An introduction to the theory of reproducing kernel Hilbert spaces (Q2798209)

The purpose of this fine monograph is two-fold. On the one hand, the authors introduce a wide audience to the basic theory of reproducing kernel Hilbert spaces (RKHS), on the other hand they present applications of this theory in a variety of areas of mathematics.NEWLINENEWLINEAccordingly, the material is split into two parts. Part~I (``General theory'') develops the basic theory of RKHS along with many examples, for which only elementary Hilbert space theory, including the closed graph theorem, is assumed as a prerequisite. Some examples, though, require familiarity with analytic functions or measure theory. This part discusses, among many other things, characterisations of reproducing kernels, interpolation and approximation, operations on kernels and vector-valued RKHS. In addition, there is a section on the Cholesky factorisation and the Schur product that comes in handy when discussing some examples like the Drury-Arveson space.NEWLINENEWLINEPart~II (``Applications and examples'') starts with a brief section on the pull-back construction and continues with a very well-crafted section on statistical learning theory. Further sections deal with Schoenberg's theory of negative definite functions and positive definite functions in relation with group representations. Another major section presents applications to integral operators with the goal of proving Mercer's theorem. It should be mentioned that the theory of compact operators is not taken as a prerequisite at this stage, but it is developed here between the lines for the operators under consideration. The last section displays some applications in the theory of stochastic processes, e.g., the Karhunen-Loève decomposition.NEWLINENEWLINEReviewer's remark: The monograph is written in a very clear and precise style, although I always frown upon formulations like ``an orthonormal basis \(\{e_n(x): n\in \mathbb{N}\}\)'' (when it is actually \(\{e_n: n\in \mathbb{N}\}\) that is the orthonormal basis) which can be met at times. I also came across two oversights: The RKHS for the min-kernel consists of functions that are absolutely continuous on each bounded interval rather than on all of \([0,\infty)\), and the definition of an iid sequence pretends that pairwise independent random variables are independent. These minor glitches aside, the authors have succeeded in arranging a very readable modern presentation of RKHS and in conveying the relevance of this beautiful theory by many examples and applications.