A reproducing kernel Hilbert space \(H\) on a domain \(E\) is a Hilbert space of complex-valued functions that satisfy a reproducing formula \[ f(p)=\left\langle f, K_p\right\rangle_{H} \] for \(p\in E\) and \(f\in H\). Here, \(K_p\) is the reproducing kernel for the space \(H\) and can be viewed as a map \(K:E\times E\to \mathbb{C}\). Equivalently, by the Riesz representation theorem, these are function spaces so that point evaluations of the function are continuous linear functionals. There are some very simple examples of reproducing kernel Hilbert spaces that almost all mathematicians have been exposed to (e.g., Bergman spaces). Reproducing kernels play a fundamental role in analytic function spaces and operator theory. They have also found applications in statistical learning theory, boundary value problems for harmonic differential equations and partial differential equations.
Here is a summary of the contents of the book. Chapter 1 gives the basic definitions and numerous examples of reproducing kernel Hilbert spaces (such as Paley-Wiener, the Bergman kernel, and Sobolev spaces). Chapter 2 then provides fundamental properties of reproducing kernel Hilbert spaces (basic properties of reproducing kernels, properties for algebraic sums of kernels, and connections with linear mappings). Chapter 3 deals with the connection between reproducing kernels and Tikhonov regularization. Chapter 4 uses the machinery developed previously to study the real inversion formula for the Laplace transform. Chapters 5, 6, and 7 are devoted to applications in ordinary differential equations, partial differential equations and integral equations. Chapter 8 studies special topics related to reproducing kernels (such as norm inequalities, inversion of matrices, sampling theory, membership problems for reproducing kernel Hilbert spaces, generalized reproducing kernels and generalized delta functions).