Integral Sphere on Unit – Mathematics Stack Exchange

I'm looking for ideas to calculate
$$J (x) = int_ {| y | = 1} f (x cdot y) dy$$
or $$x, y in mathbb R ^ n$$, $$| x |, x cdot y$$ are the Euclidean norm and dot products, and $$f (t)$$ is a real function of a real variable, eg. $$f (t) = P (1 / t) e ^ {- 1 / 2t ^ 2}$$ or $$P$$ 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 transformation of $$delta (| y | -1)$$ but I do not know where to go next …?

Thank you!

p.