Abstract. We study an intriguing question in frame theory we call Weaving Frames that is partially motivated by preprocessing of Gabor frames. Two frames {ϕ i } i∈I and {ψ i } i∈I for a Hilbert space H are woven if there are constants 0 < A B so that for every subset σ ⊂ I , the family {ϕ i } i∈σ ∪ {ψ i } i∈σ 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.Mathematics subject classification (2010): 42C15.

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The most fundamental notion in frame theory is the frame expansion of a vector. Although it is well known that these expansions are unconditionally convergent series, no characterizations of the unconditional constant were known. This has made it impossible to get accurate quantitative estimates for problems which require using subsequences of a frame. We will prove some new results in frame theory by showing that the unconditional constants of the frame expansion of a vector in a Hilbert space are bounded by B A , where A, B are the frame bounds of the frame. Tight frames thus have unconditional constant one, which we then generalize by showing that Bessel sequences have frame expansions with unconditional constant one if and only if the sequence is an orthogonal sum of tight frames. We give further results concerning frame expansions, in which we examine when B A is actually attained or not. We end by discussing the connections of this work to frame multipliers. These results hold in both real and complex Hilbert spaces.

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