Christoffel transformation for a matrix of bi-variate measures Articles uri icon

publication date

  • November 2019

start page

  • 3979

end page

  • 4005

issue

  • 8

volume

  • 13

International Standard Serial Number (ISSN)

  • 1661-8254

abstract

  • We consider the sequences of matrix bi-orthogonal polynomials with respect to the bilinear forms ⟨⋅,⋅⟩R^ and ⟨⋅,⋅⟩L^ ⟨P(z1),Q(z2)⟩R^=∫T×TP(z1)†L(z1)dmu(z1,z2)Q(z2),⟨P(z1),Q(z2)⟩L^=∫T×TP(z1)L(z1)dmu(z1,z2)Q(z2)†,P,Q∈Lp×p[z]where mu(z1,z2) is a matrix of bi-variate measures supported on T×T, with T the unit circle, Lp×p[z] is the set of matrix Laurent polynomials of size p×p and L(z) is a special polynomial in Lp×p[z]. A connection formula between the sequences of matrix Laurent bi-orthogonal polynomials with respect to ⟨⋅,⋅⟩R^, (resp. ⟨⋅,⋅⟩L^) and the sequence of matrix Laurent bi-orthogonal polynomials with respect to dmu(z1,z2) is given.

keywords

  • block cmv matrices; gauss¿borel factorization; matrix biorthogonal polynomials; matrix christoffel transformations; matrix-valued measures; nondegenerate continuous bilinear forms; quasideterminants