Coefficients of holomorphic functions defined by Borel probability measures on the unit disc - MathOverflow

Coefficients of holomorphic functions defined by Borel probability measures on the unit disc

2010-08-13T06:21:59Z

Let be $\mathcal M(\partial\mathbb D)$ denote the set of all Borel complex probability measures on $\partial\mathbb D$ (unit circle in the complex plane). Define a mapping $\Phi:\mathcal M(\partial\mathbb D)\to \ell^{\infty}(\mathbb N)$ by $$ \Phi(\mu)=(c_0,c_1,\ldots) $$ where $c_k$, are the coefficients of the Taylor expansion (around $z=0$) of the function $$ f(z)=\int_{\partial\mathbb D}\frac{1}{1-wz} d\mu(w), $$ Question: Who is the set $\Phi(\mathcal M(\partial\mathbb D))$ ? Given a sequence is there a simple criterion to verify if it belongs to this space?

Motivation: Let be $A=(a_0,a_1,\ldots)\in\Phi(\mathcal M(\mathbb D))$, $g:U\subset\mathbb C\to \mathbb C$ analytic with $$g(z)=\sum_{k=0}^{\infty}b_kz^k$$ Define $A*g:U\subset\mathbb C\to \mathbb C$ by $$ A*h(z)=\sum_{k=0}^{\infty} a_kb_kz^k. $$ The above integral representation, can be used to give a short proof of $$ \|A*h\|_{U}\leq \|h\|_{U}, $$ where $\|g\|_{U}=\sup_{z\in U}|g(z)|$ .

PS:This question it is a generalization of a question that arose in a discussion at Area 51.

Edition: I am correcting the question, because in the previous version the integrals could have no meaning. In fact, I was looking for the criterion by a line integral representation.

Answer by Zen Harper for Coefficients of holomorphic functions defined by Borel probability measures on the unit disc

2010-08-13T09:52:18Z

EDIT: this was for a different problem, but it has now been changed; so ignore this!

Too long for a comment...

If the integral is over $D$, then $f'(z)$ is (after changing variable $w$ to $\bar{w}$ in the measure $\mu$) \[ \frac{d}{dz} \int_D \frac{d\mu(w)}{1-\bar{w}z} = \int_D \frac{\bar{w} \, d\mu(w)}{(1-\bar{w}z)^2} = P_{L^2_a}(\bar{w} \mu), \] which is (formally) the $L^2_a(D)$ Bergman space projection operator applied to the measure $\bar{w} \mu$; but I don't know if this function of $z$ necessarily does lie in $L^2_a$.

So it's closely connected to Bergman space Toeplitz operators, which are known to be very tricky (much worse than on the Hardy space). Even worse, there is usually no nice way to characterise various function spaces which arise in terms of Taylor coefficients.

So your question really combines two interesting and highly non-trivial problems! I doubt there is any really simple answer.

Answer by Piero D'Ancona for Coefficients of holomorphic functions defined by Borel probability measures on the unit disc

2010-08-13T15:03:09Z

There's probably something I do not understand about your question, in case just forget my babbling. Anyway let me try: if you expand $$\frac{1}{1-wz}=\sum_{k\ge0}z^kw^k$$ and write $$f(z)=\sum_{k\ge0}z^k\int_{\partial D}w^k d\mu(w)= \sum_{k\ge0}z^k\int_0^{2\pi}e^{ik\theta}d\mu(\theta)$$ you see that the $c_k$ are the Fourier coefficients of positive order of the measure $\mu$. Now I would not go too far and say that any bounded sequence may apply, but this is well documented in several places (see e.g. the trigonometric moment problem).

EDIT: and maybe this is a good starting point.