Linear combinations of generators in multiplicatively invariant spaces

Multiplicatively invariant (MI) spaces are closed subspaces of $L^2(\Omega,\mathcal{H})$ that are invariant under multiplications of (some) functions in $L^{\infty}(\Omega)$. In this paper we work with MI spaces that are finitely generated. We prove that almost every linear combination of the generators of a finitely generated MI space produces a new set on generators for the same space and we give necessary and sufficient conditions on the linear combinations to preserve frame properties. We then apply what we prove for MI spaces to system of translates in the context of locally compact abelian groups and we obtain results that extend those previously proven for systems of integer translates in $L^2(\mathbb{R}^d)$.Comment: 13 pages. Minor changes have been made. To appear in Studia Mathematic

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

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 $\Gamma$ of $G$ acting as a group of unitary operators on $L^2(G)$. The dilation operation in the wavelet setting is replaced by integer powers of a unitary operator $\delta$ onto $L^2(G)$. We show that, under some compatibility conditions between $\delta$ and the action of the group $\Gamma$, the linear independence of the translates of any function in $L^2(G)$ by elements of $\Gamma$ implies the linear independence of affine Parseval frames in $L^2(G)$.Comment: A new revised and correct versio