WEAVING FRAMES

We study an intriguing question in frame theory we call Weaving Frames that is partially motivated by preprocessing of Gabor frames. Two frames {phi(i)}(i is an element of I) and {psi(i)}(i is an element of I) for a Hilbert space H are woven if there are constants 0 < A <= B so that for every subset sigma subset of I, the family {phi(i)}(i is an element of sigma)U{psi(i)}(i is an element of sigma c) is a frame for H with frame bounds A, B. Fundamental properties of woven frames are developed and key differences between weaving Riesz bases and weaving frames are considered. In particular, it is shown that a Riesz basis cannot be woven with a redundant frame. We also introduce an apparently weaker form of weaving but show that it is equivalent to weaving. Weaving frames has potential applications in wireless sensor networks that require distributed processing under different frames, as well as preprocessing of signals using Gabor frames.