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We obtain mproved bounds for one bit sensing. For instance, let K s denote the set of s-sparse unit vectors in the sphere S n in dimension n + 1 with sparsity parameter 0 < s < n+1 and assume that 0 < δ < 1. We show that for m δ −2 s log n s , the one-bit map x → sgnx, g j m j=1 , where g j are iid gaussian vectors on R n+1 , with high probability has δ-RIP… (More)

- Dmitriy Bilyk, Michael T Lacey, Ioannis Parissis, Armen Vagharshakyan
- 2010

Let R denote dyadic rectangles in the unit cube [0, 1] 3 in three dimensions. Let h R be the L ∞-normalized Haar function whose support is R. We show that for all integers n ≥ 1 and choices of coefficients a R ∈ {±1}, we have ∑ |R|=2 −n |R 1 |≥2 −n/2 a R h R L ∞ n 9/8. The trivial L 2 lower bound is n, and the sharp lower bound would be n 3/2. This is the… (More)

- Michael T Lacey
- 2007

Let Sω f = ω f (ξ)e ixξ dξ be the Fourier projection operator to an interval ω in the real line. Rubio de Francia's Littlewood–Paley inequality (Rubio de Francia, 1985) states that for any collection of dis-joint intervals Ω, we have ω∈Ω |Sω f | 2 1/2 p f p, 2 ≤ p < ∞. We survey developments related to this inequality, including the higher dimensional case,… (More)

In this paper we establish a surprising new identity for Parseval frames in a Hilbert space. Several variations of this result are given, including an extension to general frames. Finally, we discuss the derived results.

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