For which reproducing kernel Hilbert spaces is Pick's theorem true!
type: Item
instance of: scholarly article
title: For which reproducing kernel Hilbert spaces is Pick's theorem true!
zbMATH Open document ID: 0779.30026
DOI: 10.1007/BF01358955
author: Peter Quiggin
published in: Integral Equations and Operator Theory
publication date: 1993-06-29T00:00:00Z
review text: Let \(H(K)\) denote a general reproducing kernel Hilbert space of functions on a set \(X\) with reproducing kernel \(K\). A function \(\varphi:X\to\mathbb{C}\) is a multiplier of \(H(K)\) if \(H(K)\) is closed under multiplication by \(\varphi\). Let \(M(K)\) denote the space of multipliers on \(K\). With \(X=\mathbb{D}\), the open unit disc, and \(K\) the Szegö kernel on \(D\), \(H(K)\) is the Hardy space \(H^ 2\) and \(M(K)\) is \(H^ \infty\). The classical Pick's theorem states that given \(n\) values \(z_ i\in\mathbb{C}\) and \(n\) points \(w_ i\in\mathbb{D}\), there exists a function \(\varphi\in H^ \infty\) with \(\varphi(w_ i)=z_ i\) and \(\|\varphi\|_ \infty\leq 1\) if and only if the matrix \[ \left[{1-z_ i\overline{z_ j}\over 1-w_ i\overline{w_ j}}\right] \] is positive. In the paper the author considers the extension of Pick's theorem to \(H(K)\), and provides sufficient conditions on the kernel \(K\) for the generalized Pick's theorem to hold. These are then used to prove Pick's theorem for certain weighted Hardy and Sobolev spaces.
Mathematics Subject Classification ID: 30E05
Mathematics Subject Classification ID: 46E22
Mathematics Subject Classification ID: 47A20
zbMATH DE Number: 222166
zbMATH Keywords: reproducing kernel Hilbert space
zbMATH Keywords: Pick's theorem
reviewed by: Manfred Stoll
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 222166