Conditions on $F(y)$ for the existence of $f(x)$ where $F(y) = \int_{0}^{\infty} y \exp{[(-\frac{1}{2}(y^2 + x^2)]} I_{0}(xy) f(x) \; \mathrm{d}x$

Question

I've been working with the following integral transform:

$$F(y) = \int_{0}^{\infty} y \exp{\left[-\frac{1}{2}(y^2 + x^2)\right]} I_{0}(xy) f(x) \; \mathrm{d}x$$

where:

• $x,y$ are positive-definite so have range $[0,\infty)$

• $I_0(x)$ is the modified Bessel function of the first kind, order zero.

Specifically, using numerical methods to invert the transform to obtain $f(x)$ for a given $F(y)$. However, I'm often finding that my numerical inversion produces nonsense for some measured $F(y)$, but works fine for others. My question is therefore the following:

Are there any conditions on $F(y)$ in order for $f(x)$ to exist? If so is it possible to determine whether a given $F(y)$ meets these conditions so that $f(x)$ will exist?

Thanks for your time!