Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants
2016 (English)In: Journal of the London Mathematical Society, ISSN 0024-6107, E-ISSN 1469-7750, Vol. 94, no 3, 726-746 p.Article in journal (Refereed) Published
We study connections between orthogonal polynomials, reproducing kernel functions, and polynomials p minimizing Dirichlet-type norms vertical bar pf-1 vertical bar for a given function f. For [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, 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 D(1,fPn), 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.
Place, publisher, year, edition, pages
2016. Vol. 94, no 3, 726-746 p.
Orthogonal polynomials, kernels, zeros
Research subject Mathematics
IdentifiersURN: urn:nbn:se:su:diva-140074DOI: 10.1112/jlms/jdw057ISI: 000392842700004OAI: oai:DiVA.org:su-140074DiVA: diva2:1077301