I am to computing this limit:
$$\lim_{n\to+\infty}\int_{\mathbb{R}} \sin(nx)h(x) \,, dx$$ where $h(x)$ stays in $L^1(\mathbb{R})$.
Is it correct this method? I choose a sequence of functions $h_n(x)\in C_c^\infty$ which converges to $h$ in $L^1$ sense and I have
$$\int_{\mathbb{R}} \sin(nx)h_n(x) \, dx = \left.-\frac{\cos(nx)} n h_n(x)\right|_{-\infty}^{+\infty} + \int_{\mathbb{R}} \frac{\cos(nx)}{n}h_n'(x) \, dx$$
so
$$\left|\int_{\mathbb{R}} \sin(nx)h_n(x) \, dx\right|\le \frac Cn\to0.$$
In this way I have
$$\|\sin(nx)h(x)\|_{L^1}\le\| \sin(nx)h(x)-\sin(nx)h_n(x)\|_{L^1}+\| \sin(nx) h_n(x) \|_{L^1} \to 0$$