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APPROXIMATION OF AND BY COMPLETELY MONOTONE FUNCTIONS

Part of:Approximations and expansionsFoundations, constitutive equations, rheologyIntegral transforms, operational calculus

Published online by Cambridge University Press:  06 March 2020

R. J. LOYOpen the ORCID record for R. J. LOY[Opens in a new window] and
R. S. ANDERSSENOpen the ORCID record for R. S. ANDERSSEN[Opens in a new window]
R. J. LOY*
Affiliation:
Mathematical Sciences Institute, Hanna Neumann Building No. 145, Australian National University, CanberraACT 2601, Australia email rick.loy@anu.edu.au
R. S. ANDERSSEN
Affiliation:
Data61, CSIRO, GPO Box 1700, Canberra, ACT 2601, Australia email Bob.Anderssen@Data61.CSIRO.au
*
✉rick.loy@anu.edu.au
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Abstract

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We investigate convergence in the cone of completely monotone functions. Particular attention is paid to the approximation of and by exponentials and stretched exponentials. The need for such an analysis is a consequence of the fact that although stretched exponentials can be approximated by sums of exponentials, exponentials cannot in general be approximated by sums of stretched exponentials.

Keywords

complete and absolute monotonicitystretched exponentialapproximationcompletely monotone and Kohlrausch functionsuniform and pointwise convergence

MSC classification

Primary:41A30: Approximation by other special function classes
Secondary:41A10: Approximation by polynomials44A10: Laplace transform76A10: Viscoelastic fluids

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 61 , Issue 4 , October 2019, pp. 416 - 430
Copyright
© 2020 Australian Mathematical Society

