Functions with positive real part

Let f be a holomorphic function on the unit disc \Delta such that f(0)=1 and \Re f(z) > 0 for all z \in \Delta. If the Taylor expansion of f is

f(z) = 1 + 2a_1z+2a_2z^2+2a_3z^3+\dotsc

then a classical results states that |a_n| \leq 1 for all integer n \geq 1. Moreover, such functions are in one-to-one correspondence with probability measures on the unit circle \Gamma as follows. Let \mu be a probability measure on \Gamma and let \hat{\mu}_k \in \mathbb{C} for k \in \mathbb{Z} be its Fourier coefficients

\hat{\mu}_k = \displaystyle \int_{\Gamma} z^{-k} d\mu(z).

Then \hat{\mu}_0 = 1, |\hat{\mu}_k| \leq 1 and \hat{\mu}_{-k} is the complex conjugate of \hat{\mu}_k. If \mu has a density function \delta with respect to the Lebesgue probability measure that is holomorphic on some neighborhood of \Gamma then \delta is real-valued on \Gamma and has the Laurent expansion

\delta(z) = \displaystyle \sum_{k \in \mathbb{Z}} \hat{\mu}_k z^k.

Now define a function f_{\mu} on a neighborhood of the closed disc \overline{\Delta} as follows:

f_{\mu}(z) = 1 + 2\displaystyle \sum_{k = 1}^{\infty} \hat{\mu}_k z^k.

Then, using the fact that \overline{z} = z^{-1} for all z \in \Gamma, for all z \in \Gamma we have \Re f(z) = \delta(z) \geq 0 and therefore \Re f(z) > 0 for all z \in \Delta. It turns out that the correspondence \mu \leftrightarrow f_{\mu} between probability measures and functions with positive real part also holds for \mu that not have such a nice density function. (In general the density is a hyper function on \Gamma related to f_{\mu}). 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 g be any holomorphic function on a neighborhood of the closed disc \overline{\Delta} with Taylor expansion

g(z) = \displaystyle \sum_{k=0}^{\infty} a_kz^k.

Let \overline{g} be the holomorphic function such that \overline{g}( \overline{z} ) = \overline{g(z)}, so the coefficients of \overline{g} are the complex conjugates of those of g. Then for all z \in \Gamma

g(z) \overline{g}(z^{-1}) = g(z) \overline{g(z)} = |g(z)|^2 \geq 0.

Now the function g(z) \overline{g}(z^{-1}) has the Laurent expansion

(\displaystyle \sum_{k=0}^{\infty} a_kz^k)(\sum_{k=0}^{\infty} \overline{a}_kz^{-k}) = \sum_{k \in \mathbb{Z}} \sum_{m = 0}^{\infty} a_{m+k} \overline{a}_m z^k

where I used the convention that a_k=0 for all k < 0. Now the sequence a is square summable and so it is an element of the Hilbert space \ell^2. Let T: \ell^2 \rightarrow \ell^2 be the left shift operator, so (T^ka)_m = a_{m+k} for all k \geq 0 and indices m. Then the Laurent series above can be rewritten as

\displaystyle (\sum_{k=1}^{\infty} \langle a, T^ka \rangle z^{-k}) + ||a||^2 + (\sum_{k=1}^{\infty} \langle T^ka, a \rangle z^k)

where \langle \cdot, \cdot \rangle denotes the sesquilinear form on \ell^2. Note that by the Cauchy-Schwarz inequality |\langle T^ka, a \rangle| \leq ||T^ka|| \cdot ||a|| \leq ||a||^2. Now define a holomorphic function f_g on a neighborhood of the closed disc \overline{\Delta} by

f_g(z) = ||a||^2 + 2 \displaystyle \sum_{k=1}^{\infty} \langle T^ka, a \rangle z^k.

Then for all z \in \Gamma we have \Re f_g(z) = |g(z)|^2 \geq 0 and therefore \Re f_g(z) > 0 on the disc \Delta. The association g \mapsto f_g is not injective however, since g_1 and g_2 map to the same function if |g_1(z)|^2 = |g_2(z)|^2 on the unit circle \Gamma. In this case, both g_1 and g_2 are equal to some function g without zeroes on the disc \Delta multiplied by appropriate finite Blaschke products. This function g is then unique up to multiplication with a constant on the unit circle.

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