Uniform Approximation by Rational Functions with Prescribed Poles: Operator-Theoretic Perspective and Symmetries

In this paper, the uniform approximation of continuous functions on [0,1] by rational functions with prescribed poles and bounded multiplicities is studied. A classical theorem of Fichera characterizes density in C([0,1]) through the divergence of a conformally invariant series involving the pole distribution. A modern reformulation of this result is developed and it is given an operator-theoretic interpretation in which the approximation property is equivalent to cyclicity and to the absence of nontrivial invariant subspaces in an associated Hardy-space model. In this framework, the classical Blaschke condition emerges as the fundamental obstruction to density, linking rational approximation to the structure of model spaces and non-selfadjoint operator algebras. The density criterion is interpreted in terms of symmetry: divergence corresponds to a balanced distribution of poles compatible with the conformal geometry of the slit domain, while convergence induces symmetry breaking and the emergence of invariant structures. Numerical models illustrate the sharpness of the criterion and provide a concrete manifestation of the Blaschke obstruction and cyclicity mechanism. This new approach places Fichera’s theorem within a broader operator-theoretic and spectral framework, connecting classical approximation theory with Hardy spaces, invariant subspace theory, and modern rational approximation methods.