Abstract
The Heisenberg-Pauli-Weyl inequality is established for the Fourier transform associated with the spherical mean operator. Also, a generalization of this inequality is proved. Next, a local uncertainty principle is checked.
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References
Andersson L.E.: On the determination of a function from spherical averages. Siam.J.Math.Anal. 19, 214–232 (1988)
M.G.Cowling and J.F.Price, Generalizations of Heisenberg’s inequality in Harmonic Analysis, (Cortona, 1982)Lecture Notes in Math.992, Springer, Berlin (1983), 443-449.MR 0729369(86g :42002b).
N.G.De Bruijn, Uncertainty principles in Fourier analysis. Inequalities (Proc. Sympos. Wright-Patterson Air Force Base, Ohio, 1965), Academic Press, New York, (1967), 57-71.
A.Erdely and others, Asymptotic expansions. Dover publications, Inc.New-York, 1956.
A.Erdely and others, Tables of integral transforms, vol. I, Mc GRAW-Hill Book Compagny, INC, 1954.
Faris W.G.: Inequalities and uncertainty principles. J.Math.Phys 19, 461–466 (1987)
Fawcett J.A.: Inversion of N-dimensional spherical means. Siam. J.Appl.Math. 45, 336–341 (1985)
Hardy G.H.: A theorem concerning Fourier transforms. J.London Math.Soc 8, 227–231 (1933)
W.Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematic und Mechanik. Zeit.Physic, 43, 172 (1927);The Physical Principles of the Quantum Theory. (Dover, New York, 1949; The Univ.Chicago Press, 1930).
Helesten H., Andersson L.E.: An inverse method for the processing of synthetic aperture radar data. Inverse Problems 4, 111–124 (1987)
M.Herberthson, A Numerical Implementation of an Inversion Formula for CARABAS Raw Data, Internal report D 30430-3.2, National Defense Research Institute, FOA, Box 1165; S-581 11, Linköping, Sweden, 1986.
Lebedev N.N.: Special Functions and their applications. Dover publications, Inc. New-York (1972)
Morgan G.W.: A note on Fourier transforms. J.London.Math.Soc 9, 178–192 (1934)
Nessibi M.M., Rachdi L.T., Trimèche K.: Ranges and inversion formulas for spherical mean operator and its dual. Journal of Mathematical Analysis and Applications 196, 861–884 (1995)
Omri S., Rachdi L.T.: An L p−L q version of Morgan’s theorem associated wih Riemann-Liouville transform. International Journal of Mathematical Analysis 1, 805–824 (2007)
Price J.F.: Inequalities and local uncertainty princilpes. J.Math.Phys 24, 1711–1714 (1983)
Price J.F.: Sharp local uncertainty inequalities. Studia Math 85, 37–45 (1987)
Rachdi L.T., Trimèche K.: Weyl transforms associated with the spherical mean operator. Analysis and Applications 1, 141–164 (2003)
Rösler M., Voit M.: An uncertainty principle for Hankel Transforms. American Mathematical Society 127, 183–194 (1999)
Rösler M.: An uncertainty principle for the Dunkl transform. Bull. Austral. Math.Soc. 59, 353–360 (1999)
Shimeno N.: A note on the uncertainty principle for the Dunkl transform. J.Math.Sci.Univ.Tokyo 8, 33–42 (2001)
G.Szegö, Orthogonal polynomials. Amer.Math.Soc.Colloquium, 23, New york (1939).
Watson G.N.: A treatise on the theory of Bessel functions, 2nd ed. Cambridge University Press, London/New-York (1966)
H. Weyl, Gruppentheorie und Quantenmechanik. S.Hirzel, Leipzig, 1928, and Dover edition, New York, 1950.
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Msehli, N., Rachdi, L.T. Heisenberg-Pauli-Weyl Uncertainty Principle for the Spherical Mean Operator. Mediterr. J. Math. 7, 169–194 (2010). https://doi.org/10.1007/s00009-010-0044-1
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DOI: https://doi.org/10.1007/s00009-010-0044-1