On the Darboux transformations and sequences of p-orthogonal polynomials

2019/10/07 by Rolanía, D. Barrios, García-Ardila, J. C., Manrique, D.
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1910.03039

Abstract

For a fixed p ∈ ℕ, sequences of polynomials \P_n\, n ∈ ℕ, defined by a (p+2)-term recurrence relation are related to several topics in Approximation Theory. A (p+2)-banded matrix J determines the coefficients of the recurrence relation of any of such sequences of polynomials. The connection between these polynomials and the concept of orthogonality has been already established through a p-dimension vector of functionals. This work goes further in this topic by analyzing the relation between such vectors for the set of sequences \P_n^(j)\, n ∈ N, associated with the Darboux transformations J^(j), j=1, ..., p, of a given (p+2)-banded matrix J.

Related

• Darboux equivalence for matrix-valued orthogonal polynomials
• Darboux transforms on Band Matrices, Weights and associated Polynomials
• Associated orthogonal polynomials of the first kind and Darboux transformations
• Higher-order recurrence relations, Sobolev-type inner products and matrix factorizations
• Orthogonal matrix polynomials and higher order recurrence relations