Recovering signals from inner products involving prolate spheroidals in the presence of jitter

Author:
Dorota Dabrowska

Journal:
Math. Comp. **74** (2005), 279-290

MSC (2000):
Primary 68Q17, 94A12, 94A11; Secondary 94A20, 65G99

Published electronically:
April 16, 2004

MathSciNet review:
2085411

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 ) and jitter meant as perturbation of the ends of the integration interval (bounded by ) 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 and . We prove that jitter causes error of order , where is a bandwidth, which is similar to the error caused by jitter in the case of recovering signals from samples.

**1.**D. Dabrowska and M. A. Kowalski,*Approximating band- and energy-limited signals in the presence of jitter*, J. Complexity**14**(1998), 557-570. MR**2000d:65241****2.**C. Flammer,*Spheroidal Wave Functions*, Stanford University Press, Stanford, 1957. MR**19:689a****3.**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**30:4128****4.**B. Z. Kacewicz and M. A. Kowalski,*Approximating linear functionals on unitary spaces in the presence of bounded data errors with applications to signal recovery*, J. Adaptive Control Signal Process.**9**(1995), 19-31. MR**95m:94001****5.**-,*Recovering linear operators from inaccurate data*, J. Complexity**11**(1995), 227-239. MR**96d:65104****6.**-, ``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.**7.**M. A. Kowalski,*Optimal complexity recovery of band- and energy-limited signals*, J. Complexity**2**(1989), 239-254. MR**89i:94009****8.**-,*On approximation of band-limited signals*, J. Complexity**5**(1989), 283-302. MR**90i:94006****9.**M. A. Kowalski, K. A. Sikorski, and F. Stenger,*Selected topics in approximation and computation*, Oxford University Press, New York, 1995. MR**97k:41001****10.**M. A. Kowalski and F. Stenger,*Optimal complexity recovery of band- and energy-limited signals II*, J. Complexity**5**(1989), pp. 45-49. MR**90c:41003****11.**H. J. Landau,*The eigenvalue behavior of certain convolution equations*, Trans. Amer. Math. Soc.**115**(1965), pp. 242-256. MR**33:7888****12.**-,*Sampling, data transmission, and the Nyquist rate*, Proc. IEEE**55**(1967), pp. 1701-1706.**13.**-, ``An overview of time and frequency limiting'' in*Fourier Techniques and Applications*(J. F. Price, Ed.), Plenum, New York, 1985.**14.**H. J. Landau and H. O. Pollak,*Prolate spheroidal wave functions, Fourier analysis and uncertainty - II*, Bell System Tech. J.**40**(1961), pp. 65-84. MR**25:4147****15.**-,*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), pp. 1295-1336. MR**26:5200****16.**A. A. Melkman, ``-Widths and optimal interpolation of time- and band-limited functions'' in*Optimal Estimation in Approximating Theory*(C. A. Micchelli and T. J. Rivlin, Eds.), Plenum, New York, 1977. MR**57:956****17.**-,*-Widths and optimal interpolation of time- and band-limited functions II*, SIAM J. Math. Anal.**16**(1985), pp. 803-813. MR**86h:41026****18.**C. A. Micchelli and T. J. Rivlin, ``A survey of optimal recovery'' in*Optimal Estimation in Approximating Theory*(C. A. Micchelli and T. J. Rivlin, Eds.), Plenum, New York, 1977. MR**56:3498****19.**L. Plaskota,*Noisy Information and Computational Complexity*, Cambridge Univ. Press, Cambridge, 1996. MR**99b:65189****20.**D. Slepian,*Some asymptotic expansions for prolate spheroidal wave functions*, J. of Mathematics and Physics**44**(1965), pp. 99-143. MR**31:3640****21.**-,*On bandwidth*, Proc. IEEE**64**(1976), pp. 292-300. MR**57:2738****22.**D. Slepian and H. O. Pollak,*Prolate spheroidal wave functions, Fourier analysis and uncertainty - I*, Bell System Tech. J.**40**(1961), pp. 46-64. MR**25:4146****23.**J. F. Traub, G. W. Wasilkowski, and H. Wozniakowski,*Information Based Complexity*, Academic Press, New York, 1988. MR**90f:68085****24.**-,*Information, Uncertainty, Complexity*, Addison-Wesley, Reading, Mass., 1983. MR**85g:68031****25.**J. F. Traub and H. Wozniakowski,*A General Theory of Optimal Algorithms*, Academic Press, New York, 1980. MR**84m:68041**

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Additional Information

**Dorota Dabrowska**

Affiliation:
Faculty of Mathematics and Science, Cardinal Stefan Wyszyński University in Warsaw, ul. Dewajtis 5, 01-815 Warsaw, Poland

Email:
dabrowska@uksw.edu.pl

DOI:
http://dx.doi.org/10.1090/S0025-5718-04-01648-5

Keywords:
Problem complexity,
signal theory,
application of orthogonal functions in communication

Received by editor(s):
July 19, 2002

Received by editor(s) in revised form:
June 2, 2003

Published electronically:
April 16, 2004

Article copyright:
© Copyright 2004
American Mathematical Society