A primer on reproducing kernel Hilbert spaces.

*(English)*Zbl 1377.46015As the title indicates, this work can be considered a first approach to reproducing kernel Hilbert spaces (RKHSs). It is not assumed that the reader be familiar with Hilbert spaces. Emphasis is put on the motivations associated with the possible use of RKHSs. The authors aim to empower their readers to recognize when and how RKHSs theory can be useful in their own work. After an introduction, the work continues with a second chapter on finite-dimensional RKHSs. This is the way that the authors choose to start presenting some of the basic properties of RKHSs. The third chapter is concerned with function spaces. Here, already infinite-dimensional spaces are considered, and topological aspects of inner product spaces are analysed. This gives rise to infinite-dimensional RKHSs, in the fourth chapter, by considering their basic definitions, properties and examples. Chapter five is very short, and considers the process of embedding points into a RKHS, together with an exemplification involving the Gaussian kernel. Chapter six is concerned with applications of RKHSs to linear equations and optimization. Thus, it is described the essence of interpolation problems and the possibilities of solving linear equations through generalized inverses and RKHSs. Chapter seven considers applications to stochastic processes. Chapter eight explores the embedding of realizations of random variables into RKHSs, and chapter nine illustrates the use of embeddings in RKHSs, within signal processing applications, putting also some emphasis on the process of filtering in RKHSs.

Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)

##### MSC:

46E22 | Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) |

47B32 | Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |