Abstract
In this paper, a new distribution space
is constructed and the definition of the classical Hilbert transform is extended to it. It is shown that
is the biggest subspace of
on which the extended Hilbert transform is a homeomorphism and both the classical Hilbert transform for L p functions and the circular Hilbert transform for periodic functions are special cases of the extension. Some characterizations of the space
are given and a class of useful nonlinear phase signals is shown to be in
. Finally, the applications of the extended Hilbert transform are discussed.
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References
Mallat S. Wavelet Tour of Signal Processing, 2nd ed. San Diego: Academic Press, 1999
Cohen L. Time-Frequency Analysis. Englewoord Cliffs: Prentice-Hall, 1995
Huang N E, Shen Z, et al. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc R Soc Lond Ser A, 454: 903–995 (1998)
Chen Q, Huang N, Riemenschneider S, et al. A B-spline approach for empirical mode decompositions. Adv Comput Math, 24: 171–195 (2006)
Xu Y, Yan D. The Bedrosian identity for the Hilbert transform of product functions. Proc Amer Math Soc, 134(9): 2719–2728 (2006)
Qian T. Characterization of boundary values of functions in hardy spaces with application in signal analysis. J Integral Equations Appl, 17(2): 159–198 (2005)
Tan L, Yang L, Huang D. Necessary and sufficient conditions for the Bedrosian identity. J Integral Equations Appl, in press
Pinsky M A. Introduction to Fourier Analysis and Wavelets. Pacific Grove, CA: Brook/Cole, 2001
Gasquet C, Witomski P. Fourier Analysis and Applications: Filtering, Numerical Computation, Wavelets (Translated by Ryan R). New York: Springer-Verlag, 1998
Stein E M, Weiss G. Introduction to Fourier Analysis on Euclidean Spaces. Princeton: Princeton University Press, 1971
Beltrami E J, Wohlers M R. Distributions and the Boundary Values of Analytic Functions. New York-London: Academic Press, 1966
Bremermann H J. Some remarks on analytic representations and products of distributions. SIAM J Appl Math, 15(4): 920–943 (1967)
Orton M. Hilbert transforms, Plemelj relations, and Fourier transforms of distribution. SIAM J Math Anal, 4(4): 656–670 (1973)
Pandey J N. The Hilbert transform of Schwartz distributions. Proc Amer Math Soc, 89(1): 86–90 (1983)
Pandey J N, Chaudhry M A. The Hilbert transform of generalized functions and applications. Canad J Math, 35(3): 478–495 (1983)
Pandey J N, Chaudhry M A. The Hilbert transform of Schwartz distributions II. Math Proc Cambridge Philos Soc, 102: 553–559 (1987)
Rudin W. Real and Complex Analysis, 2nd ed. New Delhi: Tata McGraw-Hill, 1987
Folland G B. Real Analysis. New York: John Wiley & Sons, Inc., 1984
Titchmarsh E C. Introduction to the Theory of Fourier Integrals, 3rd ed. New York: Chelsea Publishing Company, 1986
Meyer Y. Ondelettes ét opératewrs, Vol. I. Paris: Hermann, 1990
Bergh J, Löfstrom J. Interpolation Spaces. Berlin-Heidelberg-New York: Springer-Verlag, 1976
Picinbono B. On instantaneous amplitude and phase of signals. IEEE Trans Signal Processing, 45(3): 552–560 (1997)
Vakman D. On the analytic signal, the teager-kaiser energy algorithm, and other methods for defining amplitude and frequency. IEEE Trans Signal Processing, 44(4): 791–797 (1996)
Bedrosian E. A product theorem for Hilbert transform. Proc IEEE, 51: 868–869 (1963)
Brown J L. Analytic signals and product theorems for Hilbert transforms. IEEE Trans Circuits Syst, CAS-21: 790–792 (1974)
Brown J L. A Hilbert transform product theorem. Proc IEEE, 74: 520–521 (1986)
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This work was supported by the National Natural Science Foundation of China (Grant Nos. 60475042, 10631080)
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Yang, L. A distribution space for Hilbert transform and its applications. Sci. China Ser. A-Math. 51, 2217–2230 (2008). https://doi.org/10.1007/s11425-008-0007-1
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DOI: https://doi.org/10.1007/s11425-008-0007-1