Reproducing Kernel Hilbert spaces for wave optics: tutorial

An introduction to the Hilbert spaces that are endowed with a reproducing kernel is presented on using the mathematical tools of Fourier optics and coherence theory. After giving the basic definition of such spaces, some examples are worked out to show how the inner product can take different forms depending on the particular function space one works with. The basic rule to build a reproducing kernel Hilbert space (RKHS) is then presented together with the basic properties of those spaces. Eigenfunctions and eigenvalues of the reproducing kernel are then illustrated and lead to the important integral representation of the reproducing kernel. The latter is used to present pseudomodal expansions and generalized forms of sampling. The concluding section offers some thoughts on the applications of RKHSs in wave optics. An appendix presents an introduction to treatments using more advanced concepts of functional analysis.