanish at $x=0$.