Skip to main content
SpringerLink
Log in

An Overview of Time and Frequency Limiting

  • Chapter
Fourier Techniques and Applications

Abstract

This note aims to motivate and sketch informally some of the work on time and frequency limiting that I have seen at close quarters. The account is entirely subjective, and is not meant to speak for the friends and collaborators — notably H.O. Pollak, D. Slepian, and B.F. Logan — from whom I learned much of this material, and who could explain it far better. I also apologize to the many other contributors for inadequate mention of their work. What follows is impressionistic and incomplete, intending, as does any brief survey, only to show the interest and charm of the area, with the hope of enticing the reader to return at greater leisure.

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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. E. Basor and H. Widom, Toeplitz and Wiener-Hopf determinants with piecewise continuous symbols,J. Functional Analysis50 (1983), 387–413.

    Article  Google Scholar 

  2. A. Beurling and P. Malliavin, On the closure of characters and zeros of entire functions,Acta Math. (Stockholm) 118 (1967), 79–93.

    Article  Google Scholar 

  3. P. DeSantis and F. Gori, On an iterative method for super-resolution, Optica Acta 22 (1975), 691–695.

    Google Scholar 

  4. P. DeSantis, F. Gori, G. Guattari, and C. Palma, Optical systems with feedback, Optica Acta 23 (1976), 505–518.

    Google Scholar 

  5. R.W. Gerchberg, Super-resolution through error energy reduction,Optica Acta21 (1974), 709–720.

    Google Scholar 

  6. A. Papoulis, A new algorithm in spectral analysis and bandlimited extrapolation,IEEE Trans.Circuits Syst., CAS-22 (1975), 735–742.

    Article  Google Scholar 

  7. F. Gori and S. Paolucci, Degrees of freedom of an optical image in coherent illumination, in the presence of observations, J.Opt.Soc. Am 65 (1975), 495–501.

    Article  Google Scholar 

  8. FA. Grunbaum, Finite convolution integral operators commuting with differential operators: some counterexamples,Num. Funct.Anal.and Optimiz. 3 (1981), 185–199.

    Article  Google Scholar 

  9. FA. Grunbaum, Band and time limiting, recursion relations and some nonlinear evolution equations,in“Special Functions: Group Theoretical Aspects and Applications,” R. Askey, T. Koornwinder and W. Schemp, eds, Reidel, Dordrecht (1984).

    Google Scholar 

  10. M. Kac, W. Murdock, and G. Szego, On the eigenvalues of certain Hermitian forms,J.Rat.Mech.Analysis2 (1953), 767–800.

    Google Scholar 

  11. JF. Kaiser, Nonrecursive digital filter design using the I0-sinh window function, Proc. 1974IEEE Symp.Circuits and Systems, 20–23.

    Google Scholar 

  12. JF. Kaiser and RW. Schafer, On the use of the I0-sinh window for spectrum analysis,IEEE Trans.Acoust.,Speech and Signal Processing, ASSP-28 (1980), 105–107.

    Article  Google Scholar 

  13. HJ. Landau and HO. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty II,Bell Syst.Tech.J.40 (1961), 65–84.

    Google Scholar 

  14. HJ. Landau and HO. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty III,Bell Syst.Tech.J.41 (1962), 1295–1336.

    Google Scholar 

  15. HJ. Landau, Sampling, data transmission, and the Nyquist rate,Proc. IEEE55 (1967), 1701–1706.

    Article  Google Scholar 

  16. HJ. Landau, Necessary density condition for sampling and interpolation of certain entire functions,Acta Math. (Stockholm) 117 (1967), 73–93.

    Article  Google Scholar 

  17. HJ. Landau, On Szego’s eigenvalue distribution theorem and non-Hermitian kernels,J.d’Analyse Math. 28 (1975), 335–357.

    Article  Google Scholar 

  18. HJ. Landau, The notion of approximate eigenvalues applied to an integral equation of laser theory,Quart.Appl.Math. 35 (1977), 165–172.

