Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 

 

Chromatic expansions in function spaces


Author: Ahmed I. Zayed
Journal: Trans. Amer. Math. Soc. 366 (2014), 4097-4125
MSC (2010): Primary 41A58, 42C15; Secondary 44A15, 42B35
Published electronically: March 24, 2014
MathSciNet review: 3206453
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Chromatic series expansions of bandlimited functions have recently been introduced in signal processing with promising results. Chromatic series share similar properties with Taylor series insofar as the coefficients of the expansions, which are called chromatic derivatives, are based on the ordinary derivatives of the function, but unlike Taylor series, chromatic series have a better rate of convergence and more practical applications.

The $ n$-th chromatic derivative $ K^n(f)$ of an analytic function $ f(t)$ is a linear combination of the ordinary derivatives $ f^{(k)}(t),\, 0\leq k\leq n,$ where the coefficients of the combination are based on systems of orthogonal polynomials. In addition to their practical applications, chromatic series expansions have useful theoretical and mathematical applications. For example, functions in the Paley-Wiener space can be completely characterized by their chromatic series expansions associated with the Legendre polynomials.

The purpose of this paper is to show that chromatic series expansions can be used to characterize other important function spaces. We show that functions in weighted Bergman spaces $ \mathfrak{B}_\gamma $ can be characterized by their chromatic series expansions that use chromatic derivatives associated with the Laguerre polynomials, while functions in the Bargmann-Segal-Foch space $ \mathfrak{F}$ can be characterized by their chromatic series expansions that use chromatic derivatives associated with the Hermite polynomials. Another goal of this article is to show that each one of these spaces has an orthonormal basis that is generated from one single function $ \psi $ by applying successive chromatic derivatives to it, that is, both $ \mathfrak{B}_\gamma $ and $ \mathfrak{F}$ have an orthonormal basis of the form $ \left \{K^n\psi \right \}_{n=0}^\infty .$


References [Enhancements On Off] (What's this?)

