Translation generated oblique dual frames by actions of locally compact groups

Abstract

KEYWORDS: Biorthogonal dual; Dual frame; Dual integrable representation; Fiberization; Frame; Gramian; Heisenberg group; Infimum cosine angle; Locally compact group; Multiplication invariant space; Nilpotent Lie group; Oblique dual; Orthogonal frame; Reproducing formula; Riesz Basis; Shift-invariant space; Translation-invariant space; Unitary Representation; Zak transform For a second countable locally compact group ", let ⇢ be a unitary representation of " acting on a separable Hilbert space H. Also for a collection of functions t't : t P Nu in H, where N is a #-finite measure space, considering the continuous frame of orbit: t⇢p$q't : $ P ", t P Nu, we discuss the various dual frames of the same form, i.e., t⇢p$q t : $ P ", t P Nu for some t P H. We provide various necessary and sufficient conditions for the characterizations of dual frames. In particular, we concentrate on the context of translation generated systems in H “ L2pG q, where translations are from closed subgroup " of the locally compact group G . Our characterization results are based on the Zak transform. When G becomes locally compact abelian (denoted by G), we discuss the same using the fiberization map. At the end, we discuss our characterizations for the SI{Z nilpotent Lie group G (denoted by G), which is considered to be a high degree of non-abelian structure.