Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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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
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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.


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Additional Information

Lei Li
Affiliation: Department of Mathematics, Duke University, Durham, NC 27708
Email: leili@math.duke.edu

Jian-Guo Liu
Affiliation: Departments of Physics and Mathematics, Duke University, Durham, NC 27708
Email: jliu@phy.duke.edu

DOI: https://doi.org/10.1090/qam/1479
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

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