Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants Articles uri icon

publication date

  • December 2016

start page

  • 726

end page

  • 746

issue

  • 3

volume

  • 94

International Standard Serial Number (ISSN)

  • 0024-6107

Electronic International Standard Serial Number (EISSN)

  • 1469-7750

abstract

  • We study connections between orthogonal polynomials, reproducing kernel functions, and polynomials P minimizing Dirichlet‐type norms ∥pf−1∥alfa for a given function f . For alfa ∈ [0,1] (which includes the Hardy and Dirichlet spaces of the disk) and general f , we show that such extremal polynomials are non‐vanishing in the closed unit disk. For negative alfa , the weighted Bergman space case, the extremal polynomials are non‐vanishing on a disk of strictly smaller radius, and zeros can move inside the unit disk. We also explain how dist Dalfa (1, f ยท Pn) , where Pn is the space of polynomials of degree at most n , can be expressed in terms of quantities associated with orthogonal polynomials and kernels, and we discuss methods for computing the quantities in question.