A note on deconvolution with completely monotone sequences and discrete fractional calculus
Authors:
Lei Li and Jian-Guo Liu
Journal:
Quart. Appl. Math. 76 (2018), 189-198
MSC (2010):
Primary 47D03
DOI:
https://doi.org/10.1090/qam/1479
Published electronically:
August 22, 2017
MathSciNet review:
3733099
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Abstract: We study in this work convolution groups generated by completely monotone sequences related to the ubiquitous time-delay memory effect in physics and engineering. In the first part, we give an accurate description of the convolution inverse of a completely monotone sequence and show that the deconvolution with a completely monotone kernel is stable. In the second part, we study a discrete fractional calculus defined by the convolution group generated by the completely monotone sequence $c^{(1)}=(1,1,1,\ldots )$, and show the consistency with time-continuous Riemann-Liouville calculus, which may be suitable for modeling memory kernels in discrete time series.
References
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References
- Wei Cai, Computational methods for electromagnetic phenomena, Cambridge University Press, Cambridge, 2013. Electrostatics in solvation, scattering, and electron transport; With a foreword by Weng Cho Chew. MR 3027264
- Bernard D. Coleman and Walter Noll, Foundations of linear viscoelasticity, Rev. Modern Phys. 33 (1961), 239–249. MR 0158605, DOI https://doi.org/10.1103/RevModPhys.33.239
- Philippe Flajolet and Robert Sedgewick, Analytic combinatorics, Cambridge University Press, Cambridge, 2009. MR 2483235
- R. Gorenflo and F. Mainardi, Fractional calculus: integral and differential equations of fractional order, Fractals and fractional calculus in continuum mechanics (Udine, 1996) CISM Courses and Lect., vol. 378, Springer, Vienna, 1997, pp. 223–276. MR 1611585
- B. G. Hansen and F. W. Steutel, On moment sequences and infinitely divisible sequences, J. Math. Anal. Appl. 136 (1988), no. 1, 304–313. MR 972601, DOI https://doi.org/10.1016/0022-247X%2888%2990133-3
- Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006. MR 2218073
- R. Kubo, The fluctuation-dissipation theorem, Reports on progress in physics 29 (1966), no. 1, 255.
- L. Li and J.-G. Liu, A generalized definition of Caputo derivatives and its application to fractional ODEs, arXiv Preprint arXiv:1612.05103v2 (2017).
- Yumin Lin and Chuanju Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys. 225 (2007), no. 2, 1533–1552. MR 2349193, DOI https://doi.org/10.1016/j.jcp.2007.02.001
- Jian-Guo Liu and Robert L. Pego, On generating functions of Hausdorff moment sequences, Trans. Amer. Math. Soc. 368 (2016), no. 12, 8499–8518. MR 3551579, DOI https://doi.org/10.1090/tran/6618
- H. M. Nussenzveig, Causality and dispersion relations, Academic Press, New York-London, 1972. Mathematics in Science and Engineering, Vol. 95. MR 0503032
- Gianpietro Del Piero and Luca Deseri, On the concepts of state and free energy in linear viscoelasticity, Arch. Rational Mech. Anal. 138 (1997), no. 1, 1–35. MR 1463802, DOI https://doi.org/10.1007/s002050050035
- Stephan Ruscheweyh, Luis Salinas, and Toshiyuki Sugawa, Completely monotone sequences and universally prestarlike functions, Israel J. Math. 171 (2009), 285–304. MR 2520111, DOI https://doi.org/10.1007/s11856-009-0050-9
- René L. Schilling, Renming Song, and Zoran Vondraček, Bernstein functions, 2nd ed., De Gruyter Studies in Mathematics, vol. 37, Walter de Gruyter & Co., Berlin, 2012. Theory and applications. MR 2978140
- O. Stenzel, The physics of thin film optical spectra, Springer, 2005.
- John S. Toll, Causality and the dispersion relation: logical foundations, Phys. Rev. (2) 104 (1956), 1760–1770. MR 0083623
- David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923
- Robert Zwanzig, Nonequilibrium statistical mechanics, Oxford University Press, New York, 2001. MR 2012558
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Additional Information
Lei Li
Affiliation:
Department of Mathematics, Duke University, Durham, NC 27708
MR Author ID:
1068344
Email:
leili@math.duke.edu
Jian-Guo Liu
Affiliation:
Departments of Physics and Mathematics, Duke University, Durham, NC 27708
MR Author ID:
233036
ORCID:
0000-0002-9911-4045
Email:
jliu@phy.duke.edu
Keywords:
Convolution group,
convolution inverse,
completely monotone sequence,
fractional calculus,
Riemann-Liouville derivative.
Received by editor(s):
July 3, 2017
Published electronically:
August 22, 2017
Article copyright:
© Copyright 2017
Brown University
