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.