Hilbert transforms are used in signal processing for creating "analytic" signals (see section 4.4 here https://www.di.ens.fr/~mallat/papiers/WaveletTourChap1-2-3.pdf for example) which are then used for defining things like instantaneous frequencies.
So, if we have a real-valued function $x(t)$ on reals, and its Hilbert transform is $y(t)$, the function $z(t) = x(t) + i y(t)$ is called the analytic signal corresponding to $x(t)$. In some applications, the signal $z(t)$ is used in place of $x(t)$, that is, it is considered a complex valued surrogate of $x(t)$.
I am interested in knowing why this is so.
Also, the expression "analytic" in analytic signal suggests it is somehow related to analytic functions in the sense of complex analysis. I am interested in knowing what the relation with analytic functions, if any, is.
Thanks.