# The Cauchy Transform, and the convergence of the Fourier-Stieltjes transforms of a sequence of measures

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?

• Is this not related to fractional calculus? Jan 12 at 0:07
• No. This is a question of the representability of holomorphic functions in terms of their boundary behavior, and the stability of this representability with respect to sequences convergent in various topologies.
– MCS
Jan 12 at 21:31
• Then not related to the Euler integral for the beta function, various methods of analytic continuation for it, Mellin transform extension, Pochhammer contour integral rep., convolution rep. Jan 12 at 22:19
• Mmm... when you put it that way, I'd say that it is related—in that the formulas involved are in an open neighborhood about those topics. The sequences of measures in question are, in fact, the (partial) time averages of a dirac delta under the adjoint of a linear operator on functions on the disk, one which admits a representation as a contour integral against a kernel—a rational function of two complex variables. I could write a small paper on the questions I currently have. The present question is my effort to reduce it to its essentials, to increase my chance of getting a response.
– MCS
Jan 13 at 0:33
• A multi-dimensional neighborhood--an extension/interpolation of the Hilbert transform, the Stieltjes-Cauchy transform in free probability theory, even perhaps the Todd operator--so a number of possible approaches. Unfortunately, I have little time to spend on it, but I'm certainly interested in any results. Jan 13 at 16:18