On the construction of Dirichlet series approximations for completely monotone functions

Loy, R.; Anderssen, R.

doi:10.1090/S0025-5718-2013-02725-1

On the construction of Dirichlet series approximations for completely monotone functions
HTML articles powered by AMS MathViewer

by R. J. Loy and R. S. Anderssen PDF

Math. Comp. 83 (2014), 835-846 Request permission

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.

References

R. S. Anderssen and C. A. Helliwell. Information recovery in molecular biology: causal modelling of regulated promoter switching experiments. J. Math. Biology, doi:10.1007/s00285-012-0536-7, 2012.
R. S. Anderssen, A. R. Davies, and F. R. de Hoog, On the interconversion integral equation for relaxation and creep, ANZIAM J. 48 (2006/07), no. (C), C346–C363 (2009). MR 2346167
R. S. Anderssen, A. R. Davies, and F. R. de Hoog. On the sensitivity of interconversion between relaxation and creep. Rheologica Acta, 47:159–167, 2008.
R. S. Anderssen, A. R. Davies, and F. R. de Hoog, On the Volterra integral equation relating creep and relaxation, Inverse Problems 24 (2008), no. 3, 035009, 13. MR 2421963, DOI 10.1088/0266-5611/24/3/035009
R. S. Anderssen, M. P. Edwards, S. A. Husain and R. J. Loy. Sums of exponentials approximations for the Kohlrausch function. In MODSIM2011, 19th Int. Congress of Modelling and Simulation, F. Chan, D. Marinova and R. S. Anderssen, eds., Modelling and Simulation Society of Australia and New Zealand, 263–269, 2011.
R. S. Anderssen, F. R. de Hoog and M. Wescott. Stability of the defect renewal equation. In MODSIM2011, 19th Int. Congress of Modelling and Simulation, F. Chan, D. Marinova and R. S. Anderssen, eds., Modelling and Simulation Society of Australia and New Zealand, 359–363, 2011.
R. S. Anderssen, Saiful A. Husain, and R. J. Loy, The Kohlrausch function: properties and applications, ANZIAM J. 45 (2003/04), no. (C), C800–C816. MR 2180338
F. R. de Hoog and R. S. Anderssen. Kernel perturbations for Volterra convolution integral equations. ANZIAM J., (CTAC 2006) 48:C249–C266, 2007.
Norm Eggert and John Lund, The trapezoidal rule for analytic functions of rapid decrease, J. Comput. Appl. Math. 27 (1989), no. 3, 389–406. MR 1026370, DOI 10.1016/0377-0427(89)90024-1
William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0210154
J. D. Ferry. Viscoelastic properties of polymers. John Wiley & Sons, New York, 1980.
G. Gripenberg, S.-O. Londen, and O. Staffans, Volterra integral and functional equations, Encyclopedia of Mathematics and its Applications, vol. 34, Cambridge University Press, Cambridge, 1990. MR 1050319, DOI 10.1017/CBO9780511662805
B. Gross, On the inversion of the Volterra integral equation, Quart. Appl. Math. 10 (1952), 74–76. MR 45299, DOI 10.1090/S0033-569X-1952-45299-9
R. Hilfer. $H$-function representations for stretched exponential relaxation and non-Debye susceptibilities in glassy systems. Phys. Rev., 65:061510, 2002.
C. P. Lindsey and G. D. Patterson. Detailed comparison of the Williams-Watts and Cole-Davidson functions. J. Chem. Phys., 73:3348–3357, 1980.
Y. Liu. Calculation of discrete relaxation modulus and creep compliance. Rheol. Acta, 38:357–364, 1999.
Y. Liu. A direct method for obtaining discrete relaxation spectra from creep data. Rheol. Acta, 40:256–260, 2001.
Yunkang Liu, Approximation by Dirichlet series with nonnegative coefficients, J. Approx. Theory 112 (2001), no. 2, 226–234. MR 1864811, DOI 10.1006/jath.2001.3589
R. J. Loy and R. S. Anderssen. Linear viscoelastic interconversion relationships. submitted, 2012.
J. R. Macdonald. Accurate fitting of immittance spectroscopy frequency-response data using the stretched exponential model. J. Non-Crystal. Solids, 212:95–116, 1997.
J. R. Macdonald. Surprising conductive- and dielectric-system dispersion differences and similarities for two Kohlrausch-related relaxation-time distributions. J. Phys.: Condensed Matter, 18:629–644, 2006.
Elliott W. Montroll and John T. Bendler, On Lévy (or stable) distributions and the Williams-Watts model of dielectric relaxation, J. Statist. Phys. 34 (1984), no. 1-2, 129–162. MR 739125, DOI 10.1007/BF01770352
A. Nikonov, A. R. Davies, and I. Emri. The determination of creep and relaxation functions from a single experiment. J. Rheol., 49:1193–1211, 2005.
Harry Pollard, The representation of $e^{-x^{\lambda }}$ as a Laplace integral, Bull. Amer. Math. Soc. 52 (1946), 908–910. MR 18286, DOI 10.1090/S0002-9904-1946-08672-3
Valéry Weber, Claude Daul, and Richard Baltensperger, Radial numerical integrations based on the sinc function, Comput. Phys. Comm. 163 (2004), no. 3, 133–142. MR 2112679, DOI 10.1016/j.cpc.2004.08.008
David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, vol. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923
M. Zhong, R. J. Loy and R. S. Anderssen. Sums of exponential approximations for the Kohlrausch function. to appear ANZIAM J., 2013.