Fundamental properties of RKHS

Abstract

In Chap. 2, which is at the heart of this book, we develop the general theory of reproducing kernels and its fundamental properties. First, we establish that any positive definite kernel corresponds uniquely to an RKHS. Then we discuss operations for RKHS. These operations will be fundamental for applications. After clarifying the basic properties, we will present the methods to construct RKHSs. One of the ideas is to use complete orthonormal systems or the Fourier integral. Finally, we discuss the relation between RKHS and linear mappings. One of the powerful properties that RKHSs enjoy is that we can characterize completely the image of linear mappings. But the proof of the results presented earlier than Sect. 2.4 is independent from Sect. 2.4. We list [209, 347, 403] for the introduction of theory of reproducing kernel Hilbert spaces.