Skip to main content
SpringerLink
Log in

An Overview of Time and Multiband Limiting

  • Chapter
  • First Online:
Book cover Excursions in Harmonic Analysis, Volume 1

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

  • 1759 Accesses

Abstract

The purpose of this chapter is to provide an up-to-date overview of time and multiband limiting somewhat parallel to Landau’s (Fourier Techniques and Applications (Kensington, 1983), pp. 201–220. Plenum, New York, 1985) overview. Particular focus is given to the theory of time and frequency limiting of multiband signals and to time-localized sampling approximations of Shannon type for band-limited signals.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The operator given by integration against the kernel \(\mathrm{{e}}^{-\mathrm{i}cst}{\nVdash }_{[-1,1]}(s)\) is often denoted by F c . One has \({F}_{c} = {F}_{a }\) when \(c = a\pi /2\) as we assume here.

References

  1. Bouwkamp, C.J.: On spheroidal wave functions of order zero. J. Math. Phys. Mass. Inst. Tech. 26, 79–92 (1947)

    Google Scholar 

  2. Boyd, J.P.: Algorithm 840: computation of grid points, quadrature weights and derivatives for spectral element methods using prolate spheroidal wave functions—prolate elements. ACM Trans. Math. Software 31, 149–165 (2005)

    Google Scholar 

  3. Candès, E.J., Romberg, J.: Quantitative robust uncertainty principles and optimally sparse decompositions. Found. Comput. Math. 6, 227–254 (2006)

    Google Scholar 

  4. Candès, E.J., Romberg, J.: Sparsity and incoherence in compressive sampling. Inverse Probl. 23, 969–985 (2007)

    Google Scholar 

  5. Candès, E.J., Romberg, J.K., Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theory 52, 489–509 (2006)

    Google Scholar 

  6. Candès, E.J., Romberg, J.K., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math. 59, 1207–1223 (2006)

    Google Scholar 

  7. Donoho, D.L., Stark, P.B.: Uncertainty principles and signal recovery. SIAM J. Appl. Math. 49, 906–931 (1989)

    Google Scholar 

  8. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vols. I, II. McGraw-Hill, New York (1953)

    Google Scholar 

  9. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Tables of Integral Transforms, vol. I. McGraw-Hill, New York (1954)

    Google Scholar 

  10. Grünbaum, F.A.: Eigenvectors of a Toeplitz matrix: discrete version of the prolate spheroidal wave functions. SIAM J. Algebraic Discrete Methods 2, 136–141 (1981)

    Google Scholar 

  11. Grünbaum, F.A.: Toeplitz matrices commuting with tridiagonal matrices. Linear Algebra Appl. 40, 25–36 (1981)

    Google Scholar 

  12. Hogan, J.A., Izu, S., Lakey, J.D.: Sampling approximations for time- and bandlimiting. Sampling Theor. Signal Image Process. 9, 91–118 (2010)

    Google Scholar 

  13. Hogan, J.A., Lakey, J.D.: Time–Frequency and Time–Scale Methods. Birkhäuser, Boston (2005)

    Google Scholar 

  14. Hogan, J.A., Lakey, J.D.: Duration and Bandwidth Limiting. Birkhäuser, Boston (2012)

    Google Scholar 

  15. Izu, S.: Sampling and Time–Frequency Localization in Paley–Wiener Spaces. PhD thesis, New Mexico State University (2009)

    Google Scholar 

  16. Jain, A.K., Ranganath, S.: Extrapolation algorithms for discrete signals with application in spectral estimation. IEEE Trans. Acoust. Speech Signal Process. 29, 830–845 (1981)

    Google Scholar 

  17. Kac, M., Murdock, W.L., Szegő, G.: On the eigenvalues of certain Hermitian forms. J. Rational Mech. Anal. 2, 767–800 (1953)

    Google Scholar 

  18. Landau, H.J.: The eigenvalue behavior of certain convolution equations. Trans. Am. Math. Soc. 115, 242–256 (1965)

    Google Scholar 

  19. Landau, H.J.: Sampling, data transmission, and the Nyquist rate. Proc. IEEE 55, 1701–1706 (1967)

    Google Scholar 

  20. Landau, H.J.: On Szegő’s eigenvalue distribution theorem and non-Hermitian kernels. J. Analyse Math. 28, 335–357 (1975)

