Let $f$ be a smooth real (or complex) valued function defined on $S^2$. Then a direct calculation shows that $$\int_{S^2}f(x)e^{ixy}\, dx=-\frac{i}{\|y\|}e^{i\|y\|}f\left(\frac{y}{\|y\|}\right)+\frac{i}{\|y\|}e^{-i\|y\|}f\left(-\frac{y}{\|y\|}\right)$$ where $y$ is a nonzero vector in $\mathbb{R}^3$ and $xy$ is the inner product of $x$ and $y$ in $\mathbb{R}^3$.
Clearly, if $f$ is a constant function, then the limit of right hand at $y=0$ equals the value of the left hand side. My question is whether it is possible to use the above equality to the find the value of the the left hand side at $y=0$, that is when there is no oscillation, for any $f$.