# Beyond Consistent Reconstructions: Optimality and Sharp Bounds for Generalized Sampling, and Application to the Uniform Resampling Problem

@article{Adcock2013BeyondCR, title={Beyond Consistent Reconstructions: Optimality and Sharp Bounds for Generalized Sampling, and Application to the Uniform Resampling Problem}, author={Ben Adcock and Anders C. Hansen and Clarice Poon}, journal={ArXiv}, year={2013}, volume={abs/1301.2831} }

Generalized sampling is a recently developed linear framework for sampling and reconstruction in separable Hilbert spaces. It allows one to recover any element in any finite-dimensional subspace given finitely many of its samples with respect to an arbitrary basis or frame. Unlike more common approaches for this problem, such as the consistent reconstruction technique of Eldar and others, it leads to numerical methods possessing both guaranteed stability and accuracy. The purpose of this paper…

## 74 Citations

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With a system of polynomials that the user is essentially free to choose, one can restore exponential accuracy in n and root-exponential accuracy in m and generalizes a result proved recently for piecewise Legendre polynmials.

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On optimal wavelet reconstructions from Fourier samples: linearity and universality of the stable sampling rate

- Mathematics, Computer ScienceArXiv
- 2012

Generalized sampling provides a nearly-optimal solution to the problem of computing wavelet coefficients of compactly supported functions from their Fourier samples, and under some mild assumptions it is shown that generalized sampling cannot be outperformed in terms of approximation quality by more than a constant factor.

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We study reconstruction operators on a Hilbert space that are exact on a given reconstruction subspace. Among those the reconstruction operator obtained by the least squares fit has the smallest…

Constrained Sampling: Optimum Reconstruction in Subspace With Minimax Regret Constraint

- Engineering, Computer ScienceIEEE Transactions on Signal Processing
- 2019

Constrained GSRP is proposed, a novel framework that minimizes the reconstruction error for inputs in a subspace, subject to a constraint on the maximum regret-error for any other signal in the entire signal space.

Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples

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- 2014

In this paper, we consider the problem of recovering a compactly supported multivariate function from a collection of pointwise samples of its Fourier transform taken nonuniformly. We do this by…

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