Continuous version of uniform continuity?

Question

Studying Fourier analysis, I faced this problem which author skipped saying "clearly". $$G_\delta(x)=e^{-\pi \delta x^2}$$ Let $f\in S(\mathbb R)$(Schwartz Space). Then $$\int_\infty ^\infty \hat{f}(\xi)G_\delta(\xi)d\xi \to \int_\infty ^\infty \hat{f}(\xi)d\xi $$ I guess this problem has something to do with uniform convergence(which is about the interchange of order of integral and limit). Is there a continuous version of uniform convergence?

Answer 1

Presumably the limit is taken as $\delta\to 0$. In this case, since $G_{\delta}\to 0$ pointwise, and $|G_{\delta}(\xi)\hat{f}(\xi)|\leq |\hat{f}(\xi)|\in L^1$ due to the fact that $\hat{f}$ is Schwartz, the dominated convergence theorem implies the convergence of the integrals.