Let \(H\) be a separable Hilbert space of complex-valued analytic functions defined on a plane domain \(E\) in the complex plane equipped with an inner product \(\langle\, ,\, \rangle\). The Hilbert space \(H\) is said to have the reproducing property provided that there exists a function \(k:E\times E\to \mathbb{C}\) such that for every \(t\in E\) we have \(k(\cdot , t)\in H\) and for every \(f\in H\), \[ f(t)=\langle k , f(\cdot , t)\rangle . \] In this case \(H\) is called a reproducing kernel Hilbert space, and the function \(k\) is called the reproducing kernel of \(H\). Classical examples of reproducing kernel Hilbert spaces and their relevant reproducing kernels are the Hardy spaces, the Bergman spaces, and the Fock spaces.
In the paper under review, the author considers a basic triple \((E, \mathcal{H}, \kappa_t)\) where \(E\) is an abstract set, \(\mathcal{H}\) is a separable Hilbert space, and \(\kappa_t\) is a mapping from \(E\) into \(\mathcal{H}\). The kernel \(k:E\times E\to \mathbb{C}\) defined by \[ k(s,t)=\langle \kappa_t , \kappa_s \rangle _{\mathcal{H}} \] is called the kernel function of \(\kappa_t\). It is known that there is one and only one Hilbert space \(R_k\) having \(k\) as its reproducing kernel. For \(f\in R_k\), \(t\in E\), and \(k(\cdot , t)\in R_k\), we have \[ f(t)=\langle f , k(\cdot , t)\rangle _{R_k}. \] The main result of the paper under review says that if \(\{s_n\}\subseteq E\) is chosen such that \(\{\varphi_n:=\mu_n\kappa_{s_n}\}\), with \(\mu_n\)’s normalizing factors, forms an orthonormal basis for \(\mathcal{H}\), then for every \(f\in R_k\) and \(t\in E\), the series \[ f(t)=\sum_{n\in\mathbb{N}} f(t_n)\mu _n^2 k(t , s_n) \] converges pointwise and in norm of \(R_k\). In the rest of the paper, the author uses this observation to argue that I. Kluvanek’s sampling theorem [Mat.-Fyz. Čas., Slovensk. Akad. Vied 15, 43–47 (1965; Zbl 0154.44403)] is a special case of the above result.
For the entire collection see [Zbl 1302.00086].