Aliasing and oblique dual pair designs for consistent sampling

Abstract

In this paper we study some aspects of oblique duality between finite sequences of vectors $\cF$ and $\cG$ lying in finite dimensional subspaces $\cW$ and $\cV$, respectively. We compute the possible eigenvalue lists of the frame operators of oblique duals to $\cF$ lying in $\cV$; we then compute the spectral and geometrical structure of minimizers of convex potentials among oblique duals for $\cF$ under some restrictions. We obtain a complete quantitative analysis of the impact that the relative geometry between the subspaces $\cV$ and $\cW$ has in oblique duality. We apply this analysis to compute those rigid rotations $U$ for $\cW$ such that the canonical oblique dual of $U\cdot \cF$ minimize every convex potential; we also introduce a notion of aliasing for oblique dual pairs and compute those rigid rotations $U$ for $\cW$ such that the canonical oblique dual pair associated to $U\cdot \cF$ minimize the aliasing. We point out that these two last problems are intrinsic to the theory of oblique duality.