Matrix Uvarov Transformation on the Unit Circle: Asymptotic Properties

Abstract

Let 𝜎 be an 𝑙×𝑙 Hermitian matrix measure supported on the unit circle. In this contribution, we study some algebraic and analytic properties of matrix orthogonal polynomials associated with the Uvarov matrix transformation of 𝜎 defined by d𝜎𝑢𝑚(𝑧)=d𝜎(𝑧)+∑𝑗=1𝑚𝐌𝑗𝛿(𝑧−𝜁𝑗), where 𝐌𝑗 is an 𝑙×𝑙 positive definite matrix, 𝜁𝑗∈ℂ with 𝜁𝑗≠𝜁𝑖 and 𝛿 is the Dirac matrix measure.