Publication: Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants
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2016-12
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Wiley
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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.
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Bénéteau, C., Khavinson, D., Sola, A. A., Liaw, C. & Seco, D. (2016). Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants. Journal of the London Mathematical Society, 94(3), pp. 726–746.