# Reference request for Stieltjes Transform

I am wondering to use Stieltjes transform to signal processing like Fourier Transform. The Fourier is known to give the frequencies of a signal but not sure what Stieltjes transform gives. I am only interested in real signals (not complex). The first question, is given a random sequence from some Gaussian PDF, can I perform Stieltes transform and obtain the original sequence form the Stieljes Transform? In other words does a random signal (real) has one to one correspondence in Stieltjes Transform domain?

Stieljes transform is about taking convolution of your signal $s(t)$ with the $\frac1 t$, so as to obtain $$C_s(z)=\int \frac{s(t)}{z-t}dt.$$ It is of course well defined at least when $z$ is a complex number with positive imaginary part. To be in a setting where you can "invert" this transform, one typically takes the real part of $C_s$ and the limit $z=x+i\epsilon$ with $\epsilon\to 0$, which gives you the Hilbert-transform $Hs(x)$ of $s$ (maybe up to a minus sign, depending on the definitions). In the frequency space, this is just multiplication by $-i \,\mathrm{sign}(x).$ The Hilbert transform shares similarity with the Fourier transform since we have $H(H(s))=-s$, see https://www.wikiwand.com/en/Hilbert_transform.
You may also be interested in the Sokhotski-Plemelj formula (real line version): https://www.wikiwand.com/en/Sokhotski%E2%80%93Plemelj_theorem. In particular it states that you can recover $s$ by taking the imaginary part of $C_s(z)$ and the same limit than above, although I suspect this is not the interpretation you're looking for.