Recently I'm reading a paper by Magyar-Stein-Wainger, Discrete analogues in harmonic analysis:Spherical averages, Annals of Mathematics, 155 (2002), 189–208.
A key ingredient in the proof of their main result is a representation formula of the surface measure $\mathrm{d}\sigma_\lambda$, where $\mathrm{d}\sigma_\lambda$ is the normalized invariant measure on the sphere $\{x\in\mathbb{R}^d:|x|=\lambda\}$ (In the paper, the dimension $d\geq5$). To be more precise, let $\delta>0$ be a parameter, and the function $\widehat{I_\lambda^{\delta}}(x)$ is defined by $$\widehat{I_\lambda^{\delta}}(x)=\int_{\mathbb{R}}\mathrm{e}^{-2\pi\mathrm{i}t}\mathrm{e}^{-\pi\delta t^2}\mathrm{e}^{-2\pi\frac{|x|^2}{\lambda^2}(1-\mathrm{i}t)}\mathrm{d}t,\text{ where }x\in\mathbb{R}^d.$$ In the paper, they concluded that there exists a constant $c_d>0$ independent of $\delta$ and $\lambda$, so that for every Schwartz function $\varphi\in\mathcal{S}(\mathbb{R}^d)$, one has $$\lim_{\delta\rightarrow0}\int_{\mathbb{R}^d}\varphi(x)\widehat{I_\lambda^{\delta}}(x)\mathrm{d}x=c_d\int_{\mathbb{R}^d}\varphi(x)\mathrm{d}\sigma_\lambda(x).$$
This limit is quite similar to the approximation to the identity in Fourier analysis. However, I cannot verify this limit, any reference or hint on this limit?
Sure one may calculate $\widehat{I_\lambda^{\delta}}$ directly: $$\widehat{I_\lambda^{\delta}}(x)=\mathrm{e}^{-2\pi\frac{|x|^2}{\lambda^2}}\delta^{-\frac {1}{2}}\mathrm{e}^{-\pi(1-|x|^2/\lambda^2)^2/\delta}.$$ In the original paper, the term $\mathrm{e}^{-\pi(1-|x|^2/\lambda^2)^2/\delta}$ was $\mathrm{e}^{-\pi(1-|x|^2/\lambda^2)/\delta}$, but I think this was a misprint. The degree of $|x|$ in the exponential term is 4, and not a Gaussian.