Approximately dual frames are studied in the Hilbert space setting. Approximate duals are easier to construct than classical dual frames, and can be tailored to yield almost perfect reconstruction. Bounds on the deviation from perfect reconstruction are obtained for approximately dual frames constructed via perturbation theory. An alternative bound is derived for the rich class of Gabor frames,...

Abstract. Frames in Hilbert spaces generalize orthonormal bases and allow stable representation of all the elements of the space via a given frame and its dual frame. Frames are not only interesting from theoretical point of view, but play significant role in signal and image processing, which leads to many applications in informatics, engineering, medicine, and many other fields. In this paper...

An interesting method called Oblique Extension Principle (OEP) has been proposed in the literature for constructing compactly supported MRA tight and dual wavelet frames with high vanishing moments and high frame approximation orders. Many compactly supported MRA wavelet frames have been recently constructed from scalar refinable functions via OEP. Despite the great flexibility and popularity o...

We provide explicit constructions of particularly convenient dual pairs of Gabor frames. We prove that arbitrary polynomials restricted to sufficiently large intervals will generate Gabor frames, at least for small modulation parameters. Unfortunately, no similar function can generate a dual Gabor frame, but we prove that almost any such frame has a dual generated by a B-spline. Finally, for fr...

Weighted and controlled frames have been introduced recently to improve the numerical efficiency of iterative algorithms for inverting the frame operator. In this paper we develop systematically these notions, including their mutual relationship. We will show that controlled frames are equivalent to standard frames and so this concept gives a generalized way to check the frame condition, while ...

We use several fundamental results which characterize frames for a Hilbert space to give natural generalizations of Hilbert space frames to general Banach spaces. However, we will see that all of these natural generalizations (as well as the currently used generalizations) are equivalent to properties already extensively developed in Banach space theory. We show that the dilation characterizati...

We incorporate the new theory of equivariant moving frames for Lie pseudo-groups into Vessiot’s method of group foliation of differential equations. The automorphic system is replaced by a set of reconstruction equations on the pseudo-group jets. The result is a completely algorithmic and symbolic procedure for finding both invariant and non-invariant solutions of differential equations admitti...