Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-05T14:04:44.943Z Has data issue: false hasContentIssue false
  • English
  • Français

Uncertainty principles invariant under the fractional Fourier transform

Published online by Cambridge University Press:  17 February 2009

David Mustard
David Mustard
Affiliation:
School of Mathematics, University of N.S.W., P. O. Box 1, Kensington, Australia2033.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Uncertainty principles like Heisenberg's assert an inequality obeyed by some measure of joint uncertainty associated with a function and its Fourier transform. The more groups under which that measure is invariant, the more that measure represents an intrinsic property of the underlying object represented by the given function. The Fourier transform is imbedded in a continuous group of operators, the fractional Fourier transforms, but the Heisenberg measure of overall spread turns out not to be invariant under that group. A new family is developed of measures that are invariant under the group of fractional Fourier transforms and that obey associated uncertainty principles. The first member corresponds to Heisenberg's measure but is generally smaller than his although equal to it in special cases.

Type
Research Article
Information
The ANZIAM Journal , Volume 33 , Issue 2 , October 1991 , pp. 180 - 191
Copyright
Copyright © Australian Mathematical Society 1991

References

[1] Bargmann, V., “On a Hilbert space of analytic functions and an associated integral transform, Part I.” Comm. Pure Appl. Math. 14 (1961) 187–214.CrossRefGoogle Scholar
[2] Condon, E. U., “Immersion of the Fourier transform in a continuous group of functional transformations” Proc. Nat. Acad. Sci., U.S.A. 23 (1937) 158–164.Google Scholar
[3] Cowling, M. G. and Price, J. F., “Bandwidth versus time-concentration: the Heisenberg- Pauli-Weyl inequality”, SIAM J. Math. Anal. 15 (1984) 151–165.Google Scholar
[4] Cowling, M. G. and Price, J. F., “Generalizations of Heisenberg's inequality”, Harmonic Analysis; Proceedings 1982, Mauceri, G., Ricci, F. and Weiss, G. (eds) Lecture Notes in Mathematics 992, Springer-Verlag, Berlin (1983).Google Scholar
[5] Dym, H. and McKean, H. P., Fourier Series and Integrals (Academic Press, New York, 1972).Google Scholar
[6] Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., Higher Transcendental Functions Vol. II. (McGraw Hill, New York, 1953).Google Scholar
[7] Hirschman, I. I. Jr, “A note on entropy.” Amer. J. Math. 79 (1957) 152–156.CrossRefGoogle Scholar
[8] Landau, H. J. and Pollak, H. O., “Prolate spheroidal wave functions, Fourier analysis and uncertainty (2)”, Bell System Tech. J. 40 (1961) 65–84.CrossRefGoogle Scholar
[9] Landau, H. J., “An overview of time and frequency limiting”, Fourier Techniques and Applications (ed. Price, J. F.), (Plenum, New York, 1985).Google Scholar
[10] Miller, W. Jr, Lie Theory and Special Functions (Academic Press, New York, 1968).Google Scholar
[11] Mustard, D., Lie group imbeddings of the Fourier transform. Applied Mathematics Preprint AM87/14, School of Mathematics, U.N.S.W. (1987).Google Scholar
[12] Mustard, D., The fractional Fourier transform and the Wigner distribution, Applied Mathematics Preprint AM89/6, School of Mathematics, UNSW (1989).Google Scholar
[13] Pollak, H. O. and Slepian, D., “Prolate spheroidal wave functions, Fourier analysis and uncertainty (1)”, Bell System Tech. J. 40 (1961) 43–64.Google Scholar
[14] Price, J. F., “Inequalities and local uncertainty principles”, J. Math. Physics 24 (1983) 1711–1714.CrossRefGoogle Scholar
[15] Price, J. F., “Uncertainty principles and sampling theorems”, in Fourier Techniques and Applications, (ed. Price, J. F.), (Plenum, New York, 1985).CrossRefGoogle Scholar
[16] Vilenkin, N. Ja., Special Functions and Theory of Group Representations, Izd. Nauka., Moscow, (1965) (in Russian); Special Functions and the Theory of Group Representations, AMS Transl. Vol 22, Providence, R.I. (1968) (English Transl.)Google Scholar