An Introduction to the Theory of Reproducing Kernel Hilbert Spaces

This is a brief but well-constructed introduction to the construction and use of reproducing kernels for integral transforms. The coverage is brief and to the point and does not waste much space in proving required theorems from functional analysis or complex function theory. At the same time it does indeed have an “introductory” feel about it. Rather than focus solely on the kernel side of things it begins with defining a “reproducing kernel Hilbert space” (RKHS in the text) as a subset of the set of all functions from an arbitrary set \(X\) to a field \(F\). In fact, an RKHS is just a Hilbert space that contains a bounded evaluation functional. Several helpful examples (and non-examples) occur next addressing \(L_2[0,1]\), Sobolev spaces, Hardy spaces and so forth. The classic Bergman and Szego kernels make their appearance as the text moves us into the function spaces of classic complex analysis. The exposition is easy to follow and relatively straightforward throughout.

The introductory material is followed by chapters introducing basic operations on kernels — complexification and tensor products of operators as well as considerations of interpolation and approximation. The remaining 75 pages are devoted to applications of kernels and their spaces to integral operators, Fock space, negative-definite functions, machine learning and stochastic processes.

The exposition is good (although anyone learning to really learn more about machine learning should look elsewhere) and newly introduced terms are carefully “boxed-in” (a habit which is sometimes carried to excess). The sections are devoted to showing the utility of what used to be known as “the kernel trick” and showing that a deep and beautiful theory underlies it. Anyone looking for a nice introduction to this theory need look no further.

Jeff Ibbotson Smith Teaching Chair in Mathematics Phillips Exeter Academy