    Google Scholar 

  21. Landau, H.J.: An overview of time and frequency limiting. In: Fourier Techniques and Applications (Kensington, 1983), pp. 201–220. Plenum, New York (1985)

    Google Scholar 

  22. Landau, H.J.: On the density of phase-space expansions. IEEE Trans. Inform. Theor. 39, 1152–1156 (1993)

    Google Scholar 

  23. Landau, H.J., Widom, H.: Eigenvalue distribution of time and frequency limiting. J. Math. Anal. Appl. 77, 469–481 (1980)

    Google Scholar 

  24. Rokhlin, V., Xiao, H.: Approximate formulae for certain prolate spheroidal wave functions valid for large values of both order and band-limit. Appl. Comput. Harmon. Anal. 22, 105–123 (2007)

    Google Scholar 

  25. Slepian, D.: Prolate spheroidal wave functions, Fourier analysis, and uncertainty. V - The discrete case. ATT Tech. J. 57, 1371–1430 (1978)

    Google Scholar 

  26. Slepian, D.: Some comments on Fourier analysis, uncertainty and modeling. SIAM Rev. 25, 379–393 (1983)

    Google Scholar 

  27. Slepian, D., Pollak, H.O.: Prolate spheroidal wave functions, Fourier analysis and uncertainty, I. Bell System Tech. J 40, 43–63 (1961)

    Google Scholar 

  28. Walter, G.G.: Differential operators which commute with characteristic functions with applications to a lucky accident. Complex Variables Theor. Appl. 18, 7–12 (1992)

    Google Scholar 

  29. Walter, G.G., Shen, X.A.: Sampling with prolate spheroidal wave functions. Sampl. Theor. Signal Image Process. 2, 25–52 (2003)

    Google Scholar 

  30. Widom, H.: Asymptotic behavior of the eigenvalues of certain integral equations, II. Arch. Rational Mech. Anal. 17, 215–229 (1964)

    Google Scholar 

  31. Xiao, H., Rokhlin, V., Yarvin, N.: Prolate spheroidal wavefunctions, quadrature and interpolation. Inverse Probl. 17, 805–838 (2001)

    Google Scholar 

  32. Xu, W.Y., Chamzas, C.: On the periodic discrete prolate spheroidal sequences. SIAM J. Appl. Math. 44, 1210–1217 (1984)

    Google Scholar 

Download references

Acknowledgments

Several of the ideas presented here developed while the author was on sabbatical at Washington University in St. Louis. He is very grateful to the Department of Mathematics at WUSTL for its hospitality. Special thanks go to Guido Weiss and Ed Wilson. Jeff Hogan and Scott Izu contributed substantially to the ideas presented here.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM, 88003–8001, USA

    Joseph D. Lakey

Authors
  1. Joseph D. Lakey
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joseph D. Lakey .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, University of Maryland, Norbert Wiener Center, College Park, 20742-0001, Maryland, USA

    Travis D. Andrews

  2. Norbert Wiener Center, Department of Mathematics, University of Maryland, College Park, 20742, Maryland, USA

    Radu Balan

  3. Department of Mathematics, University of Maryland, College Park, 20742-4015, Maryland, USA

    John J. Benedetto

  4. Department of Mathematics, University of Maryland, College Park, 20742-4015, Maryland, USA

    Wojciech Czaja

  5. , Department of Mathematics, University of Maryland, College Park, 20742, Maryland, USA

    Kasso A. Okoudjou

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Birkhäuser Boston

About this chapter

Cite this chapter

Lakey, J.D. (2013). An Overview of Time and Multiband Limiting. In: Andrews, T., Balan, R., Benedetto, J., Czaja, W., Okoudjou, K. (eds) Excursions in Harmonic Analysis, Volume 1. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8376-4_5

Download citation

Publish with us

Policies and ethics

Search

Navigation