user90189

Dear @fedja, what do you mean by $\int e^{i\lambda|x|^2}\,dx$ diverges at infinity? The function $e^{i\lambda|x|^2}$ has absolute value 1 for real $\lambda$, but do oscillate strongly. And as $\lambda\to\infty$ the integral goes to zero as $1/|\lambda|^{n/2}$. Thank you for your comment.

Instead of Mathematica, you can try to calculate the inner integral, it is classical and should be in google. Look for Fresnel functions. You can calculate it by residues, using a "pizza"-like path, with the corner at zero.

I think it is easier to use polar coordinates, the integral is just $\int_{S^{n-1}}\int_0^\infty r^{n-1}\delta(n-r^2)\,drd\sigma$. I'm looking at your calculation, but I don't see the big problem, but the inner integral should be something of the order $1/\sqrt{|\lambda|}$. This is an oscillatory integral, so you should be careful, even more if you try to use Fubini.

Hint: use polar coordinates. Heuristically, $\delta(n-|x|^2)$ is like a function accumulated at the sphere of radius $\sqrt{n}$, so the integral should be related to the area of $\sqrt{n}S^{n-1}$.