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On the construction of Dirichlet series approximations for completely monotone functions


Authors: R. J. Loy and R. S. Anderssen
Journal: Math. Comp. 83 (2014), 835-846
MSC (2010): Primary 41A30, 41A29; Secondary 65R20, 76A10
DOI: https://doi.org/10.1090/S0025-5718-2013-02725-1
Published electronically: June 11, 2013
MathSciNet review: 3143694
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Abstract: In a series of papers, Liu established and analysed conditions under which completely monotone ( $ \mathcal {CM}$) functions can be approximated by finite Dirichlet series with positive coefficients. Motivated by a representation theorem of Pollard for Kohlrausch functions, a constructive procedure and proof is given for $ \mathcal {CM}$ functions which are the Laplace transform of absolutely continuous finite positive measures. The importance of this result, which is new even for Kohlrausch functions, is that it allows accurate approximations to be generated for the Laplace transform of such $ \mathcal {CM}$ functions which can then be utilized in various ways including the approximate solution of the interconversion relationship of rheology and its generalization for the solution of Volterra integral equations of the first kind.


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Additional Information

R. J. Loy
Affiliation: Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia
Email: rick.loy@anu.edu.au

R. S. Anderssen
Affiliation: CSIRO Mathematics, Informatics and Statistics, GPO Box 664, Canberra, ACT 2601, Australia
Email: bob.anderssen@csiro.au

DOI: https://doi.org/10.1090/S0025-5718-2013-02725-1
Keywords: Finite Dirichlet series, approximation, completely monotone functions, relaxation modulus, interconversion equation, linear viscoelasticity
Received by editor(s): January 25, 2012
Received by editor(s) in revised form: May 29, 2012, and June 12, 2012
Published electronically: June 11, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.