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P. Berger, K. Gröchenig, G. Matz:
"Sampling and Reconstruction in Distinct Subspaces Using Oblique Projections";
Journal of Fourier Analysis and Applications, 1 (2018), 1 - 33.

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 operator norm, and therefore is most stable with respect to noisy measurements. We then construct the operator with the smallest possible quasi-optimality constant, which yields the most stable reconstruction with respect to a systematic error appearing before the sampling process (model uncertainty). We describe how to vary continuously between the two reconstruction methods, so that we can trade stability for quasi-optimality. As an application we study the reconstruction of a compactly supported function from nonuniform samples of its Fourier transform.

Sampling theory; Stable reconstruction; Frame theory; Oblique projections; Weighted least squares

https://publik.tuwien.ac.at/files/publik_276652.pdf