Aliasing and oblique dual pair designs for consistent sampling

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.Comment: 23 page

Frames, semi-frames, and Hilbert scales

Given a total sequence in a Hilbert space, we speak of an upper (resp. lower) semi-frame if only the upper (resp. lower) frame bound is valid. Equivalently, for an upper semi-frame, the frame operator is bounded, but has an unbounded inverse, whereas a lower semi-frame has an unbounded frame operator, with bounded inverse. For upper semi-frames, in the discrete and the continuous case, we build two natural Hilbert scales which may yield a novel characterization of certain function spaces of interest in signal processing. We present some examples and, in addition, some results concerning the duality between lower and upper semi-frames, as well as some generalizations, including fusion semi-frames and Banach semi-frames.Comment: 27 pages; Numerical Functional Analysis and Optimization, 33 (2012) in press. arXiv admin note: substantial text overlap with arXiv:1101.285

Compare and contrast between duals of fusion and discrete frames

Fusion frames are valuable generalizations of discrete frames. Most concepts of fusion frames are shared by discrete frames. However, the dual setting is so complicated. In particular, unlike discrete frames, two fusion frames are not dual of each other in general. In this paper, we investigate the structure of the duals of fusion frames and discuss the relation between the duals of fusion frames with their associated discrete frames.Comment: 12 page

Nullspaces and frames

In this paper we give new characterizations of Riesz and conditional Riesz frames in terms of the properties of the nullspace of their synthesis operators. On the other hand, we also study the oblique dual frames whose coefficients in the reconstruction formula minimize different weighted norms.Comment: 16 page

Frames of translates with prescribed fine structure in shift invariant spaces

For a given finitely generated shift invariant (FSI) subspace \cW\subset L^2(\R^k) we obtain a simple criterion for the existence of shift generated (SG) Bessel sequences E(\cF) induced by finite sequences of vectors \cF\in \cW^n that have a prescribed fine structure i.e., such that the norms of the vectors in \cF and the spectra of S_{E(\cF)} is prescribed in each fiber of \text{Spec}(\cW)\subset \T^k. We complement this result by developing an analogue of the so-called sequences of eigensteps from finite frame theory in the context of SG Bessel sequences, that allows for a detailed description of all sequences with prescribed fine structure. Then, given $0<\alpha_1\leq \ldots\leq \alpha_n$ we characterize the finite sequences \cF\in\cW^n such that $\|f_i\|^2=\alpha_i$, for $1\leq i\leq n$, and such that the fine spectral structure of the shift generated Bessel sequences E(\cF) have minimal spread (i.e. we show the existence of optimal SG Bessel sequences with prescribed norms); in this context the spread of the spectra is measured in terms of the convex potential P^\cW_\varphi induced by \cW and an arbitrary convex function $\varphi:\R_+\rightarrow \R_+$.Comment: 31 pages. Accepted in the JFA. This revised version has several changes in the notation and the organization of the text. There exists text overlap with arXiv:1508.01739 in the preliminary section