The generalized operator-based Prony method is an important tool for describing signals which can be written as finite linear combinations of eigenfunctions of certain linear operators. On the other hand, Bernoulli's algorithm and its generalizations can be used to recover the parameters of rational functions belonging to finite-dimensional subspaces of $H_2$ Hardy-Hilbert spaces. In this work, we discuss several results related to these methods. We discuss a rational variant of the generalized operator-based Prony method and show that in fact, any Prony problem can be treated this way. This realization establishes the connection between Prony and Bernoulli methods and allows us to address some well-known numerical pitfalls. Several numerical experiments are provided to showcase the usefulness of the introduced methods. These include problems related to the identification of time-delayed linear systems and parameter recovery problems in reproducing kernel Hilbert spaces.

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