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Analytic Kramer kernels, Lagrange-type interpolation series and de Branges spaces. (English) Zbl 1270.46022

Let \(K(x,t)\) be a complex-valued function defined on \(I\times D,\) where \(x\in I\), \(t\in D\), \(I\) is an interval and \(D\) is an open set in \(\mathbb{R}^d\) such that \(K(.\, ,t)\in L^2(I)\) for every \(t\). The Kramer sampling theorem states that, if there exists a sequence \(\left\{t_n\right\}\) such that \(\left\{K(x,t_n)\right\}\subset D\) is a complete orthogonal set of functions in \(L^2(I),\) then for any function \[ f(t)=\int_I F(x)K(x,t) dx, \quad \text{ where } F\in L^2(I), \tag{1} \] we have the sampling expansions \[ f(t) =\sum_n f(t_n) S_n(t), \tag{2} \] where \[ S_n(t)=\frac{\int_I K(x,t) \overline{K}(x,t_n)\,dx}{\int_I \left|{K}(x,t_n)\right|^2 \,dx}. \] This theorem has been generalized in a number of different ways. For example, the condition that \(\left\{K(x,t_n)\right\}\) is a complete orthogonal set of functions in \(L^2(I)\) may be replaced by a weaker condition, such as requiring that the set \(\left\{K(x,t_n)\right\}\) be a Riesz basis. Many papers have been written to show that the kernel function \(K(x,t)\) can arise from boundary value problems involving differential and integral equations and that the sampling series (2) in certain cases is a Lagrange-type interpolation series.
The paper under review provides another extension of the Kramer sampling theorem. It investigates the case where the set \(D\) is the complex plane and \(L^2(I)\) is replaced by a separable Hilbert space \({\mathcal H}\), with \(K(.,z)\) being an \({\mathcal H}\)-valued function satisfying the condition that \(K(.,z_n)=a_n x_n \), where \(z_n\in D\), \(\left\{a_n\right\}\) is a sequence of complex numbers and \(\left\{x_n\right\}\) is a Riesz basis of \({\mathcal H}\). In this case, the inner product (1) is replaced by the inner product \(f_x(z)=\langle K(z),x\rangle_{{\mathcal H}}\). The set of all such \(f_x\) is a reproducing-kernel Hilbert space of entire functions, which is denoted by \({\mathcal H}_K\).
The authors first show under what conditions the sampling theorem holds for the space \({\mathcal H}_K\) and then derive necessary and sufficient conditions for the sampling series to be a Lagrange-type interpolation series. In the last section, they focus on the relationship between the space \({\mathcal H}_K\) and de Branges spaces of entire functions and derive conditions under which the space \({\mathcal H}_K\) is a de Branges space. The results are interesting and fill a gap in the literature.

MSC:

46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
94A20 Sampling theory in information and communication theory
46E20 Hilbert spaces of continuous, differentiable or analytic functions
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