Discrete frames for L2(Rn2) arising from tiling systems on GLn(R)

A discrete frame for L-2(R-d) is a countable sequence {e(j)}(j is an element of J) in L-2(R-d) together with real constants 0 < A <= B < infinity such that A parallel to f parallel to(2)(2) <= Sigma(j is an element of J)vertical bar < f, e(j)>vertical bar(2) <= B parallel to f parallel to(2)(2), for all f is an element of L-2(R-d). 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 M-n(R) proportional to GL(n)(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. (C) 2021 Elsevier Inc. All rights reserved.