I was studying on the book Random Fields and Geometry (R.J Adler, J.E. Taylor) when I came up with this probability lemma:
Let $X,Y$ be two independent, standard (with mean$=0$ and variance=$Id_k$), $k$-dimensional Gaussian vectors and $f,g:\mathbb{R}^k\to \mathbb{R}$ two $C^2$ bounded functions. Then: $$ Cov(f(X),g(X))=\int_0^1\mathbb{E}\left[\nabla f(X)\cdot \nabla g\left(\alpha X+\sqrt{1-\alpha^2}Y\right)\right]d \alpha.$$
The proof is really straightforward for $f(x)=e^{it\cdot x}$ and $g(x)=e^{is\cdot x}$ for $t,s,x\in\mathbb{R}^k.$
Then, the authors state that well-known approximation results help in order to conclude.
My question now is: which approximation results are used to generalize the result to general $C^2$ functions? I couldn’t find any Fourier analysis result that would help me.
Thanks to everybody.