RT Journal Article
T1 Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants
A1 Bénéteau, Catherine
A1 Khavinson, Dmitry
A1 Sola, Alan A.
A1 Liaw, Constanze
A1 Seco Forsnacke, Daniel
AB We study connections between orthogonal polynomials, reproducing kernel functions, and polynomials P minimizing Dirichlet‐type norms ∥pf−1∥α for a given function f . For α ∈ [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 α , 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 Dα (1, f · 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.
PB Wiley
SN 0024-6107
YR 2016
FD 2016-12
LK https://hdl.handle.net/10016/32632
UL https://hdl.handle.net/10016/32632
LA eng
NO This work was supported by NSF under the grant DMS1500675. DS was supported by ERC Grant 2011-ADG-20110209 from EU programme FP2007-2013 and MEC Projects MTM2014-51824-P and MTM2011-24606.
DS e-Archivo
RD 18 jul. 2024