Reproducing kernel Hilbert spaces via sampling in discrete spaces

This paper explores the reproducing property of some Hilbert spaces of functions from set X to some linear space E and discusses the essential ideas behind such space along with their applications in sampling theory and machine learning. We extend the idea of E-valued reproducing kernel Hilbert spaces to scalar case as considering X×E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\times E$$\end{document} instead of the set X, where E is a linear space. Our investigation focuses on vector-valued reproducing kernel function spaces without the condition of Hilbert spaces on E. We concentrate on two concepts: E-valued reproducing kernels and relative E-valued reproducing kernels, and in particular we characterize reproducing kernel Hilbert spaces by multipliers and then focus on finite graphs and find Laplace operator.