Discrete frames for [formula omitted] arising from tiling systems on [formula omitted

Abstract

A discrete frame for L2(Rd) is a countable sequence {ej}j∈J in L2(Rd) together with real constants 0<A≤B<∞ such thatA‖f‖22≤∑j∈J|〈f,ej〉|2≤B‖f‖22, for all f∈L2(Rd). We present a method of sampling continuous frames, which arise from square-integrable representations of affine-type groups, to create discrete frames for high-dimensional signals. Our method relies on partitioning the ambient space by using a suitable “tiling system”. We provide all relevant details for constructions in the case of Mn(R)⋊GLn(R), although the methods discussed here are general and could be adapted to some other settings. Finally, we prove significantly improved frame bounds over the previously known construction for the case of n=2.