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.