Comments:

• The schemes implemented in this website correspond to j=0 (Backward Euler), j=1 (Pade 1-2), and j=3 (Pade 3-4) when X is the real line. The scheme SD is under current research. Until today, we do not know if it inverts the Laplace transform. However, as you can see, the numerical experiments suggest that the SD scheme provides a very fast approximation. In the appendix of [JaDi08] you can find a Mathematica code for implementing the schemes associated with the operator given in (4) for any natural number j.
• The schemes presented in this website have by default the Laplace transform of f(t)=Sin(t). However, you can replace it by any other Laplace transform. In fact, the implementations show that the operators given by (4) could also invert the Laplace transform of functions that are piecewise continuous but this remains subject of future research. 
• The subdiagonal Padé approximants are not the only rational functions that can be considered in order to obtain inversion formulas for the Laplace transform, see [Jar08, JaDi08,JNO08].
• If the function f is a polynomial of degree 2j+1 then the operator given by (4) calculates the exact function f in terms of its Laplace transform for every natual number n since in this case the 2j+2-derivative of f is zero. Thus, M=0 in (6).
• The error estimate obtained in (6) can be improved for certain functions with an analytic extension over a sector containing the positive real line. In fact, in that case, the error estimate is uniform in time; for details see [JNO08]. 
• Finally, it seems important to note that the results not only hold for bounded continuous functions but also for exponentially bounded continuous functions.

References:

The following references try to compile the results behind Theorem 1.

[ABHN01] W. Arendt, C. Batty, M. Hieber, and F. Neubrander. Vector-valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics 96. Birkhäuser -Verlag, 2001.

[BrTh79] P. Brenner and V. Thomeé. On rational approximations of semigroups. SIAM J. Numer. Anal. 16 (1979), 683-694.

[Ehl73] B. Ehle. A-stable methods and Padé approximations of the exponential. SIAM J. Math. Anal. 4 (1973), 671-680.

[EnNa00] K. Engel and R. Nagel. One-parameter Semigroups for Linear Evolution Equations. Graduate Text in Mathematics 194 Springer-Verlag, 2000.

[Far04] B. Farkas. Perturbations of bi-continuous semigroups. Dissertation. Eötvös Loránd University, 2003.

[Gol85] J. Goldstein. Semigroups of Linear Operators and Applications. Oxford mathematical monographs, 1985.

[HeKa79] R. Hersh and T. Kato. High-accuracy stable difference schemes for well-posed initial value problems. SIAM J. Numer. Anal. 16 (1979), 670-682.

[HNW78] E. Hairer, S.P. Nørsett, and G. Wanner. Order stars and stability theorems. BIT 18 (1978), 475-489.

[Ise82] A. Iserles. Composite exponential approximations. Math. Comp. 38 (1982), 99-112.

[Jar08] P. Jara. Rational approximation schemes for bi-continuous semigroups. J. Math. Anal. Appl.  344  (2008),  no. 2, 956--968.    

[JNO] P. Jara,  F. Neubrander, and K. Ozer. Rational inversion formulas for the Laplace transform. Submitted.    

[JaDi08] P. Jara, Rational approximation schemes for solutions of abstract Cauchy problems and evolution equations. Dissertation. Louisiana State University, 2008.    

[Kün01] F. Kühnemund. Bi-continuous semigroups on spaces with two topologies: theory and applications. Dissertation. Universität Tübingen, 2001.

[Wid46] D. Widder. The Laplace Transform. Princeton Univeristy Press. 1946.


Last update: December 2009