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Local sampling theorems for spaces generated by splines with arbitrary knots

Author: Wenchang Sun
Journal: Math. Comp. 78 (2009), 225-239
MSC (2000): Primary 65T40; Secondary 41A58, 42A65, 42C15, 94A20
Published electronically: June 25, 2008
MathSciNet review: 2448704
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Abstract | References | Similar Articles | Additional Information

Abstract: Most of the known results on sampling theorems, e.g., regular and irregular sampling theorems for band-limited functions, are concerned with global sampling. That is, to recover a function at a point or on an interval, we have to know all the samples, which are usually infinitely many. On the other hand, local sampling, which invokes only finitely many samples to reconstruct a function on a bounded interval, is practically useful since we only need to consider a function on a bounded interval in many cases and hardware can process only finitely many samples. In this paper, we give a characterization of local sampling sequences for spaces generated by B-splines with arbitrary knots.

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  • 1. A. Aldroubi, Non-uniform weighted average sampling and reconstruction in shiftinvariant and wavelet spaces, Appl. Comp. Harmonic Anal., 13(2002), 151-161. MR 1942749 (2003i:42045)
  • 2. A. Aldroubi and K. Gröchenig, Beuling-Landau-type theorems for non-uniform sampling in shift invariant spline spaces, J. Fourier Anal. Appl., 6 (2000), 93-103. MR 1756138 (2001i:42057)
  • 3. A. Aldroubi and M. Unser, Sampling procedures in function spaces and asymptotic equivalence with Shannon's sampling theory, Numer. Funct. Anal. and Optimiz., 15 (1994), 1-21. MR 1261594 (95a:94002)
  • 4. N. Atreas, J.J. Benedetto and C. Karanikas, Local sampling for regular wavelet and Gabor expansions, Sampl. Theory Signal Image Process, 2(2003), 1-24. MR 2002854 (2004k:42050)
  • 5. A. Beurling, in L. Carleson, Ed., A. Beurling Collected Works, Vol. 2, Birkhäuser, Boston, 1989, 341-365.
  • 6. P.L. Butzer and J. Lei, Approximation of signals using measured sampled values and error analysis, Commun. Appl. Anal., 4(2000), 245-255. MR 1752849 (2001i:94033)
  • 7. O. Christensen, An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 2003. MR 1946982 (2003k:42001)
  • 8. C.K. Chui, An Introduction to Wavelets, Academic Press, New York, 1992. MR 1150048 (93f:42055)
  • 9. I. Daubechies, Ten Lectures on Wavelets, SIAM Philadelphia, 1992. MR 1162107 (93e:42045)
  • 10. H. Feichtinger and K. Gröchenig, Theory and practice of irregular sampling, in ``Wavelets: Mathematics and Applications'' (J. Benedetto and M. Frazier, eds.), 305-363, CRC press Inc., 1994. MR 1247520 (94i:94008)
  • 11. T.N. Goodman and S.L. Lee, Wavelets of multiplicity $ r$, Trans. Amer. Math. Soc., 342(1994), 307-324. MR 1232187 (94k:41016)
  • 12. K. Gröchenig, Reconstruction algorithms in irregular sampling, Math. Comput., 59(1992), 181-194. MR 1134729 (93a:41025)
  • 13. S. Jaffard, A density criterion for frames of complex expotentials, Michigan Math. J., 38(1991), 339-348. MR 1116493 (92i:42001)
  • 14. A.J.E.M. Janssen, The Zak transform and sampling theorem for wavelet subspaces, IEEE Trans. Signal processing, 41 (1993), 3360-3364.
  • 15. H. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, ACTA Math., 117(1967), 37-52. MR 0222554 (36:5604)
  • 16. Y. Liu, Irregular sampling for spline wavelet subspaces, IEEE Trans. Inform. Theory, 42 (1996), 623-627. MR 1381369 (97b:94006)
  • 17. P.R. Massopust, D.K. Ruch And P.J. Van Fleet, On the support properties of scaling vectors, Appl. Comput. Harmon. Anal., 3(1996), 229-238. MR 1400081 (97i:42018)
  • 18. J. Ortega-Cerdà and K. Seip, Fourier frames, Ann. of Math., 155(2002), 789-806. MR 1923965 (2003k:42055)
  • 19. L. Schumaker, Spline Functions: Basic Theory, Wiley-Interscience, Boston, 1981. MR 606200 (82j:41001)
  • 20. C.E. Shannon, Communication in the presence of noise, Proc. IRE., 37(1949), 10-21. MR 0028549 (10:464e)
  • 21. W. Sun, X. Zhou, Average sampling theorems for shift invariant subspaces, Sci. China Ser. E 43(2000) 524-530. MR 1799696 (2001m:94030)
  • 22. W. Sun and X. Zhou, Average sampling in spline subspaces, Appl. Math. Letters, 15(2002) 233-237. MR 1880763 (2003a:94026)
  • 23. W. Sun and X. Zhou, Reconstruction of functions in spline subspaces from local averages, Proc. Amer. Math. Soc. 131(2003) 2561-2571. MR 1974656 (2004f:42057)
  • 24. W. Sun and X. Zhou, Characterization of local sampling sequences in spline subspaces, Adv. Comput. Math., to appear.
  • 25. G. Walter, A sampling theorem for wavelet subspaces, IEEE Trans. Inform. Theory, 38 (1992), 881-884. MR 1162226 (93e:94006)
  • 26. R.G. Wiley, Recovery of band-limited signals from unequally spaced samples, IEEE Trans. Comm., 26 (1978), 135-137.
  • 27. J. Xian, Weighted sampling and signal reconstruction in spline subspaces, Signal Processing, 86(2006), 331-340.
  • 28. R.M. Young, An Introduction to Non-Harmonic Fourier Series, Academic, New York, 1980. MR 591684 (81m:42027)
  • 29. C. Zhao and P. Zhao, Sampling theorem and irregular sampling theorem for multiwavelet subspaces, IEEE Trans. Signal Proc., 53(2005), 705-713. MR 2117324
  • 30. X. Zhou, W. Sun, On the sampling theorem for wavelet subspaces, J. Fourier Anal. Appl., 5 (1999), 347-354. MR 1700088 (2000i:42025)

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Additional Information

Wenchang Sun
Affiliation: Department of Mathematics and LPMC, Nankai University, Tianjin 300071, China

Keywords: Local sampling theorems, local sampling sequences, spline subspaces, irregular sampling, periodic nonuniform sampling
Received by editor(s): December 6, 2006
Received by editor(s) in revised form: October 25, 2007
Published electronically: June 25, 2008
Additional Notes: This work was supported partially by the National Natural Science Foundation of China (10571089 and 60472042), the Program for New Century Excellent Talents in Universities, and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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