# American Institute of Mathematical Sciences

November  2013, 33(11&12): 5305-5317. doi: 10.3934/dcds.2013.33.5305

## Rational approximations of semigroups without scaling and squaring

 1 Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, United States 2 Department of Mathematics, Roger Williams University, Bristol, RI 02809, United States 3 Mathematisches Institut, Universität Tübingen, Tübingen, 72076, Germany

Received  January 2012 Published  May 2013

We show that for all $q\ge 1$ and $1\le i \le q$ there exist pairwise conjugate complex numbers $b_{q,i}$ and $\lambda_{q,i}$ with $\mbox{Re} (\lambda_{q,i}) > 0$ such that for any generator $(A, D(A))$ of a bounded, strongly continuous semigroup $T(t)$ on Banach space $X$ with resolvent $R(\lambda,A) := (\lambda I-A)^{-1}$ the expression $\frac{b_{q,1}}{t}R(\frac{\lambda_{q,1}}{t},A) + \frac{b_{q,2}}{t}R(\frac{\lambda_{q,2}}{t},A) + \cdots + \frac{b_{q,q}}{t}R(\frac{\lambda_{q,q}}{t},A)$ provides an excellent approximation of the semigroup $T(t)$ on $D(A^{2q-1})$. Precise error estimates as well as applications to the numerical inversion of the Laplace transform are given.
Citation: Frank Neubrander, Koray Özer, Teresa Sandmaier. Rational approximations of semigroups without scaling and squaring. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5305-5317. doi: 10.3934/dcds.2013.33.5305
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##### References:
 [1] T. M. Apostol, "Mathematical Analysis,", Addison-Wesley, (1974).   Google Scholar [2] W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, "Vector-Valued Laplace Transforms and Cauchy Problems,", $2^{nd}$ edition Monographs in Mathematics, (2011).   Google Scholar [3] P. Brenner and V. Thomée, On rational approximation of semigroups,, SIAM J. Numer. Anal., 16 (1979), 683.  doi: 10.1137/0716051.  Google Scholar [4] B. L. Ehle, $A$-stable methods and Padé approximations to the exponential function,, SIAM J. Math. Anal., 4 (1973), 671.  doi: 10.1137/0504057.  Google Scholar [5] J. A. Goldstein, "Semigroups of Operators and Applications,", Oxford University Press, (1985).   Google Scholar [6] William Harrison, Ph.D thesis,, Louisiana State University, (2012).   Google Scholar [7] R. Hersh and T. Kato, High-accuracy stable difference schemes for wellposed initial value problems,, SIAM J. Numer. Anal., 16 (1979), 670.  doi: 10.1137/0716050.  Google Scholar [8] P. Jara, Rational approximation schemes for bi-continuous semigroups,, J. Math. Anal. Appl., 344 (2008), 956.  doi: 10.1016/j.jmaa.2008.02.068.  Google Scholar [9] P. Jara, F. Neubrander and K. Özer, Rational inversion of the Laplace transform,, Journal of Evolution Equations, ().  doi: 10.1007/s00028-012-0139-1.  Google Scholar [10] Mihály Kovács, "On Qualitative Properties and Convergence of Time-Discretization Methods for Semigroups,", Ph. D. Thesis, (2004).   Google Scholar [11] M. Kovács, On the convergence of rational approximations of semigroups on intermediate spaces,, Math. Comp., 76 (2007), 273.  doi: 10.1090/S0025-5718-06-01905-3.  Google Scholar [12] M. Kovács and F. Neubrander, On the inverse Laplace-Stieltjes transform of $A$-stable rational functions,, New Zealand J. Math., 36 (2007), 41.   Google Scholar [13] Koray Özer, "Laplace Transform Inversion and Time-Discretization Methods for Evolution Equations,", Ph.D. thesis, (2008).   Google Scholar [14] M. H. Padé, Sur répresentation approchée d'une fonction par des fractionelles,, Ann. de l'Ecole Normale Superieure, 9 (1892).   Google Scholar [15] O. Perron, "Die Lehre von den Kettenbrüchen,", Chelsea Pub. Co., (1950).   Google Scholar [16] Armin Reiser, "Time Discretization for Evolution Equations,", Diplomarbeit, (2008).   Google Scholar [17] D. V. Widder, "The Laplace Transform,", Princeton University Press, (1946).   Google Scholar [18] Teresa Sandmaier, "Implizite und Explizite Approximationsverfahren,", Wissenschaftliche Arbeit, (2010).   Google Scholar
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