References

Anderssen, R. S. and Loy, R. J., “Rheological implications of completely fading memory”, J. Rheology 46 (2002) 1459–1472; doi:10.1122/1.1514203.CrossRefGoogle Scholar
Anderssen, R. S., Edwards, M. P., Husain, S. A. and Loy, R. J., “Sums of exponentials approximations for the Kohlrausch function”, in: MODSIM2011, 19th Int. Congress on Modelling and Simulation (eds Chan, F., Marinova, D. and Anderssen, R. S.), (Modelling and Simulation Society of Australia and New Zealand, 2011) 263–269; ISBN: 978-0-9872143-1-7. doi:10.36334/modsim.2011.A3.anderssen.Google Scholar
Anderssen, R. S., Husain, S. A. and Loy, R. J., “The Kohlrausch function: properties and applications”, ANZIAM J. 45 (2004) C800–816; doi:10.21914/anziamj.v45i0.924.CrossRefGoogle Scholar
Bernstein, S., “Sur les fonctions absolument monotones”, Acta Math. 52 (1929) 1–66; doi:10.1007/BF02592679.CrossRefGoogle Scholar
Bauer, H., Measure and integration theory, Volume 26 of De Gruyter Stud. Math. (De Gruyter, Berlin, 2001); doi:10.1515/9783110866209.CrossRefGoogle Scholar
Berg, C. and Forst, G., Potential theory on locally compact abelian groups, Ergeb. Math. Grenzgebeite, Band 97 (Springer, Berlin, 1975); doi:10.1007/978-3-642-66128-0.CrossRefGoogle Scholar
Berry, G. C. and Plazek, D. J., “On the use of stretched exponential functions for both linear viscoelastic creep and stress relaxation”, Rheol. Acta 36 (1997) 320–329; doi:10.1007/BF00366673.CrossRefGoogle Scholar
de Gennes, P.-G., “Relaxation anomalies in linear polymer melts”, Macromolecules 35 (2002) 3785–3786; doi:10.1021/ma012167y.CrossRefGoogle Scholar
Dobreva, A., Gutzow, I. and Schmelzer, J., “Stress and time dependence of relaxation and the Kohlrausch stretched exponent formula”, J. Non-Cryst. Solids 209 (1997) 257–263; doi:10.1016/S0022-3093(96)00565-0.CrossRefGoogle Scholar
Fancey, K. S., “A mechanical model for creep, recovery and stress relaxation in polymeric materials”, J. Mater. Sci. 40 (2005) 4827–4831; doi:10.1007/s10853-005-2020-x.CrossRefGoogle Scholar
Ferry, J. D., Viscoelastic properties of polymers (John Wiley & Sons, New York, 1980).Google Scholar
Gripenberg, G., Londen, S. O. and Staffans, O. J., Volterra integral and functional equations (Cambridge University Press, Cambridge, 1990); doi:10.1017/cbo9780511662805.CrossRefGoogle Scholar
Hughes, B. D., Random walks and random environments, Volume 1 (Oxford University Press, Oxford, 1995); doi:10.2307/2533883.Google Scholar
Johnson, W. P., “The curious history of Faà di Bruno’s formula”, Amer. Math. Monthly 109 (2002) 2963–2972; doi:10.2307/2695352.Google Scholar
Liu, Y., “Approximation by Dirichlet series with nonnegative coefficients”, J. Approx. Theory 112 (2001) 226–234; doi:10.1006/jath.2001.3589.CrossRefGoogle Scholar
Loy, R. J. and Anderssen, R. S., “On the construction of Dirichlet series approximations for completely monotone functions”, Math. Comput. 83 (2014) 835–846; doi:10.1090/S0025-5718-2013-02725-1.CrossRefGoogle Scholar
Loy, R. J. and Anderssen, R. S., “$L^{p}$ approximation of completely monotone functions”, J. Approx. Theory 248 (2019) 105301; doi:10.1016/j.jat.2019.105301.CrossRefGoogle Scholar
Maraldi, M., Molari, L., Molari, G. and Regazzi, N., “Time-dependent mechanical properties of straw bales used for construction”, Biosystems Eng. 172 (2018) 75–83; doi:10.1016/j.biosystemseng.2018.05.014.CrossRefGoogle Scholar
Megginson, R. E., Am introduction to Banach space theory, Volume 183 of Grad. Texts in Math. (Springer, New York, 1998); doi:10.1007/978-1-4612-0603-3.CrossRefGoogle Scholar
Paulsen, J. D. and Nagel, S. R., “A model for approximately stretched-exponential relaxation with continuously varying stretching exponents”, J. Stat. Phys. 167 (2017) 749–762; doi:10.1007/s10955-017-1723-0.CrossRefGoogle Scholar
Pollard, H., “The representation of $e^{-x^{\unicode[STIX]{x1D706}}}$ as a Laplace integral”, Bull. Amer. Math. Soc. 52 (1946) 908–910; doi:10.1090/S0002-9904-1946-08672-3.CrossRefGoogle Scholar
Pólya, G. and Szegö, G., Problems and theorems in analysis I (Springer, Berlin, 1978); doi:10.1007/978-3-642-61983-0.Google Scholar
Sasaki, N., Yakayama, Y., Yoshikawa, M. and Enyo, A., “Stress relaxation of bone and bone collagen”, J. Biomech. 26 (1993) 1369–1376; doi:10.1016/0021-9290(93)90088-V.CrossRefGoogle ScholarPubMed
Schiavi, A. and Prato, A., “Evidences of non-linear short-term stress relaxation in polymers”, Polymer Testing 59 (2017) 220–229; doi:10.1016/j.polymertesting.2017.01.030.CrossRefGoogle Scholar
Schiff, J. L., Normal families, Universitext (Springer, New York, 1993); doi:10.1007/978-1-4612-0907-2.CrossRefGoogle Scholar
Schilling, E. L., Song, R. and Vondraček, Z., Bernstein functions, theory and applications, Volume 183 of De Gruyter Stud. Math. (De Gruyter, Berlin, 2012); doi:10/1515/9783110269338.CrossRefGoogle Scholar
Widder, D. V., The Laplace transform (Princeton University Press, Princeton, NJ, 1946); doi:10.1515/9781400876457.Google Scholar
Zhong, M., Loy, R. J. and Anderssen, R. S., “Approximating the Kohlrausch function by sums of exponentials”, ANZIAM J. 54 (2013) 306–323; doi:10.21914/anziamj.v54i0.5539.Google Scholar