Let $C\left(\mathbb{R}/\mathbb{Z}\right)$ denote the Banach space of continuous, $1$-periodic complex-valued functions on the unit interval, let $M\left(\mathbb{R}/\mathbb{Z}\right)$ denote its dual, the space of finite omplex Borel measures on $\mathbb{R}/\mathbb{Z}$, and let $\mathcal{A}\left(\mathbb{D}\right)$ denote the vector space space of all holomorphic functions $f:\mathbb{D}\rightarrow\mathbb{C}$ on the open unit disk. We equip $\mathcal{A}\left(\mathbb{D}\right)$ with the topology of compact convergence on $\mathbb{D}$—that is, uniform convergence on every compact subset of $\mathbb{D}$. Next, letting $\omega$ be a positive real number, define the order $\omega$ Cauchy transform $\mathscr{C}_{\omega}$ as the linear operator $\mathscr{C}_{\omega}:M\left(\mathbb{R}/\mathbb{Z}\right)\rightarrow\mathcal{A}\left(\mathbb{D}\right)$ given by: $$\mathscr{C}_{\omega}\left\{ d\mu\right\} \left(z\right)\overset{\textrm{def}}{=}\int_{0}^{1}\frac{d\mu\left(t\right)}{\left(1-e^{-2\pi it}z\right)^{\omega}},\textrm{ }\forall\left|z\right|<1,\textrm{ }\forall\mu\in M\left(\mathbb{R}/\mathbb{Z}\right)$$ Finally, for any $\mu\in M\left(\mathbb{R}/\mathbb{Z}\right)$, let $\hat{\mu}:\mathbb{Z}\rightarrow\mathbb{C}$ denote the Fourier coefficients of $\mu$/ Fourier-Stieltjes transform of $\mu$: $$\hat{\mu}\left(n\right)\overset{\textrm{def}}{=}\int_{0}^{1}e^{-2\pi int}d\mu\left(t\right),\textrm{ }\forall n\in\mathbb{Z}$$
I've been doing quite a bit of reading about Cauchy transforms (fractional or otherwise), but I haven't been able to find much regarding the behavior of the transform with respect to the Fourier coefficients of sequences of elements in $M\left(\mathbb{R}/\mathbb{Z}\right)$. Specifically, let $\left\{ \mu_{m}\right\} _{m\geq1}$ be a sequence in $M\left(\mathbb{R}/\mathbb{Z}\right)$, and let:$$f_{m}\left(z\right)\overset{\textrm{def}} {=}\mathscr{C}_{\omega}\left\{ d\mu_{m}\right\} \left(z\right),\textrm{ }\forall m\geq1$$
Now, suppose that:
I. As $m\rightarrow\infty$, the $f_{m}$s converge compactly over $\mathbb{D}$ to a limit $f\in\mathcal{A}\left(\mathbb{D}\right)$.
II. There is a function $c:\mathbb{Z}\rightarrow\mathbb{C}$ so that: $$\lim_{m\rightarrow\infty}\sup_{n\in\mathbb{Z}}\left|c\left(n\right)-\hat{\mu}_{m}\left(n\right)\right|=0$$
With these hypotheses, does it then follow that there is a measure $d\mu$ so that both:
i. $c=\hat{\mu}$
ii. $f=\mathscr{C}_{\omega}\left\{ d\mu\right\}$
That is to say: are the pointwise limit of the Fourier coefficients of the $\mu_{m}$s the Fourier coefficients of a measure $\mu$, and is $f$ the Cauchy transform of this $\mu$?
Additionally, would it make any difference if it was known that the $f$s also converged compactly outside of the closed unit disk?
Finally, if, for each $m$, $d\mu_{m}\left(t\right)=\phi_{m}\left(t\right)dt$, where there is a $p\in\left(1,\infty\right)$ so that $\phi_{m}\in L^{p}\left(\mathbb{R}/\mathbb{Z}\right)$ for all $m$, is $d\mu$ of the form $d\mu\left(t\right)=\phi\left(t\right)dt$ for some $\phi\in$ $L^{p}\left(\mathbb{R}/\mathbb{Z}\right)$, where $\mu=c$, where $c$ is as described above?
