I'm reading Stein and Shakarchi's Fourier Analysis, and have a question about the proof of Fourier inversion for Schwartz class functions.
In the proof below, I have two questions. Here, $K_\delta(x)=\delta^{-1/2}e^{-\pi x^2/\delta}$.
First, why does the integral $\int f(x)K_\delta (x)dx$ goes to $f(0)$ as $\delta$ tends to $0$ since $K_\delta$ is a good kernel? I've attached all the previous propositions in the text as well, but I can't see why it must follow because $K_\delta$ is a good kernel. My guess is, using Corollary 1.7, we have $(f*K_\delta) (0)=\int f(x)K_\delta (0-x) dx \to f(0)$ uniformly as $\delta \to 0$ and $K_\delta$ is even so $K_\delta (-x)=K_\delta (x)$ and we can get the result. But this requires $K_\delta$ being even and not just a good kernel.
Finally, how does the second integral, $\int \hat{f}(\xi)G_\delta (\xi)d\xi$ converges to $\int \hat{f}(\xi)d\xi$ as $\delta \to 0$? I think we are supposed to change the order of $\lim_{\delta \to 0}$ and the integration, but how are we guaranteed to do this?
For the good kernel part you don't need symmetry of the kernel. As it integrates to one the claim follows from the more general result that for $f$ continuous and uniformly bounded we have $$ \lim_{\delta\rightarrow 0} \frac{1}{\sqrt{\delta}} \int_{\Bbb R} \left(f(x)-f(0)\right)\ e^{-\pi x^2/\delta} dx = 0 $$ Given $\epsilon>0$ find $\eta$ so that $|x|<\eta \Rightarrow |f(x)-f(0)|<\epsilon/2$ and then $\delta$ so that the contribution from the integral over $|x|\geq \epsilon$ (the condition (iii)) is smaller than $\epsilon/2$
For the second part the easiest is to note that $\hat{f}$ is $L^1$ and use Dominated convergence (since $G_\delta(x)$ goes pointwise to 1). But you may also give an $\epsilon,\delta$ - proof by hand:
The function $\hat{f}$ is integrable, say $I=\int|\hat{f}|<+\infty$ so given $\epsilon>0$ first find $M$ so that $$\int_{|\xi|>M} |\hat{f}(\xi)| d\xi < \epsilon/2$$ Now find $\delta>0$ so that $$\sup_{|\xi|\leq M} (1-G_\delta(\xi)) < \frac{\epsilon}{2 I}$$ Then $$ \left|\int_{|\xi|\leq M}\ \hat{f}(\xi)(1-G_\delta(\xi)) d\xi \right|<\epsilon/2 $$ Combining with the above, noting that $0< G_\delta\leq 1$ we obtain the result.