Unconditional convergence constants of g-frame expansions

In this paper, we prove that the unconditional constants of the g-frame expansion in a Hilbert space are bounded by [Formula: see text], where [Formula: see text], [Formula: see text] are the frame bounds of the g-frames. It follows that tight g-frames have unconditional constant one. Then we generalize this to a classification of such g-frames by showing that a g-Bessel sequence has unconditional constant one if it is an orthogonal sum of g-tight frames. We also obtain a new result under which a g-Bessel sequence is a g-frame from the view of unconditional constant. Finally, we prove similar results for cross g-frame expansions as long as the cross g-frame expansions stay uniformly bounded away from zero.

Sums of A Pair of Orthogonal Frames

Frames are more stable as compared to bases under the action of a bounded linear operator. Sums of different frames under the action of a bounded linear operator are studied with the help of analysis, synthesis and frame operators. A simple construction of frames from the existing ones under the action of such an operator is presented here. It is shown that a frame can be added to its alternate dual frames, yielding a new frame. It is also shown that the sum of a pair of orthogonal frames is a frame. This provides an easy construction of a frame where the frame bounds can be computed easily. Moreover, for a pair of orthogonal frames, the necessary and sufficient condition is presented for their alternate dual frames to be orthogonal. This allows for an easy construction of a large number of new frames.

The Sums of a Pair of Orthogonal Frames

A sum of different frames under the action of a bounded linear operator is studied with the help of analysis, synthesis and frame operators. In particular, it is shown that the sum of a pair of orthogonal frames is a frame. This provides an easy construction of a frame where the frame bounds can be computed easily. For a pair of orthogonal frames, the necessary and sufficient condition is presented for their alternate duals to be orthogonal.

Filter banks on discrete abelian groups

In this work, we provide polyphase, modulation, and frame theoretical analyses of a filter bank on a discrete abelian group. Thus, multi-dimensional or cyclic filter banks as well as filter banks for signals in [Formula: see text] or [Formula: see text] spaces are studied in a unified way. We obtain perfect reconstruction conditions and the corresponding frame bounds in this particular context.

Generalized Gramians: Creating frame vectors in maximal subspaces

A frame is a system of vectors [Formula: see text] in Hilbert space [Formula: see text] with properties which allow one to write algorithms for the two operations, analysis and synthesis, relative to [Formula: see text], for all vectors in [Formula: see text]; expressed in norm-convergent series. Traditionally, frame properties are expressed in terms of an [Formula: see text]-Gramian, [Formula: see text] (an infinite matrix with entries equal to the inner product of pairs of vectors in [Formula: see text]); but still with strong restrictions on the given system of vectors in [Formula: see text], in order to guarantee frame-bounds. In this paper, we remove these restrictions on [Formula: see text], and we obtain instead direct-integral analysis/synthesis formulas. Applications are given to reproducing kernel Hilbert spaces, and to random fields.

VECTOR-VALUED WEYL–HEISENBERG WAVELET FRAME

This paper deals with the theory of frames for vector-valued Weyl–Heisenberg wavelets (VVWHW). We derive frame and the corresponding frame bounds for VVWHW.