arXiv:2611.14555v1 Announce Type: new Abstract: We study representing systems of Cauchy kernels or dilations of a given function in Hardy spaces. We give a complete geometric description of countable sets $\Lambda \subset \mathbb{D}$ such that the system $\{\hat{k}_{\lambda}\}_{\lambda \in \Lambda}$ of normalized Cauchy kernels forms a generalized frame in the Hardy space $H^p(\mathbb{D})$ with $p \in (1, \infty)$ with respect to a particular coefficient space. For a wide class of functions we construct representing systems of their dilations in $H^2$. We also obtain a linear method for expanding functions in series of Cauchy kernels in $H^2$.
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