A Generalized Sampling Theorem for Stable Reconstructions in Arbitrary Bases
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We introduce a generalized framework for sampling and reconstruction in separable Hilbert spaces. Specifically, we establish that it is always possible to stably reconstruct a vector in an arbitrary Riesz basis from sufficiently many of its samples in any other Riesz basis. This framework can be viewed as an extension of the well-known consistent reconstruction technique (Eldar et al.). However, whilst the latter imposes stringent assumptions on the reconstruction basis, and may in practice be unstable, our framework allows for recovery in any (Riesz) basis in a manner that is completely stable.
Whilst the classical Shannon Sampling Theorem is a special case of our theorem, this framework allows us to exploit additional information about the approximated vector (or, in this case, function), for example sparsity or regularity, to design a reconstruction basis that is better suited. Examples are presented illustrating this procedure.
KeywordsSampling theory Stable reconstruction Shannon sampling theorem Infinite matrices Hilbert space Wavelets
Mathematics Subject Classification94A20 65T99 47A99 42C40 42A10
The authors would like to thank Emmanuel Candès and Hans G. Feichtinger for valuable discussions and input.
- 1.Adcock, B., Hansen, A.C.: Generalized sampling and infinite dimensional compressed sensing. Technical report NA2011/02, DAMTP, University of Cambridge (submitted) Google Scholar
- 2.Adcock, B., Hansen, A.C.: Generalized sampling and the stable and accurate reconstruction of piecewise analytic functions from their Fourier coefficients. Technical report NA2011/12, DAMTP, University of Cambridge (submitted) Google Scholar
- 3.Adcock, B., Hansen, A.C.: Sharp bounds, optimality and a geometric interpretation for generalised sampling in Hilbert spaces. Technical report NA2011/10, DAMTP, University of Cambridge (submitted) Google Scholar
- 4.Adcock, B., Hansen, A.C.: Stable reconstructions in Hilbert spaces and the resolution of the Gibbs phenomenon. Appl. Comput. Harmon. Anal. (to appear) Google Scholar
- 5.Adcock, B., Hansen, A.C., Herrholz, E., Teschke, G.: Generalized sampling: extension to frames and ill-posed problems. Technical report NA2011/17, DAMTP, University of Cambridge (submitted) Google Scholar
- 6.Adcock, B., Hansen, A.C., Herrholz, E., Teschke, G.: Generalized sampling, infinite-dimensional compressed sensing, and semi-random sampling for asymptotically incoherent dictionaries. Technical report NA2011/13, DAMTP, University of Cambridge (submitted) Google Scholar
- 9.Böttcher, A.: Infinite matrices and projection methods. In: Lectures on Operator Theory and Its Applications, Waterloo, ON, 1994. Fields Inst. Monogr., vol. 3, pp. 1–72. Amer. Math. Soc., Providence (1996) Google Scholar
- 29.Nyquist, H.: Certain topics in telegraph transmission theory. Trans. AIEE 47, 617–644 (1928) Google Scholar
- 34.Whittaker, E.T.: On the functions which are represented by the expansions of the interpolation theory. Proc. R. Soc. Edinb. 35, 181–194 (1915) Google Scholar