Linear independence of translates implies linear independence of affine Parseval frames on LCA groups

2014/11/05 by Sandra Saliani, Saliani, Sandra
Computer Science · Mathematics · #42C40 #Digital Filter Design and Implementation #FOS: Mathematics #Functional Analysis (math.FA) #Image and Signal Denoising Methods #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.1411.1252

openalex publication_date 2014/11/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Motivated by Bownik and Speegle's result on linear independence of wavelet Parseval frames, we consider affine systems (analogous to wavelet systems) defined on a second countable, locally compact abelian group G, where the translations are replaced by the action of a countable, discrete subgroup Γ of G acting as a group of unitary operators on L²(G). The dilation operation in the wavelet setting is replaced by integer powers of a unitary operator δ onto L²(G). We show that, under some compatibility conditions between δ and the action of the group Γ, the linear independence of the translates of any function in L²(G) by elements of Γ implies the linear independence of affine Parseval frames in L²(G).

Related

• The orthonormal dilation property for abstract Parseval wavelet frames
• The unitary extension principle on locally compact abelian groups
• Orthogonality and duality of frames over locally compact abelian groups
• Reproducing formulas for generalized translation invariant systems on locally compact abelian groups
• Frames generated by actions of countable discrete groups