Let $f \in \mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. We set $$\forall k \in \mathbb{Z}, \quad \forall x \in \mathbb{R}, \quad \psi_{k}(x) = \psi(x+k)$$ where $\psi(x) = \frac{\sin(\pi x)}{\pi x}$ for $x \neq 0$ and $\psi(0) = 1$. Let $(d_{k})_{k \in \mathbb{Z}}$ be the sequence of complex numbers from V.C. Prove that the sequence of functions $\left(\sum_{k=-n}^{n} d_{k} \psi_{k}\right)_{n \in \mathbb{N}}$ converges uniformly to $f$ on $\mathbb{R}$.
Let $f \in \mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. We set
$$\forall k \in \mathbb{Z}, \quad \forall x \in \mathbb{R}, \quad \psi_{k}(x) = \psi(x+k)$$
where $\psi(x) = \frac{\sin(\pi x)}{\pi x}$ for $x \neq 0$ and $\psi(0) = 1$.
Let $(d_{k})_{k \in \mathbb{Z}}$ be the sequence of complex numbers from V.C. Prove that the sequence of functions $\left(\sum_{k=-n}^{n} d_{k} \psi_{k}\right)_{n \in \mathbb{N}}$ converges uniformly to $f$ on $\mathbb{R}$.