H-Matrix approximation for the operator exponential with applications

108 I. P. Gavrilyuk et al.
It follows from (2.13) that
(6.10) ‖f‖ H 1 (D d ) ≤ 2c
∫ ∞
−∞
which together with (6.5) and (6.9) implies
‖η N (f,h)‖ ≤c √ [ exp(−πd/h)
π
which completes the proof.
e −αx2 dx = 2c √ α
√ π
√ α sinh (πd/h)
+ exp[−α(N +1)2 h 2 ]
αh(N +1)
Now, we conclude with the proof of Theorem 2.4.
Proof. First, we note that one can choose as integration path any parabola
(6.11) Γ b = {z = a k η2 + b + iη : η ∈ (−∞, ∞),k >1,b0
suchthat for |ν|