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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?

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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.

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Maybe the following reference will be useful https://arxiv.org/abs/1105.0060 (Signal Processing in Large Systems: a New Paradigm, by R. Couillet, M. Debbah). See also chapter 3 in the book "Random Matrix Methods for Wireless Communications" by the same authors (https://www.amazon.com/Random-Matrix-Methods-Wireless-Communications/dp/1107011639) and the following MO question: Intuitive understanding of the Stieltjes transform

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  • $\begingroup$ I am familiar with the references and gone through some of them. IMHO, op is not asking in that direction. Those direction are through eigenvalues of Random matrix, OP is clearly comparing through Fourier. Is OP's idea is wrong it should be noted why. $\endgroup$ – Creator Dec 24 '17 at 10:22

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