For a function $h \in C([-1,1])$, we denote by $\widetilde{h}$ the following $2\pi$-periodic function: $$\tilde{h} : \begin{cases} \mathbb{R} \rightarrow \mathbb{R} \\ \theta \mapsto h(\cos(\theta)) \end{cases}$$ The Fourier coefficients of $\widetilde{h}$ are given by: $$a_0(\widetilde{h}) = \frac{1}{2\pi} \int_{-\pi}^{\pi} \widetilde{h}(t)\, dt, \quad a_n(\widetilde{h}) = \frac{1}{\pi} \int_{-\pi}^{\pi} \widetilde{h}(t) \cos(nt)\, dt, \quad b_n(\widetilde{h}) = \frac{1}{\pi} \int_{-\pi}^{\pi} \widetilde{h}(t) \sin(nt)\, dt.$$ Let $f \in C^\infty([-1,1])$. Show that the sequence $(a_n(\widetilde{f}))_{n \in \mathbb{N}}$ has rapid decay. What is the value of $b_n(\widetilde{f})$?
For a function $h \in C([-1,1])$, we denote by $\widetilde{h}$ the following $2\pi$-periodic function:
$$\tilde{h} : \begin{cases} \mathbb{R} \rightarrow \mathbb{R} \\ \theta \mapsto h(\cos(\theta)) \end{cases}$$
The Fourier coefficients of $\widetilde{h}$ are given by:
$$a_0(\widetilde{h}) = \frac{1}{2\pi} \int_{-\pi}^{\pi} \widetilde{h}(t)\, dt, \quad a_n(\widetilde{h}) = \frac{1}{\pi} \int_{-\pi}^{\pi} \widetilde{h}(t) \cos(nt)\, dt, \quad b_n(\widetilde{h}) = \frac{1}{\pi} \int_{-\pi}^{\pi} \widetilde{h}(t) \sin(nt)\, dt.$$
Let $f \in C^\infty([-1,1])$.
Show that the sequence $(a_n(\widetilde{f}))_{n \in \mathbb{N}}$ has rapid decay. What is the value of $b_n(\widetilde{f})$?