    Google Scholar 

  19. HJ. Landau and H. Widom, The eigenvalue distribution of time and frequency limiting,J.Math.Anal.and Appl. 77 (1980), 469–481.

    Article  Google Scholar 

  20. C. Lindberg, J. Park, and DJ. Thomson, Optimal spectral windows for normal mode time series, American Geophysical Union, Fall meeting, San Francisco, Dec. 1983.

    Google Scholar 

  21. B.F. Logan, “High-Pass Functions”, to appear.

    Google Scholar 

  22. AA. Melkman, N-widths and optimal interpolation of time- and band-limited functions, “Optimal Estimation in Approximation Theory”, CA. Micchelli and TJ. Rivlin, eds, Plenum, New York (1977), 55–68.

    Google Scholar 

  23. A.A. Melkman, N-widths and optimal interpolation of time- and band-limited functions II, to appear.

    Google Scholar 

  24. B.J. Patz and D.L. Van Buren, A FORTRAN computer program for calculating the prolate spheroidal angular functions of the first kind,Naval Research Laboratory Report4414 (1981), Washington, D.C.

    Google Scholar 

  25. HO. Pollak and D. Slepian, Prolate spheroidal wave functions, Fourier analysis and uncertainty I,Bell Syst.Tech. J. 40 (1961) 43–64.

    Google Scholar 

  26. JF. Price, Inequalities and local uncertainty principles, J.Math.Phys. 24 (1983), 1711–1714; Sharp local uncertainty inequalities (submitted).

    Article  Google Scholar 

  27. RM. Redheffer, Completeness of the set of complex exponentials,Advances in Math. 24 (1977), 1–62.

    Article  Google Scholar 

  28. F. Riesz and B. Sz-Nagy, “Functional Analysis,” Ungar, New York (1971).

    Google Scholar 

  29. D. Slepian, Prolate spheroidal wave functions, Fourier analysis and uncertainty IV,Bell Syst.Tech.J.43 (1964), 3009–3057.

    Google Scholar 

  30. D. Slepian, Analytic solution of two apodization problems, J. Opt.Soc. Am. 55 (1965), 110–115.

    Article  Google Scholar 

  31. D. Slepian, Some asymptotic expansions for prolate spheroidal functions,J.Math,and Phys. 44 (1965), 99– 140.

    Google Scholar 

  32. D. Slepian, On bandwidth,Proc.IEEE63 (1976), 292–300.

    Article  Google Scholar 

  33. D. Slepian, Prolate spheroidal wave functions, Fourier analysis and uncertainty V,Bell Syst.Tech.J.57 (1978), 1371–1430.

    Google Scholar 

  34. D. Slepian, Some comments on Fourier analysis, uncertainty and modeling,SIAM Review25 (1983), 379–393.

    Article  Google Scholar 

  35. DJ. Thomson, Spectrum estimation and harmonic analysis,Proc.IEEE70 (1982), 1055–1096.

    Article  Google Scholar 

  36. A.L. Van Buren, A FORTRAN computer program for calculating the linear prolate functions,Naval Research Laboratory Report7994 (1976), Washington, D.C.

    Google Scholar 

  37. H. Widom, On a class of integral operators with discontinuous symbol, in: “Toeplitz Centennial (Tel-Aviv, 1981), Operator Theory: Adv. Appl., 4,” I. Gohberg, ed., Birhauser, Basel-Boston, (1982), 477–500.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. A.T. & T. Bell Laboratories, Murray Hill, New Jersey, USA

    H. J. Landau

Authors
  1. H. J. Landau
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Mathematics, University of New South Wales, Kensington, New South Wales, Australia

    John F. Price

Rights and permissions

Reprints and permissions

Copyright information

© 1985 Plenum Press, New York

About this chapter

Cite this chapter

Landau, H.J. (1985). An Overview of Time and Frequency Limiting. In: Price, J.F. (eds) Fourier Techniques and Applications. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-2525-3_12

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-2525-3_12

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4612-9525-9

  • Online ISBN: 978-1-4613-2525-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation