H-Matrix approximation for the operator exponential with applications

90 I. P. Gavrilyuk et al.
where the integrals are finite for t>0. Furthermore, we have
dT
dt + LT = 1 ∫
−ze −zt (zI −L) −1 dz
2πi Γ
( ∫
)
1
+ L e −zt (zI −L) −1 dz
2πi Γ
= − 1 ∫
ze −zt (zI −L) −1 dz
2πi Γ
+ 1 ∫
ze zt (zI −L) −1 dz =0,
2πi Γ
i.e., T (t) = exp(−tL) satisfies the differential equation (2.4). This completes
the proof.
The parametrised integral (2.5) can be represented in the form
(2.6) exp(−tL) = 1 ∫ ∞
F (η, t)dη
2πi −∞
with
F (η, t) =e −zt −1 dz
(zI −L)
dη , z =ãη2 + b − iη.
2.4 The computational scheme and the convergence analysis
Following [25], we construct a quadrature rule for the integral in (2.6) by
using the Sinc approximation on (−∞, ∞). For1 ≤ p ≤∞, introduce the
family H p (D d ) of all operator-valued functions, which are analytic in the
infinite strip D d ,
(2.7) D d = {z ∈ C : −∞ < Rez