Theoretical Foundations of Matrix Continued Fraction Interpolation (MCFI)

We introduce Matrix Continued Fraction Interpolation (MCFI), a framework for constructing rational matrix-valued interpolants. Unlike traditional elementwise methods, which fit entries independently and often yield inconsistent pole structures, MCFI produces a single rational matrix function with a shared scalar denominator. We rigorously define the method using a generalized matrix Thiele continued fraction expansion and prove that this common-denominator structure prevents spurious artifacts in cross-channel quantities. We establish conditions for existence and uniqueness and derive local stability indicators to monitor noise amplification. Furthermore, we extend the framework to include adaptive degree selection via smoothing functionals and cross-validation, as well as preservation of algebraic structures for data in inverse-closed subspaces. Numerical experiments demonstrate that MCFI significantly outperforms independent scalar interpolation in terms of stability, noise resilience, and physical consistency.