Explicit construction of frames and pairs of dual frames on locally compact abelian groups

Abstract

Locally compact abelian (LCA) groups form a natural setting for frame theory in signal processing: while classical frame theory was developed in the setting of \( L^2({\mathbb {R}}),\) concrete applications have to be implemented on \({\mathbb {R}}^d,\) two scenarios that are covered under the setup of LCA groups. Due to rapid progress over the past few years, it is known within the mathematical community that large parts of the standard frame theory for \( L^2({\mathbb {R}})\) can be carried over to the setting of LCA groups. We show that this also holds on the level of explicit constructions, which is a crucial issue for applications within signal processing; in fact we provide explicit constructions of frames and pairs of dual frames on LCA groups. Special attention will be given to the so-called elementary LCA groups, i.e., groups that are tensor products of \({\mathbb {R}},\) \({\mathbb {Z}},\) the torus group \({{\mathbb {T}}},\) and the finite group \({\mathbb {Z}}_N.\) The frames will have the form of a Fourier-like system, i.e., the Fourier transform of a generalized shift-invariant system. The constructions are based on a partition of unity; in particular we provide a general and flexible approach for constructing a partition of unity that makes it easy to control key features of the underlying functions, e.g., support size and continuity. The results apply to arbitrary LCA groups with a supply of “sufficiently fine” lattices, a condition that is satisfied for all elementary LCA groups. Concrete constructions of pairs of dual frames for several elementary LCA groups are provided as illustration. In the case of a system of functions generated by a single lattice, such as a Gabor system, the general results become particularly transparent.