ABSTRACT

This chapter presents multivariate Paley–Wiener space theory. Its reproducing Hilbert space kernel structure is used to develop spline interpolatory sampling. Many extensions of the Shannon-type sampling theorems can be studied in more detail. The approach opens the perspective to introduce Paley–Wiener splines and to discuss some of their significant applications, such as the multivariate antenna theory. There are diverse important spline concepts in the context of lattice point sampling, mostly based on (iterated) univariate approaches. In the line of W. Freeden, the chapter proposes a specific spline understanding resulting from the reproducing kernel structure of the Paley–Wiener space. Additional spline methods (such as spline smoothing of noisy data, best approximation of linear functionals, Sard’s Theorem, Schönberg’s Theorem, etc.) can be realized in a canonical way by use of the reproducing property of the kernel. The chapter explains a generalization of the multivariate antenna problem.