Let $f\left(x\right)=x\cos \left(α x\right)$ at $\left[-π,π\right]$ for $λ \in \mathbb{R}$. Let $S$ be $f\left(x\right)$'s corresponding Fourier Series.
Prove or disprove: 1) if $λ \in \mathbb{Z}$ then $S$ Pointwise-converges to $f\left(x\right)$.
2) If $λ = m + \dfrac{1}{2}$ for $m \in \mathbb{Z}$ then $S$ Pointwise-converges to $f\left(x\right)$.
Thoughts -
Since $f\left(x\right)$ is continuous (multiplication of continuous functions) in $\left[-π,π\right]$ we could use Dirichlet's theorem and just say that $S$ should converge to $f\left(x\right)$, it just seems a little bit naive. What am i missing ?
Thanks in advance!
For any $\lambda \in \mathbb R$ the function $f$ is continuous and it is of bounded variation. Hence the Fourier series converges to $f$ at every point. It is not true that the Fourier series of any continuous function converges pointwise. Dirichlet's Theorem only gives convergence almost everywhere, not everywhere.