Sobolev orthogonal polynomials, Gauss-Borel factorization and perturbations

We present a comprehensive class of Sobolev bi-orthogonal polynomial sequences, which emerge from a moment matrix with an LU factorization. These sequences are associated with a measure matrix defining the Sobolev bilinear form. Additionally, we develop a theory of deformations for Sobolev bilinear forms, focusing on polynomial deformations of the measure matrix. Notably, we introduce the concepts of Christoffel-Sobolev and Geronimus-Sobolev transformations. The connection formulas between these newly introduced polynomial sequences and existing ones are explicitly determined.