Let $f : \mathbb{R}_{+} \rightarrow \mathbb{C}$ be a continuous function and zero outside a segment. We define the function $\mathcal{L}(f)$ (Laplace transform of $f$) on $\mathbb{R}$ by $$\forall x \in \mathbb{R}, \quad \mathcal{L}(f)(x) = \int_{0}^{+\infty} f(t) e^{-xt} \mathrm{d}t$$ We will admit that $\mathcal{L}(f)$ is of class $C^{\infty}$ on $\mathbb{R}$ and that $$\forall x \in \mathbb{R}, \quad \forall n \in \mathbb{N}, \quad (\mathcal{L}(f))^{(n)}(x) = (-1)^{n} \int_{0}^{+\infty} f(t) t^{n} e^{-xt} \mathrm{d}t$$ In the remainder of this part, we shall assume that $$\lim_{n \rightarrow +\infty} \sum_{0 \leqslant k \leqslant \lfloor nx \rfloor} \frac{(n\lambda)^{k}}{k!} e^{-n\lambda} = \frac{1}{2} \quad \text{if } x = \lambda$$ VI.C.1) Let $x \in \mathbb{R}_{+}$. Prove that $$\lim_{n \rightarrow +\infty} \sum_{0 \leqslant k \leqslant \lfloor nx \rfloor} (-1)^{k} \frac{n^{k}}{k!} (\mathcal{L}(f))^{(k)}(n) = \int_{0}^{x} f(y) \mathrm{d}y$$ VI.C.2) Deduce that the map $\mathcal{L} : f \mapsto \mathcal{L}(f)$ is injective on the set of complex-valued functions, continuous on $\mathbb{R}_{+}$ and zero outside a bounded interval.
Let $f : \mathbb{R}_{+} \rightarrow \mathbb{C}$ be a continuous function and zero outside a segment. We define the function $\mathcal{L}(f)$ (Laplace transform of $f$) on $\mathbb{R}$ by
$$\forall x \in \mathbb{R}, \quad \mathcal{L}(f)(x) = \int_{0}^{+\infty} f(t) e^{-xt} \mathrm{d}t$$
We will admit that $\mathcal{L}(f)$ is of class $C^{\infty}$ on $\mathbb{R}$ and that
$$\forall x \in \mathbb{R}, \quad \forall n \in \mathbb{N}, \quad (\mathcal{L}(f))^{(n)}(x) = (-1)^{n} \int_{0}^{+\infty} f(t) t^{n} e^{-xt} \mathrm{d}t$$
In the remainder of this part, we shall assume that
$$\lim_{n \rightarrow +\infty} \sum_{0 \leqslant k \leqslant \lfloor nx \rfloor} \frac{(n\lambda)^{k}}{k!} e^{-n\lambda} = \frac{1}{2} \quad \text{if } x = \lambda$$
VI.C.1) Let $x \in \mathbb{R}_{+}$. Prove that
$$\lim_{n \rightarrow +\infty} \sum_{0 \leqslant k \leqslant \lfloor nx \rfloor} (-1)^{k} \frac{n^{k}}{k!} (\mathcal{L}(f))^{(k)}(n) = \int_{0}^{x} f(y) \mathrm{d}y$$
VI.C.2) Deduce that the map $\mathcal{L} : f \mapsto \mathcal{L}(f)$ is injective on the set of complex-valued functions, continuous on $\mathbb{R}_{+}$ and zero outside a bounded interval.