Mean Value Theorems and Concentration Operators in Bargmann and Bergman Space

Abstract

A frame of wavelets corresponds to a discrete set of points in the plane or in the upper half plane. The density of a frame, if such a number exists, will mean the density of this set with respect to the appropriate geometry (in the “ax+b” case this is the hyperbolic geometry.) Assuming no regularity condition we ask the following question: Must the density in any part of the plane exceed some lower bound, depending only on the analyzing wavelet (the “dual” question is for interpolation, then we ask for the existence of an upper bound)? Assuming regularity, we know that $\frac{1}{{2\pi }}$ is a critical density in the Weyl-Heisenberg case [2, pp. 37–47]. In the “ax+b” case it is known that such a density corresponding to the Nyqui.st rate does not exist [2, pp. 69–71]. The following discussion could be applied to yield sane more information about this topic.