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- W2018778093 abstract "Consider the problem of recovering a measure $mu $ supported on a lattice of span $Delta $, when measurements are only available concerning the Fourier Transform $hat mu (omega )$ at frequencies $|omega | leqslant Omega $. If $Omega $ is much smaller than the Nyquist frequency ${pi / Delta }$ and the measurements are noisy, then, in general, stable recovery of $mu $ is impossible. In this paper it is shown that if, in addition, we know that the measure $mu $ satisfies certain sparsity constraints, then stable recovery is possible. Say that a set has Rayleigh index less than or equal to R if in any interval of length ${{4pi } / Omega } cdot R$ there are at most R elements. Indeed, if the (unknown) support of $mu $ is known, a priori, to have Rayleigh index at most R, then stable recovery is possible with a stability coefficient that grows at most like $Delta ^{ - 2R - 1} $ as $Delta to 0$. This result validates certain practical efforts, in spectroscopy, seismic prospecting, and astronomy, to provide superresolution by imposing support limitations in reconstruction. The results amount to inequalities for interpolation of entire functions of exponential type from values at special point sets which are irregular, yet internally balanced, uniformly discrete, and of uniform density 1." @default.
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- W2018778093 date "1992-09-01" @default.
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- W2018778093 title "Superresolution via Sparsity Constraints" @default.
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- W2018778093 doi "https://doi.org/10.1137/0523074" @default.
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