I'm attacking a homework problem, which I have reduced to the following:
Let Schwartz function $f \in \mathcal{S}^1(\mathbb{R})$ be nonnegative and $\|f\|_{L^1} = 1$. Assume further that $\int_\mathbb{R} x f(x) dx = 0$. Then for any Schwartz function $\phi$, $$ \lim_{N \to \infty} \int_\mathbb{R} [(\hat{f})^N(\tfrac{\xi}{N}) - 1]\, \phi(\xi)\, d\xi = 0, $$ where the Fourier transform $\hat{f}$ here is normalized as $$ \hat{f}(\xi) = \int_\mathbb{R} e^{-i\xi x} f(x) dx. $$
I can show that $|\,\hat{f}| \le 1$ with equality only at the origin $\xi = 0$. So I further reduced to the following (pointwise convergence) using dominated convergence:
Under the same assumptions about $f$, for all $\xi \in \mathbb{R}$, $$ \lim_{N \to \infty} (\hat{f})^N(\tfrac{\xi}{N}) = 1. $$
But here I got stuck. The two $N$'s in the limit compete with each other, and I didn't see any workaround. Could anyone help with this? Thanks in advance.
I finally got this.
Since $f$ is Schwartz, $\hat{f}$ is also Schwartz (in particular, $C^\infty$), so we may take the Taylor series of $\hat{f}$ near the origin. Through interchanging $D_\xi$ and $x$, we can easily show that $\hat{f}(0) = 1$, $(\hat{f})'(0)$ (by $\int xf(x)dx = 0$), and $(\hat{f})''(0) = \int x^2 f(x)dx$ which is some finite constant, say $J$. Then $$ (\hat{f})^N(\xi/N) = (1+J(\xi/N)^2+o((\xi/N)^2))^N. $$ At last, checking for some fixed $M$ ($J$ plus some small constant) that $$ \lim_{N \to \infty} (1 + M N^{-2})^N = 1 $$ is a simple matter of l'Hopital.