## 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.