Rational approximations of semigroups without scaling and squaring

Frank Neubrander; Koray Özer; Teresa Sandmaier

doi:10.3934/dcds.2013.33.5305

Article Contents

2013, Volume 33, Issue 11&12: 5305-5317. Doi: 10.3934/dcds.2013.33.5305

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Rational approximations of semigroups without scaling and squaring

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

Received: January 2012

Published: November 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 65R10, 47A58, 44A10, 41A20, 41A25.

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