Sampling and Reconstruction in Distinct Subspaces Using Oblique Projections
For researchers in signal processing and approximation theory, this provides a theoretical framework for balancing stability and robustness in reconstruction, though the results are incremental extensions of known oblique projection concepts.
The paper studies reconstruction operators in Hilbert spaces that are exact on a given subspace, deriving operators with minimal norm (least squares) and minimal quasi-optimality constant for model uncertainty, and shows how to continuously trade off between stability and quasi-optimality. An application to reconstructing compactly supported functions from nonuniform Fourier samples is provided.
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 is the most stable 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.