Local sampling theorems for spaces generated by splines with arbitrary knots

Sun, Wenchang

doi:10.1090/S0025-5718-08-02151-0

Local sampling theorems for spaces generated by splines with arbitrary knots
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Math. Comp. 78 (2009), 225-239 Request permission

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.

References

Akram Aldroubi, Non-uniform weighted average sampling and reconstruction in shift-invariant and wavelet spaces, Appl. Comput. Harmon. Anal. 13 (2002), no. 2, 151–161. MR 1942749, DOI 10.1016/S1063-5203(02)00503-1
Akram Aldroubi and Karlheinz Gröchenig, Beurling-Landau-type theorems for non-uniform sampling in shift invariant spline spaces, J. Fourier Anal. Appl. 6 (2000), no. 1, 93–103. MR 1756138, DOI 10.1007/BF02510120
Akram Aldroubi and Michael Unser, Sampling procedures in function spaces and asymptotic equivalence with Shannon’s sampling theory, Numer. Funct. Anal. Optim. 15 (1994), no. 1-2, 1–21. MR 1261594, DOI 10.1080/01630569408816545
N. Atreas, J. J. Benedetto, and C. Karanikas, Local sampling for regular wavelet and Gabor expansions, Sampl. Theory Signal Image Process. 2 (2003), no. 1, 1–24. MR 2002854
A. Beurling, in L. Carleson, Ed., A. Beurling Collected Works, Vol. 2, Birkhäuser, Boston, 1989, 341–365.
Paul L. Butzer and Junjiang Lei, Approximation of signals using measured sampled values and error analysis, Commun. Appl. Anal. 4 (2000), no. 2, 245–255. MR 1752849
Ole Christensen, An introduction to frames and Riesz bases, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2003. MR 1946982, DOI 10.1007/978-0-8176-8224-8
Charles K. Chui, An introduction to wavelets, Wavelet Analysis and its Applications, vol. 1, Academic Press, Inc., Boston, MA, 1992. MR 1150048
Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107, DOI 10.1137/1.9781611970104
Hans G. Feichtinger and Karlheinz Gröchenig, Theory and practice of irregular sampling, Wavelets: mathematics and applications, Stud. Adv. Math., CRC, Boca Raton, FL, 1994, pp. 305–363. MR 1247520
T. N. T. Goodman and S. L. Lee, Wavelets of multiplicity $r$, Trans. Amer. Math. Soc. 342 (1994), no. 1, 307–324. MR 1232187, DOI 10.1090/S0002-9947-1994-1232187-7
Karlheinz Gröchenig, Reconstruction algorithms in irregular sampling, Math. Comp. 59 (1992), no. 199, 181–194. MR 1134729, DOI 10.1090/S0025-5718-1992-1134729-0
S. Jaffard, A density criterion for frames of complex exponentials, Michigan Math. J. 38 (1991), no. 3, 339–348. MR 1116493, DOI 10.1307/mmj/1029004386
A.J.E.M. Janssen, The Zak transform and sampling theorem for wavelet subspaces, IEEE Trans. Signal processing, 41 (1993), 3360–3364.
H. J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967), 37–52. MR 222554, DOI 10.1007/BF02395039
Youming Liu, Irregular sampling for spline wavelet subspaces, IEEE Trans. Inform. Theory 42 (1996), no. 2, 623–627. MR 1381369, DOI 10.1109/18.485731
Peter R. Massopust, David K. Ruch, and Patrick J. Van Fleet, On the support properties of scaling vectors, Appl. Comput. Harmon. Anal. 3 (1996), no. 3, 229–238. MR 1400081, DOI 10.1006/acha.1996.0018
Joaquim Ortega-Cerdà and Kristian Seip, Fourier frames, Ann. of Math. (2) 155 (2002), no. 3, 789–806. MR 1923965, DOI 10.2307/3062132
Larry L. Schumaker, Spline functions: basic theory, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. MR 606200
Claude E. Shannon, Communication in the presence of noise, Proc. I.R.E. 37 (1949), 10–21. MR 28549
Wenchang Sun and Xingwei Zhou, Average sampling theorems for shift invariant subspaces, Sci. China Ser. E 43 (2000), no. 5, 524–530. MR 1799696
Wenchang Sun and Xingwei Zhou, Average sampling in spline subspaces, Appl. Math. Lett. 15 (2002), no. 2, 233–237. MR 1880763, DOI 10.1016/S0893-9659(01)00123-9
Wenchang Sun and Xingwei Zhou, Reconstruction of functions in spline subspaces from local averages, Proc. Amer. Math. Soc. 131 (2003), no. 8, 2561–2571. MR 1974656, DOI 10.1090/S0002-9939-03-07082-5
W. Sun and X. Zhou, Characterization of local sampling sequences in spline subspaces, Adv. Comput. Math., to appear.
Gilbert G. Walter, A sampling theorem for wavelet subspaces, IEEE Trans. Inform. Theory 38 (1992), no. 2, 881–884. MR 1162226, DOI 10.1109/18.119745
R.G. Wiley, Recovery of band-limited signals from unequally spaced samples, IEEE Trans. Comm., 26 (1978), 135–137.
J. Xian, Weighted sampling and signal reconstruction in spline subspaces, Signal Processing, 86(2006), 331–340.
Robert M. Young, An introduction to nonharmonic Fourier series, Pure and Applied Mathematics, vol. 93, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 591684
Chun Zhao and Ping Zhao, Sampling theorem and irregular sampling theorem for multiwavelet subspaces, IEEE Trans. Signal Process. 53 (2005), no. 2, 705–713. MR 2117324, DOI 10.1109/TSP.2004.840810
Xingwei Zhou and Wenchang Sun, On the sampling theorem for wavelet subspaces, J. Fourier Anal. Appl. 5 (1999), no. 4, 347–354. MR 1700088, DOI 10.1007/BF01259375