Abstract
In this paper, starting from a function analytic in a neighborhood of the unit disk and based on Bessel functions, we construct a family of generalized multivariate sinc functions, which are radial and named radial Bessel-sinc (RBS) functions being time-frequency atoms with nonlinear phase. We obtain a recursive formula for the RBS functions in ℝd with d being odd. Based on the RBS function, a corresponding sampling theorem for a class of non-bandlimited signals is established. We investigate a class of radial functions and prove that each of these functions can be extended to become a monogenic function between two parallel planes, where the monogencity is taken to be of the Clifford analysis sense.
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References
Chen Q, Li L, Malonek H. The construction of generalized B-spline low-pass filters related to possibility density. Sci China Math, 2012, 55,12: 2481–2491
Chen Q, Qian T. Sampling theorem and multiscale specturm based on nonlinear Foureir atoms. Appl Anal, 2009, 88: 903–919
Chen Q, Wang Y B, Wang Y. A sampling theorem for non-bandlimited signals using generalized sinc functions. Comput Math Appl, 2008, 56: 1650–1661
Feichtinger H G, Gröchenig K. Error analysis in regular and irregular sampling theory. Appl Anal, 1993, 50: 167–189
Gröchenig K, Heil C, Walnut D. Nonperiodic sampling and the local three squares theorem. Ark Mat, 2000, 38: 77–92
García A G, Villalón P G. On the truncation error of generalized sampling expansions in shift-invariant spaces. Sampl Theory Signal Image Process, 2007, 6: 53–69
Heil C, Colella D. Matrix refinement equations: Existence and uniqueness. J Fourier Anal Appl, 1996, 2: 363–377
Heil C, Koo Y Y, Lim J K. Duals of frame sequences. Acta Appl Math, 2009, 107: 75–90
Kou K I, Qian T. The Paley-Wiener theorem in R n with the Clifford analysis setting. J Funct Anal, 2002, 189: 227–241
Kou K I, Qian T. Shannon sampling in the Clifford analysis setting. Z Anal Anwendungen, 2005, 24: 825–842
Li Y, Chen Q, Qian T, et al. Sampling error analysis and some properties of non-bandlimited signals that are reconstructed by generalized sinc functions. Appl Anal, doi: 10.1080/00036811.2013.769135
Li C, McIntosh A, Qian T. Clifford algebras, Fourier transforms, and singular convolution operators on Lipschitz surfaces. Rev Mat Iberoamericana, 1994, 10: 665–695
Watson G N. A Treatise on the Theory of Bessel Functions, 2nd Ed. Cambridge: Cambridge University Press, 1966
Woodward P M. Probability and Information Theory with Applications to Radar. London: Pergamon Press, 1953
Yang S, Peng L. Raising approximation order of refinable vector by increasing multiplicity. Sci China Ser A, 2006, 49: 86–97
Yang S, Peng L. Construction of high order balanced multiscaling functions via PTST. Sci China Ser F, 2006, 49: 504–515
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Chen, Q., Qian, T. & Li, Y. Shannon-type sampling for multivariate non-bandlimited signals. Sci. China Math. 56, 1915–1934 (2013). https://doi.org/10.1007/s11425-013-4650-9
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DOI: https://doi.org/10.1007/s11425-013-4650-9