Let $f \in \mathcal{S}$. We assume that $\mathcal{F}(f)$ is integrable on $\mathbb{R}$. For every positive natural number $n$, we set $$I_{n} = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) \theta\left(\frac{\xi}{n}\right) \mathrm{d}\xi \quad J_{n} = \int_{-\infty}^{+\infty} f\left(\frac{t}{n}\right) \mathcal{F}(\theta)(t) \mathrm{d}t$$ Prove that $\forall n \in \mathbb{N}^{*}, I_{n} = J_{n}$. We will admit the Fubini formula: $$\int_{-\infty}^{+\infty} \left(\int_{-\infty}^{+\infty} f(t) \theta\left(\frac{\xi}{n}\right) e^{-2\pi\mathrm{i} t\xi} \mathrm{d}\xi\right) \mathrm{d}t = \int_{-\infty}^{+\infty} \left(\int_{-\infty}^{+\infty} f(t) \theta\left(\frac{\xi}{n}\right) e^{-2\pi\mathrm{i} t\xi} \mathrm{d}t\right) \mathrm{d}\xi$$
Let $f \in \mathcal{S}$. We assume that $\mathcal{F}(f)$ is integrable on $\mathbb{R}$. For every positive natural number $n$, we set
$$I_{n} = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) \theta\left(\frac{\xi}{n}\right) \mathrm{d}\xi \quad J_{n} = \int_{-\infty}^{+\infty} f\left(\frac{t}{n}\right) \mathcal{F}(\theta)(t) \mathrm{d}t$$
Prove that $\forall n \in \mathbb{N}^{*}, I_{n} = J_{n}$.
We will admit the Fubini formula:
$$\int_{-\infty}^{+\infty} \left(\int_{-\infty}^{+\infty} f(t) \theta\left(\frac{\xi}{n}\right) e^{-2\pi\mathrm{i} t\xi} \mathrm{d}\xi\right) \mathrm{d}t = \int_{-\infty}^{+\infty} \left(\int_{-\infty}^{+\infty} f(t) \theta\left(\frac{\xi}{n}\right) e^{-2\pi\mathrm{i} t\xi} \mathrm{d}t\right) \mathrm{d}\xi$$