For which reproducing kernel Hilbert spaces is Pick's theorem true!
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.
scientific article; zbMATH DE number 222166
scientific article