Let be a holomorphic function on the unit disc such that and for all . If the Taylor expansion of is
then a classical results states that for all integer . Moreover, such functions are in one-to-one correspondence with probability measures on the unit circle as follows. Let be a probability measure on and let for be its Fourier coefficients
Then , and is the complex conjugate of . If has a density function with respect to the Lebesgue probability measure that is holomorphic on some neighborhood of then is real-valued on and has the Laurent expansion
Now define a function on a neighborhood of the closed disc as follows:
Then, using the fact that for all , for all we have and therefore for all . It turns out that the correspondence between probability measures and functions with positive real part also holds for that not have such a nice density function. (In general the density is a hyper function on related to ). The correspondence that I sketched here is usually presented as a convexity result in functional analysis, but I think that the direct link to density functions is quite instructive.
There is however also a different way to parameterize functions with a positive real part but it is much less known (I’d be interested in any reference in fact). Let be any holomorphic function on a neighborhood of the closed disc with Taylor expansion
Let be the holomorphic function such that , so the coefficients of are the complex conjugates of those of . Then for all
Now the function has the Laurent expansion
where I used the convention that for all . Now the sequence is square summable and so it is an element of the Hilbert space . Let be the left shift operator, so for all and indices . Then the Laurent series above can be rewritten as
where denotes the sesquilinear form on . Note that by the Cauchy-Schwarz inequality . Now define a holomorphic function on a neighborhood of the closed disc by
Then for all we have and therefore on the disc . The association is not injective however, since and map to the same function if on the unit circle . In this case, both and are equal to some function without zeroes on the disc multiplied by appropriate finite Blaschke products. This function is then unique up to multiplication with a constant on the unit circle.