I am looking for ideas to calculate $$ J(x) = \int_{|y| = 1} f(x\cdot y) dy $$ where $x,y\in\mathbb R^n$, $|x|,x\cdot y$ are the Euclidean norm and dot products, and $f(t)$ is a real function of a real variable, e.g. $f(t) = P(1/t)e^{-1/2t^2}$ where $P$ is a polynomial...
My attempt: write $J(x) = \int_{|y|=1} \int_{-\infty}^{\infty} f(t) \delta(t - x\cdot y) dt dy$, then use Fubini (?) to write $$ J(x) = \int_{-\infty}^\infty f(t) \int_{|y|=1} \delta(t - x\cdot y)dy dt $$ The inner integral is now the Radon transform of $\delta(|y|-1)$ but I am not sure where to go next...?
Thanks!
p.
Assume $x$ is a unit vector. Pick an orthogonal transformation $U$ that maps $x$ to the vector $e_n=(0,0,\dots,1)$. Then the invariance of the measure implies $$\int_{|y|=1}f(x\cdot y)\,dy=\int_{|y|=1}f(Ux,Uy)\,d(Uy)=\int_{|y|=1}f(y_n)\,dy$$ where $y_n=y\cdot e_n$ is the last coordinate of $y$. Now, writing each point $y$ on the sphere $\mathbb{S}^{n-1}$ as $y=(\sin\varphi\xi,\cos\varphi)$ where $\xi\in\mathbb{S}^{n-2}$ and $0\leq\varphi\leq\pi$, then $dy=d\xi\sin^{n-2}\varphi d\varphi$, so that $$\int_{\mathbb{S}^{n-1}}f(y_n)\,dy=\int_{\mathbb{S}^{n-2}}\int_0^{\pi}f(\cos\varphi)\sin^{n-2}\varphi\,d\varphi\,d\xi=c_n\int_0^{\pi}f(\cos\varphi)\sin^{n-2}\varphi$$ where $c_n$ is the surface area of the sphere $\mathbb{S}^{n-2}$. Thus, the integral over the sphere has been reduced to a one-dimensional integral, and it does not depend on $x$.