Pick interpolation in several variables (Q2838966)

The classical Pick interpolation problem in one dimension is generalised to the several variable framework. Given \(z_1,\dots,z_n\) distinct points in the open unit disc \(\mathbb{D}\) and \(w_1,\dots,w_n\) complex numbers, the classical Pick interpolation theorem states the existence of a holomorphic function \(f\) in \(\mathbb{D}\) with \(f(z_i)=w_i\) for \(i=1,\dots,n\) and such that \(|f(z)|\leq 1\) for \(z\in\mathbb{D}\), if and only if the matrix \([(1-w_i\overline{w}_{j})(1-z_i\overline{z}_{j})^{-1}]^n_{i,j=1}\) is positive semidefinite.NEWLINENEWLINEWrite \(\Omega\) for the unit ball \(\mathbb{B}_{d}\) or the polydisc \(\mathbb{D}^{d}\) in \(\mathbb{C}^{d}\)and let \(H^{\infty}(\Omega)\) stand for the algebra of bounded analytic functions on \(\Omega\). Let \(H^{2}(\Omega)\) be the closure of the multivariable analytic polynomials in \(L^2(\partial\Omega,\theta)\), where \(\partial\Omega\) is the distinguished boundary of \(\Omega\). The Bergman space \(L_{a}^{2}(\Omega)\) is defined as the set of analytic functions on \(\Omega\) in \(L^{2}(\Omega,\mu)\).NEWLINENEWLINEThe main results in the present work are the following ones. A first result (Theorem 1.1) states that, if \(z_1,\dots,z_n\in\Omega\), \(w_1,\dots,w_n\in\mathbb{C}\), and \(\mathcal{A}\) is any weak\(^\star\)-closed subalgebra of \(H^{\infty}(\Omega)\), then there is a function \(\varphi\in\mathcal{A}\) with \(|\varphi(z)|\leq 1\) for every \(z\in\Omega\) and \(\varphi(z_i)=w_i\) for \(i=1,\dots,n\) if and only if the matrix NEWLINE\[NEWLINE[(1-w_i\overline{w_j})k^{\nu}(z_i,z_j)]_{i,j=1}^{n}NEWLINE\]NEWLINE is positive semidefinite for every measure of the form \(\nu=|f|^{2}\theta\), where \(\theta\) is the Lebesgue measure on \(\partial\Omega\) and \(f\in H^{2}(\Omega)\).NEWLINENEWLINEHere, \(k^{\nu}\) is the associated positive kernel function on a reproducing kernel Hilbert space (Section 2).NEWLINENEWLINEThe second main result (Theorem 1.2) is analogous to the first, where \(\Omega\) is chosen among the bounded domains in \(\mathbb{C}^{d}\) and \(\nu=|f|^{2}\mu\), where \(\mu\) is the Lebesgue measure on \(\Omega\), and \(f\in L^2_{a}(\Omega)\).