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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 I the integrals could not make sensehave no meaning. In fact, I was looking for the criterion by a line integral representation.

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Let be $\mathcal M(\mathbb M(\partial\mathbb D)$ denote the set of all Borel complex probability measures on $\mathbb \partial\mathbb D$ (unit disc circle in the complex plane). Define a mapping $\Phi:\mathcal M(\mathbb M(\partial\mathbb D)\to \ell^{\infty}(\mathbb N)$ 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 $\Phi(\mathcal M(\mathbb M(\partial\mathbb D))$ ? 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 I the integrals could not make sense. I was looking for the criterion by a line integral representation.

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# CoeficientsCoefficients of Holomorphicholomorphic functions defined by Borel ProbabilityMeasuresprobabilitymeasures on the unit disc.

Let be $\mathcal M(\mathbb D)$ denote the set of all borel Borel complex probability measures on $\mathbb D$ (unit disc in the complex plane). Define a mapping $\Phi:\mathcal M(\mathbb D)\to \ell^{\infty}(\mathbb N)$ by $$\Phi(\mu)=(c_0,c_1,\ldots)$$ where $c_k$, are the coefficients of the Taylor Expansion 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 $\Phi(\mathcal M(\mathbb D))$ ? 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.

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