Recovering signals from inner products involving prolate spheroidals in the presence of jitter
HTML articles powered by AMS MathViewer
- by Dorota Da̧browska;
- Math. Comp. 74 (2005), 279-290
- DOI: https://doi.org/10.1090/S0025-5718-04-01648-5
- Published electronically: April 16, 2004
- PDF | Request permission
Abstract:
The paper deals with recovering band- and energy-limited signals from a finite set of perturbed inner products involving the prolate spheroidal wavefunctions. The measurement noise (bounded by $\delta$) and jitter meant as perturbation of the ends of the integration interval (bounded by $\gamma$) are considered. The upper and lower bounds on the radius of information are established. We show how the error of the best algorithms depends on $\gamma$ and $\delta$. We prove that jitter causes error of order $\Omega ^{\frac {3}{2}}\gamma$, where $[-\Omega ,\Omega ]$ is a bandwidth, which is similar to the error caused by jitter in the case of recovering signals from samples.References
- Dorota Da̧browska and Marek A. Kowalski, Approximating band- and energy-limited signals in the presence of jitter, J. Complexity 14 (1998), no. 4, 557–570. MR 1659000, DOI 10.1006/jcom.1998.0490
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
- W. H. J. Fuchs, On the eigenvalues of an integral equation arising in the theory of band-limited signals, J. Math. Anal. Appl. 9 (1964), 317–330. MR 173921, DOI 10.1016/0022-247X(64)90017-4
- Bolesław Z. Kacewicz and Marek A. Kowalski, Approximating linear functionals on unitary spaces in the presence of bounded data errors with applications to signal recovery, Internat. J. Adapt. Control Signal Process. 9 (1995), no. 1, 19–31. MR 1314291, DOI 10.1002/acs.4480090104
- Bolesław Z. Kacewicz and Marek A. Kowalski, Recovering linear operators from inaccurate data, J. Complexity 11 (1995), no. 2, 227–239. MR 1334235, DOI 10.1006/jcom.1995.1009
- —, “Recovering signals from inaccurate data” in Curves and Surfaces in Computer Vision and Graphics II (M. J. Silbermann and H. D. Tagare, Eds.), Proc. SPIE, Vol. 1610, Int. Soc. Opt. Eng., Bellingham, WA, (1992), pp. 68–74.
- Marek A. Kowalski, Optimal complexity recovery of band- and energy-limited signals, J. Complexity 2 (1986), no. 3, 239–254. MR 922815, DOI 10.1016/0885-064X(86)90004-X
- Marek A. Kowalski, On approximation of band-limited signals, J. Complexity 5 (1989), no. 3, 283–302. MR 1018020, DOI 10.1016/0885-064X(89)90026-5
- Marek A. Kowalski, Krzysztof A. Sikorski, and Frank Stenger, Selected topics in approximation and computation, Oxford University Press, New York, 1995. MR 1418861
- Marek A. Kowalski and Frank Stenger, Optimal complexity recovery of band- and energy-limited signals. II, J. Complexity 5 (1989), no. 1, 45–59. MR 990811, DOI 10.1016/0885-064X(89)90012-5
- H. J. Landau, The eigenvalue behavior of certain convolution equations, Trans. Amer. Math. Soc. 115 (1965), 242–256. MR 199745, DOI 10.1090/S0002-9947-1965-0199745-4
- —, Sampling, data transmission, and the Nyquist rate, Proc. IEEE 55 (1967), pp. 1701–1706.
- —, “An overview of time and frequency limiting” in Fourier Techniques and Applications (J. F. Price, Ed.), Plenum, New York, 1985.
- H. J. Landau and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty. II, Bell System Tech. J. 40 (1961), 65–84. MR 140733, DOI 10.1002/j.1538-7305.1961.tb03977.x
- H. J. Landau and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty. III. The dimension of the space of essentially time- and band-limited signals, Bell System Tech. J. 41 (1962), 1295–1336. MR 147686, DOI 10.1002/j.1538-7305.1962.tb03279.x
- Avraham A. Melkman, $n$-widths and optimal interpolation of time- and band-limited functions, Optimal estimation in approximation theory (Proc. Internat. Sympos., Freudenstadt, 1976) The IBM Research Symposia Series, Plenum, New York-London, 1977, pp. 55–68. MR 460967
- Avraham A. Melkman, $n$-widths and optimal interpolation of time- and band-limited functions. II, SIAM J. Math. Anal. 16 (1985), no. 4, 803–813. MR 793923, DOI 10.1137/0516060
- Charles A. Micchelli and Theodore J. Rivlin (eds.), Optimal estimation in approximation theory, The IBM Research Symposia Series, Plenum Press, New York-London, 1977. MR 445154
- Leszek Plaskota, Noisy information and computational complexity, Cambridge University Press, Cambridge, 1996. MR 1446005, DOI 10.1017/CBO9780511600814
- David Slepian, Some asymptotic expansions for prolate spheroidal wave functions, J. Math. and Phys. 44 (1965), 99–140. MR 179392
- David Slepian, On bandwidth, Proc. IEEE 64 (1976), no. 3, 292–300. MR 462765
- D. Slepian and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty. I, Bell System Tech. J. 40 (1961), 43–63. MR 140732, DOI 10.1002/j.1538-7305.1961.tb03976.x
- J. F. Traub, G. W. Wasilkowski, and H. Woźniakowski, Information-based complexity, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1988. With contributions by A. G. Werschulz and T. Boult. MR 958691
- Joseph Frederick Traub, G. W. Wasilkowski, and H. Woźniakowski, Information, uncertainty, complexity, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983. MR 680041
- Joe Fred Traub and H. Woźniakowsi, A general theory of optimal algorithms, ACM Monograph Series, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 584446
Bibliographic Information
- Dorota Da̧browska
- Affiliation: Faculty of Mathematics and Science, Cardinal Stefan Wyszyński University in Warsaw, ul. Dewajtis 5, 01-815 Warsaw, Poland
- Email: dabrowska@uksw.edu.pl
- Received by editor(s): July 19, 2002
- Received by editor(s) in revised form: June 2, 2003
- Published electronically: April 16, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Math. Comp. 74 (2005), 279-290
- MSC (2000): Primary 68Q17, 94A12, 94A11; Secondary 94A20, 65G99
- DOI: https://doi.org/10.1090/S0025-5718-04-01648-5
- MathSciNet review: 2085411