Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Irregular sampling, Toeplitz matrices,
and the approximation of entire functions
of exponential type

Author: Karlheinz Gröchenig
Journal: Math. Comp. 68 (1999), 749-765
MSC (1991): Primary 30E05, 30E10, 42A10, 94A12
MathSciNet review: 1613711
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In many applications one seeks to recover an entire function of exponential type from its non-uniformly spaced samples. Whereas the mathematical theory usually addresses the question of when such a function in $L^2(\mathbb{R})$ can be recovered, numerical methods operate with a finite-dimensional model. The numerical reconstruction or approximation of the original function amounts to the solution of a large linear system. We show that the solutions of a particularly efficient discrete model in which the data are fit by trigonometric polynomials converge to the solution of the original infinite-dimensional reconstruction problem. This legitimatizes the numerical computations and explains why the algorithms employed produce reasonable results. The main mathematical result is a new type of approximation theorem for entire functions of exponential type from a finite number of values. From another point of view our approach provides a new method for proving sampling theorems.

References [Enhancements On Off] (What's this?)

  • 1. J. J. Benedetto. Frame decompositions, sampling, and uncertainty principle inequalities. In ``Wavelets: Mathematics and Applications'', J. Benedetto, M. Frazier, eds., pp. 247-304, CRC Press, 1993. MR 94i:94005
  • 2. A. Beurling. Collected Works. Vol. 2, Harmonic Analysis, L. Carleson, ed., Birkhäuser, Boston, 1989, pp. 341-365. MR 92k:01046b
  • 3. A. Beurling, P. Malliavin. On the closure of characters and the zeros of entire functions. Acta Math. 118 (1967), 79-95. MR 35:654
  • 4. P. L. Butzer, W. Splettstößer, R. Stens. The sampling theorem and linear prediction in signal analysis. Jber.d. Dt. Math.-Verein. 90 (1987), 1-70. MR 89b:94006
  • 5. R. Duffin, A. Schaeffer. A class of nonharmonic Fourier series. Trans. Amer. Math. Soc. 72(1952), 341-366. MR 13:839a
  • 6. H. G. Feichtinger, K. Gröchenig. Theory and practice of irregular sampling. In ``Wavelets: Mathematics and Applications'', J. Benedetto, M. Frazier, eds., pp. 305-363, CRC Press, 1993. MR 94i:94008
  • 7. H. G. Feichtinger, K. Gröchenig, T. Strohmer. Efficient numerical methods in non-uniform sampling theory. Num. Math. 69(4) (1995), 423-440. MR 95j:65011
  • 8. K. Gröchenig, Reconstruction algorithms in irregular sampling. Math. Comp. 59 (1992), 181-194. MR 93a:41025
  • 9. K. Gröchenig, A discrete theory of irregular sampling. Lin. Alg. Appl. 193 (1993), 129-150. MR 94m:94005
  • 10. K. Gröchenig. Finite and Infinite-Dimensional Models of Non-Uniform Sampling. Proc. SampTA 97, Aveiro, Portugal, June 1997, pp. 285-290.
  • 11. G. Hardy, J. E. Littlewood, G. Pólya. Inequalities. 2nd Ed., Cambridge Univ. Press. 1952. MR 13:727e
  • 12. S. Jaffard. A density criterion for frames of complex exponentials. Michigan Math. J. 38 (1991), 339-348. MR 92i:42001
  • 13. H. Landau. Necessary density conditions for sampling and interpolation of certain entire functions. Acta Math. 117 (1967), 37-52. MR 36:5604
  • 14. H. Landau. Extrapolating a band-limited function from its samples taken in a finite interval. IEEE Trans. Information Theory 32(4) (1986), 464-470.
  • 15. B. S. Pavlov. Basicity of an exponential system and Muckenhaupt's condition. Sov. Math. Dokl. 20 (1979), 655-659. MR 84j:42042
  • 16. L. Reichel, G. S. Ammar, W. B. Gragg. Discrete least squares approximation by trigonometric polynomials. Math. Comp. 57 (1991), 273-289. MR 91j:65027
  • 17. K. Seip. On the connection between exponential bases and certain related sequences in $L^2(-\pi, \pi)$. J. Functional Anal. 130 (1995), 131-160. MR 96d:46030
  • 18. T. Strohmer. Efficient methods for digital signal and image reconstruction from non-uniform samples. Ph. D. Thesis, University of Vienna, 1993.
  • 19. R. Young. An Introduction to Nonharmonic Fourier Series. Academic Press, New York. 1980. MR 81m:42027
  • 20. A. I. Zayed. Advances in Shannon's sampling theory. CRC Press, 1993. MR 95f:94008

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 30E05, 30E10, 42A10, 94A12

Retrieve articles in all journals with MSC (1991): 30E05, 30E10, 42A10, 94A12

Additional Information

Karlheinz Gröchenig
Affiliation: Department of Mathematics, The University of Connecticut, Storrs, CT. 06269-3009

Keywords: Entire functions of exponential type, irregular sampling, Toeplitz matrices, approximation by trigonometric polynomials
Received by editor(s): October 25, 1996
Additional Notes: This work was partially supported by NSF grant DMS-9306430.
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society