Oblique dual frames in Hilbert spaces

In this paper, we consider oblique dual frames and particularly, we obtain some of their characterizations. Among other things, special attention is devoted to the study of the effect of perturbations of frame sequences on their oblique duals. In particular, as a surprising result, we show that a frame sequence is uniquely determined by the set of its oblique dual.

Extension of Bessel sequences to oblique dual frame sequences and the minimal projection

An extension of two Bessel sequences to oblique dual frame sequences and its applications to shift-invariant spaces are considered. The best-known situation where this kind of extension is necessary is the construction of a pair of biorthogonal multiresolution analyses, where two generating sets whose shifts are only assumed to be Bessel sequences are given. This extension naturally leads to consideration of the ‘minimal projection’ extending two closed subspaces. The existence or non-existence of the minimal projection is discussed.

OBLIQUE DUAL FRAMES IN FINITE-DIMENSIONAL HILBERT SPACES

A frame can be completed to a tight frame by adding some additional vectors. However, for the purpose of computational efficiency, we need to put restrictions to the number of added vectors. In this paper we propose a constructive method that allows to extend a given frame to an oblique dual frame pair such that the number of added vectors is in general much smaller than the number of added vectors used to extend it to a tight frame. We also present a uniqueness characterization and several equivalent characterizations for an oblique dual frame pair, it turns out that oblique dual frame pair provides more flexibilities than alternate dual frame pair.

GENERAL FRAMEWORK FOR CONSISTENT SAMPLING IN HILBERT SPACES

We introduce a general framework for consistent linear reconstruction in infinite-dimensional Hilbert spaces. We study stable reconstructions in terms of Riesz bases and frames, and generalize the notion of oblique dual frames to infinite-dimensional frames. As we show, the linear reconstruction scheme coincides with the so-called oblique projection, which turns into an ordinary orthogonal projection when adapting the inner product. The inner product of interest is, in general, not unique. We characterize the inner products and corresponding positive operators for which the new geometrical interpretation applies.

GENERAL FRAMEWORK FOR CONSISTENT SAMPLING IN HILBERT SPACES

We introduce a general framework for consistent linear reconstruction in infinite-dimensional Hilbert spaces. We study stable reconstructions in terms of Riesz bases and frames, and generalize the notion of oblique dual frames to infinte-dimensional frames. As we show, the linear reconstruction scheme coincides with the so-called oblique projection, which turns into an ordinary orthogonal projection when adapting the inner product. The inner product of interest is, in general, not unique. We characterize the inner products and the corresponding positive operators for which this geometrical interpretation applies.