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Uncertainty Principles and Sampling Theorems

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Fourier Techniques and Applications

Abstract

In this lecture I will be dealing with two separate topics both of which are important for a wide range of applications of Fourier analysis. The first is that of uncertainty. Roughly speaking, it says that the more we concentrate a signal in time, the greater is its bandwidth, and vice versa. (The first explicit statement of this reciprocity is in the mathematics of the quantum mechanical uncertainty principle of Heisenberg, and so the name “uncertainty” is usually applied to all results of this nature.)

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References

  1. W. Beckner, Inequalities in Fourier analysis, Ann. of Math. 102: 159–182 (1975).

    Article  Google Scholar 

  2. J.J. Benedetto, An inequality associated with the uncertainty principle (submitted).

    Google Scholar 

  3. A.L. Cauchy, Mémoire sur diverses formules d’analyse, C.R. Acad. Sci. Paris. 12: 283–298 (1841).

    Google Scholar 

  4. M.G. Cowling and J.F. Price, Bandwidth versus time-concentration: the Heisenberg-Pauli-Weyl inequality, SIAM J. Math. Anal. 15: 151–165 (1984).

    Article  Google Scholar 

  5. M.G. Cowling and J.F. Price, Generalisations of Heisenberg’s inequality,in Harmonic Analysis; Proceedings 1982, G. Mauceri, F. Ricci and G. Weiss, eds, Lecture Notes in Mathematics 992, Springer-Verlag, Berlin (1983).

    Google Scholar 

  6. W.G. Faris, Inequalities and uncertainty principles, J. Math. Phys. 19: 461–466 (1978).

    Article  Google Scholar 

  7. W.H.J. Fuchs, On the eigenvalues of an integral equation arising in the theory of band-limited signals, J. Math. Anal. Appl 9: 317–330 (1964).

    Article  Google Scholar 

  8. D. Gabor, Theory of communication, J. Inst. Electr. Engnrs. 93(3): 292–457 (1946).

    Google Scholar 

  9. G.H. Hardy, A theorem concerning Fourier transforms, J. London Math. Soc. 8: 227–231 (1933).

    Article  Google Scholar 

  10. . I.I. Hirschman Jr., A note on entropy,Amer.J.Math. 79: 152 – 156 (1957).

    Google Scholar 

  11. A.J. Jerri, The Shannon sampling theorem - its various extensions and applications, Proc. IEEE 65: 1565–1596 (1977).

    Article  Google Scholar 

  12. I. Kluvanek, Sampling theorem in abstract harmonic analysis, Mat.-Fyz. Casopsis Sloven. Akad. Vied. 15: 43–48 (1965).

    Google Scholar 

  13. V.A. Kotel’nikov, On the transmission capacity of “ether” and wire in electrocommunications, Izd. Red. Upr. Svyazi RKKA (Moscow), 1933.

    Google Scholar 

  14. H.J. Landau, H.O. Pollak and D. Slepian, Prolate spheroidal wave functions: Fourier analysis and uncertainty, Bell System Tech. J.: I, 40: 43-64(1961); II, 40: 65-84(1961); III, 41: 1295-133(1962); IV, 43: 3009-3057(1964); V, 57: 1371–1430(1978).

    Google Scholar 

  15. H. Nyquist, Certain topics in telegraph transmission theory, AIEE Trans. 47: 617–644 (1928).

    Google Scholar 

  16. J. Pearl, Time, frequency, sequency, and their uncertainty relations, IEEE Trans. Inform. Theory 19: 225–229 (1973).

    Article  Google Scholar 

  17. D.P. Petersen and D. Middleton, Sampling and reconstruction of wave-number-limited functions in N-dimensional Euclidean spaces, Inform, and Control 5: 279–323 (1962).

    Article  Google Scholar 

  18. J.F. Price, Inequalities and local uncertainty principles, J. Math. Phys. 24: 1711–1714 (1983).

    Article  Google Scholar 

  19. J.F. Price, Minimum conditions for the sampling theorem (submitted).

    Google Scholar 

  20. J.F. Price and P.C. Racki, Local uncertainty inequalities for Fourier series,Proc.Amer.Math.Soc. (to appear).

    Google Scholar 

  21. C.E. Shannon, Communication in the presence of noise, Proc. IRE 37: 10–21 (1949).

    Article  Google Scholar 

  22. C.E. Shannon and W. Weaver, “The Mathematical Theory of Communication,” University of Illinois Press, Urbana (1949).

    Google Scholar 

  23. D. Slepian, On bandwidth, Proc. IEEE 64: 292–300 (1976).

    Article  Google Scholar 

  24. G.W. Stewart, Problems suggested by an uncertainty principle in acoustics, J. Acoustical Soc. Amer. 2: 325–329 (1931).

    Article  Google Scholar 

  25. E.T. Whittaker, On the functions which are represented by the expansions of the interpolatory theory. Proc. Roy. Soc. Edinburgh 35: 181–194 (1915).

    Google Scholar 

  26. J.F. Price, Sharp local uncertainty conditions (submitted).

    Google Scholar 

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Authors and Affiliations

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

    John F. Price

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  1. John F. Price
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Editors and Affiliations

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

    John F. Price

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© 1985 Plenum Press, New York

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Price, J.F. (1985). Uncertainty Principles and Sampling Theorems. In: Price, J.F. (eds) Fourier Techniques and Applications. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-2525-3_3

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  • DOI: https://doi.org/10.1007/978-1-4613-2525-3_3

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4612-9525-9

  • Online ISBN: 978-1-4613-2525-3

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