Let $S^n$ be an $n$-dimentional unit sphere.
Consider $f: S^n \longrightarrow R_+$ even continuous function.
Denote $$ F(f):=\int_0^{\infty}\int_{S^n}f(y)g\left(\frac{|xy|}{t}\right)dy\frac{dt}{t^{n+1}}, $$ where $x \in S^n, \, t>0$ and function $g$ is such that $$ \int_{0}^{\infty}s^jg(s)ds=0, \quad j=0,2,4,\ldots, 2\left[(n-1)/2\right] $$ $$ \int_1^{\infty}s^{\alpha}|g(s)|ds< \infty, \quad \alpha>n-1. $$
Find Fourier Transform of $F$.