  • [1] V. Bargmann, P. Butera, L. Girardello, and John R. Klauder, On the completeness of the coherent states, Rep. Mathematical Phys. 2 (1971), no. 4, 221–228. MR 0290680
  • [2] V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform. Part II. A family of related function spaces. Application to distribution theory, Comm. Pure Appl. Math. 20 (1967), 1–101. MR 0201959
  • [3] V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure Appl. Math. 14 (1961), 187–214. MR 0157250
  • [4] J. Byrnes, Local signal reconstruction via chromatic differentiation filter banks, Conference Record of the Thirty-Fifth Asilomar conference on Signals, Systems and Computers 2001, Vol. 1 (2001), 568-572.
  • [5] M. Cushman and T. Herron, ``The general theory of chromatic derivatives'', Kromos Technology Technical Report (2001).
  • [6] M. Cushman, M. Narasimha, and P.P. Vaidyanathan, Finite-channel chromatic derivative filter banks, IEEE Signal Processing Letters, Vol. 10 (1), (2003), 15-17.
  • [7] Charles F. Dunkl and Yuan Xu, Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, vol. 81, Cambridge University Press, Cambridge, 2001. MR 1827871
  • [8] Peter Duren and Alexander Schuster, Bergman spaces, Mathematical Surveys and Monographs, vol. 100, American Mathematical Society, Providence, RI, 2004. MR 2033762
  • [9] T. Herron and J. Byrnes, ``Families of orthogonal differential operators for signal procesing'', Kromos Technology Technical Report (2001).
  • [10] V. Foch, Verallgemeinerung und Lösung der Diracschen statistischen Gleichung, Z. Phys., Vol. 49 (1928), pp. 339--357.
  • [11] I. Gradshteyn and I. Ryzhik, Tables of Integrals, Series, and Products, Academic Press, New York (1965).
  • [12] Aleksandar Ignjatovic and Ahmed I. Zayed, Multidimensional chromatic derivatives and series expansions, Proc. Amer. Math. Soc. 139 (2011), no. 10, 3513–3525. MR 2813383, 10.1090/S0002-9939-2011-10789-5
  • [13] A. Ignjatovic, Frequency estimation using time domain methods based on robust differential operators, 2010 IEEE 10th International Conference on Signal Processing (ICSP) (2010) 151-154.
  • [14] Aleksandar Ignjatović, Chromatic derivatives, chromatic expansions and associated spaces, East J. Approx. 15 (2009), no. 3, 263–302. MR 2741822
  • [15] Aleksandar Ignjatović, Local approximations based on orthogonal differential operators, J. Fourier Anal. Appl. 13 (2007), no. 3, 309–330. MR 2334612, 10.1007/s00041-006-6085-y
  • [16] A. Ignjatovic, Numerical differentiation and signal processing, Proc. Intenational Conference Information, Communications and Signal Processing (ICICS), Singapore (2001).
  • [17] A. Ignjatovic, Local approximations and signal processing , Kromos Technology Technical Report, Los Altos (2001).
  • [18] A. Ignjatovic, Numerical differentiation and signal processing, Kromos Technology Technical Report, Los Altos (2001).
  • [19] A. Ignjatovic and N. Carlin, Signal processing with local behavior, Provisional Patent Application, (60)-143,074 (1999), US Patent Office, Patent issued as US 6313778, June 2001.
  • [20] A. J. E. M. Janssen, Bargmann transform, Zak transform, and coherent states, J. Math. Phys. 23 (1982), no. 5, 720–731. MR 655886, 10.1063/1.525426
  • [21] A. J. E. M. Janssen, Gabor representation of generalized functions, J. Math. Anal. Appl. 83 (1981), no. 2, 377–394. MR 641340, 10.1016/0022-247X(81)90130-X
  • [22] John R. Klauder and Bo-Sture Skagerstam (eds.), Coherent states, World Scientific Publishing Co., Singapore, 1985. Applications in physics and mathematical physics. MR 826247
  • [23] John R. Klauder and E. C. G. Sudarshan, Fundamentals of quantum optics, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0231591
  • [24] N. N. Lebedev, Special functions and their applications, Dover Publications, Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman; Unabridged and corrected republication. MR 0350075
  • [25] Giorgio Mantica and Sandro Vaienti, The asymptotic behaviour of the Fourier transforms of orthogonal polynomials. I. Mellin transform techniques, Ann. Henri Poincaré 8 (2007), no. 2, 265–300. MR 2314448, 10.1007/s00023-006-0308-2
  • [26] Giorgio Mantica and Davide Guzzetti, The asymptotic behaviour of the Fourier transforms of orthogonal polynomials. II. L.I.F.S. measures and quantum mechanics, Ann. Henri Poincaré 8 (2007), no. 2, 301–336. MR 2314450, 10.1007/s00023-006-0309-1
  • [27] Vladimir A. Marchenko, Sturm-Liouville operators and applications, Operator Theory: Advances and Applications, vol. 22, Birkhäuser Verlag, Basel, 1986. Translated from the Russian by A. Iacob. MR 897106
  • [28] M.A. Naimark, Linear Differential Operators I, George Harrap, London, 1967.
  • [29] M. Narasimha, A. Ignjatovic, and P. Vaidyanathan, Chromatic derivative filter banks, IEEE Signal Processing Lett., Vol. 9 (7), (2002), pp. 215-216.
  • [30] B. Savkovic, Decorrelation properties of chromatic derivative signal representation, IEEE Signal Processin Letters, Vol. 17 (8), (2010), pp. 770-773.
  • [31] G. Szegö, Orthogonal Polynomials, Amer. Math. Soc., Providence, RI (1975).
  • [32] E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations. Part I, Second Edition, Clarendon Press, Oxford, 1962. MR 0176151
  • [33] P. Vaidyanathan, A. Ignjatovic, and M. Narasimha, New sampling expansions of bandlimited signals based on chromatic derivatives, Proc. 35th Asilomar Conf. Signals, Systems and Computers, Monterey (2001), pp. 558-562.
  • [34] Gilbert G. Walter and Xiaoping Shen, A sampling expansion for nonbandlimited signals in chromatic derivatives, IEEE Trans. Signal Process. 53 (2005), no. 4, 1291–1298. MR 2128248, 10.1109/TSP.2005.843745
  • [35] Gilbert G. Walter, Chromatic series with prolate spheroidal wave functions, J. Integral Equations Appl. 20 (2008), no. 2, 263–280. MR 2418070, 10.1216/JIE-2008-20-2-263
  • [36] Ahmed I. Zayed, Chromatic expansions of generalized functions, Integral Transforms Spec. Funct. 22 (2011), no. 4-5, 383–390. MR 2801291, 10.1080/10652469.2010.541059
  • [37] Ahmed I. Zayed, Generalizations of chromatic derivatives and series expansions, IEEE Trans. Signal Process. 58 (2010), no. 3, 1638–1647. MR 2730105, 10.1109/TSP.2009.2038415
  • [38] Ahmed I. Zayed, Advances in Shannon’s sampling theory, CRC Press, Boca Raton, FL, 1993. MR 1270907

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 41A58, 42C15, 44A15, 42B35

Retrieve articles in all journals with MSC (2010): 41A58, 42C15, 44A15, 42B35


Additional Information

Ahmed I. Zayed
Affiliation: Department of Mathematical Sciences, DePaul University, Chicago, Illinois 60614
Email: azayed@condor.depaul.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-05991-6
Keywords: Chromatic derivatives, chromatic series expansions, weighted Bergman spaces, the Bargmann-Segal-Foch space, the Bargmann transform
Received by editor(s): August 17, 2011
Received by editor(s) in revised form: July 20, 2012
Published electronically: March 24, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.