Abstract
Let ℤ2N={0, ..., 2N-1} denote the group of integers modulo 2N, and let L be the space of all real functions of ℤ2N which are supported on {0,...N−1}. The spectral phase of a function f:ℤ2N→ℝ is given by φf(k)=arg\(\hat f(k)\) for k ∈ ℤ2N, where\(\hat f\) denotes the discrete Fourier transforms of f.
For a fixed s∈L let Ks denote the cone of all f:ℤ2N→ℝ which satisfy φf ≡ φs and let Ms be its linear span. The angle αs between Ms and L determines the convergence rate of the signal restoration from phase algorithm of Levi and Stark [3]. Here we prove the following conjectures of Urieli et al. [7] who verified them for the N≤3 case:
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1.
α (Ms, L)≤π/4 for a generic s∈L.
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2.
If s∈L is geometric, i.e., s(j)=qj for 0≤j≤N−1 where ±1≠q∈ℝ, then α(Ms, L)=π/4.
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Communicated by Todd Quinto
Acknowledgments and Notes. Nir Cohen-Supported by CNPq grant 300019/96-3. Roy Meshulam-Research supported by the Fund for the Promotion of Research at the Technion.
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Cohen, N., Meshulam, R. A geometrical problem arising in a signal restoration algorithm. The Journal of Fourier Analysis and Applications 4, 643–650 (1998). https://doi.org/10.1007/BF02479671
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DOI: https://doi.org/10.1007/BF02479671