Numerische Mathematik

, Volume 69, Issue 4, pp 423-440

Efficient numerical methods in non-uniform sampling theory

  • Hans G. FeichtingerAffiliated withDepartment of Mathematics, University of Vienna, Strudlhofgasse 4, A-1090 Wien, Austria E-mail: FEI{\tt @}TYCHE.MAT.UNIVIE.AC.AT, STROHMER{\tt @}TYCHE.MAT.UNIVIE.AC.AT
  • , Karlheinz Gr\"ochenigAffiliated withDepartment of Mathematics, The University of Connecticut, Storrs, CT. 06269-3009, USA E-mail: GROCH{\tt @}MATH.UCONN.EDU
  • , Thomas StrohmerAffiliated withDepartment of Mathematics, University of Vienna, Strudlhofgasse 4, A-1090 Wien, Austria E-mail: FEI{\tt @}TYCHE.MAT.UNIVIE.AC.AT, STROHMER{\tt @}TYCHE.MAT.UNIVIE.AC.AT

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Summary.

We present a new ``second generation" reconstruction algorithm for irregular sampling, i\@.e\@. for the problem of recovering a band-limited function from its non-uniformly sampled values. The efficient new method is a combination of the adaptive weights method which was developed by the two first named authors and the method of conjugate gradients for the solution of positive definite linear systems. The choice of "adaptive weights" can be seen as a simple but very efficient method of preconditioning. Further substantial acceleration is achieved by utilizing the Toeplitz-type structure of the system matrix. This new algorithm can handle problems of much larger dimension and condition number than have been accessible so far. Furthermore, if some gaps between samples are large, then the algorithm can still be used as a very efficient extrapolation method across the gaps.

Mathematics Subject Classification (1991): 42A15, 65D05, 65D10, 65F10