Summary: Distance Functions for Reproducing Kernel Hilbert Spaces
N. Arcozzi, R. Rochberg, E. Sawyer, and B. D. Wick
Abstract. Suppose H is a space of functions on X. If H is a Hilbert space
with reproducing kernel then that structure of H can be used to build distance
functions on X. We describe some of those and their interpretations and
interrelations. We also present some computational properties and examples.
1. Introduction and Summary
If H is a Hilbert space with reproducing kernel then there is an associated set,
X, and the elements of H are realized as functions on X. The space H can then
be used to define distance functions on X. We will present several of these and
discuss their interpretations, interrelations and properties. We find it particularly
interesting that these ideas interface with so many other areas of mathematics.
Some of our computations and comments are new but many of the details
presented here are known, although perhaps not as well known as they might be.
One of our goals in this note is to bring these details together and place them in
unified larger picture. The choices of specific topics however reflects the recent
interests of the authors and some relevant topics get little or no mention.
The model cases for what we discuss are the hyperbolic and pseudohyperbolic
distance functions on the unit disk D. We recall that material in the next section. In
the section after that we introduce definitions, notation, and some basic properties