RATIONAL APPROXIMATIONS OF SEMIGROUPS WITHOUT SCALING AND SQUARING

. We show that for all q ≥ 1 and 1 ≤ i ≤ q there exist pairwise conjugate complex numbers b q,i and λ q,i with Re( λ 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 ( λ,A ) := ( λI − A ) − 1 the expression b q, 1 t R ( λ q, 1 t ,A ) + b q, 2 t R ( λ q, 2 t ,A ) + ··· + b q,q t R ( λ q,q t ,A ) provides an excellent approximation of the semigroup T ( t ) on D ( A 2 q − 1 ). Precise error estimates as well as applications to the numerical inversion of the Laplace transform are given.

1. Introduction. In this paper we prove a new variant of the following key result on the approximation of strongly continuous semigroups (see [5]). Theorem 1.1. (Hersh-Kato, Brenner-Thomée). Let r be an A-stable rational approximation scheme of the exponential of order m and (A, D(A)) be the generator of a strongly continuous semigroup T (t) of type (M, 0). Then there exists a constant C depending solely on r such that ||r( t n A) n x − T (t)x|| ≤ M Ct m+1 1 n m ||A m+1 x|| for all n ∈ N, t ≥ 0, and x ∈ D(A m+1 ).
A slightly less sharp result appeared first in a 1979 paper of Reuben Hersh and Tosio Kato in the SIAM Journal of Numerical Analysis [7]. In the same issue, Philip Brenner and Vidar Thomée [3] could weaken the regularity assumptions and show the theorem as stated above. In this paper, they also found error estimates for ||r( t n A) n x − T (t)x|| for x ∈ D(A k ), where 0 ≤ k ≤ m + 1. In 2007, this result was again improved by Mihály Kovács [11] giving error estimates for x ∈ F, where F are interpolation spaces between D(A m+1 ) and X. In all these contributions, no estimates are provided for the constant C. This is being done in the dissertation of William Harrison [6] for the case where r is an A-stable rational Padé-approximation of the exponential or order m. To keep the presentation of the material as readable as possible, we consider here mainly subdiagonal rational Padé-approximations r m of order m and highlight the case n = 1. In particular, we will show that there are constants C m which are rapidly decreasing to zero as m → ∞ such that ||r m (tA)x − T (t)x|| ≤ M C m t m+1 ||A m+1 x|| for all t ≥ 0 and x ∈ D(A m+1 ).
2. Preliminaries. Let r = P Q be an A-stable rational approximation to the exponential function of order m; i.e., P and Q are polynomials with p := deg(P ) ≤ deg(Q) =: q, and (i) |r(z) − e z | ≤ C m |z| m+1 for |z| sufficiently small, and (ii) |r(z)| ≤ 1 for Re(z) ≤ 0. It is a well-known result of Padé [14] that m ≤ p+q for all rational approximations to the exponential function. The rational approximations of maximal order m = p + q are called Padé approximations. They are of the form r = P Q , where Moreover, for every Padé approximation r(z) = P (z) Q(z) of the exponential of order m = p + q we have that (see, for example, [15], Section 75 (Die Exponentialfunktion), or [18]). As shown in [4], Padé approximations are A-stable if and only if q − 2 ≤ p ≤ q. A rational Padé approximation r(z) = P (z) Q(z) is called subdiagonal if p = q − 1. In particular, a subdiagonal Padé approximation is always A-stable, of odd approximation order m = 2q −1 = 2p+1, and the polynomial Q(·) has q distinct roots λ i with Re(λ i ) > 0 such that Moreover, if q is even, then all roots λ i are pairwise complex conjugates, and if q is odd the the largest root λ q is real and all the other roots pairwise complex conjugates. In particular, If q ≤ 15 and q − 2 ≤ p ≤ q, then p ≤ (Re(λ 1 ) · . . . · Re(λ q )) 1/q . If 16 ≤ q ≤ 28 and q − 2 ≤ p ≤ q, then p − 1 ≤ (Re(λ 1 ) · . . . · Re(λ q )) 1/q . For a proof, see [18]. Finally, a subdiagonal Padé approximation r(z) = P (z) Q(z) has the representation In particular, if r is a subdiagonal Padé approximation of order m = 2q − 1 and if A is the generator of a bounded strongly continuous T (t), then it follows from the Hille-Phillips functional calculus that for all t > 0 3. Approximations of semigroups without scaling and squaring. We now come to the main result of this paper. Our goal is to approximate semigroups T (t) in terms of sums of resolvents of the form (2), where r is an appropriately chosen subdiagonal Padé approximation of order m = 2q − 1.

FRANK NEUBRANDER, KORAYÖZER AND TERESA SANDMAIER
It can be shown that [12] or [16], Lemma III.6). Thus for 1 ≤ k ≤ m and x ∈ D(A k+1 ), k consecutive integrations by parts give As a consequence of (3), one obtains for t ≥ 0 and x ∈ D(A k+1 ). In order to estimate I k [α − H t ] L 1 (R + ) , one may use the following Fourier representation of I k [α − H t ] (for details, see [12] or [16], Lemma III.6): for all s ∈ (0, ∞), where From [16], the representation Recall that the complex inversion formula (see [17], Chapter II, Theorem 7.6a) states that if α ∈ N BV 0 [0, ∞) and Then it follows from (6) and the complex inversion formula that Using the A-stability of r as well as Cauchy's Theorem, one obtains

RATIONAL APPROXIMATIONS OF SEMIGROUPS 5309
With an induction argument, one can show that (7) implies (5) and thus .

Simple substitutions yield
where Recall that if f ∈ L 2 (R) and g : s → sf (s) ∈ L 2 (R), then Carlson's inequality implies that f ∈ L 1 (R) and . For an absolutely continuous function g ∈ L 2 with g ∈ L 2 , it is well known that 2 . Furthermore, Parseval's identity, F[g] 2 = g 2 for g ∈ L 2 (R), shows that 2 . Using these properties and inequalities, it follows from (8) that Now, returning to (4), we obtain where It is advantageous to use the following representation due to O. Perron [15] (see also [18]): where z ∈ C and z = λ 1 , . . . , λ q . Thus In order to estimate h(s), we use that . Thus In order to estimate Ψ m 2 2 , the following representation of the Beta function from [1] (Section 11.21) is needed. For p, q > −1 and Γ(n + 1) = n!,
If 0 ≤ k ≤ m−1, the previous proof can be modified to get estimates for constants C m,k such that for all t ≥ 0 and x ∈ D(A k+1 ) (see [6]). As long as q = m − p ≤ k < m, the constants C m,k still converge rapidly to zero as m gets larger, although the speed of convergence slows down with decreasing k. If k < q, then the situation becomes less clear and it remains an open problem whether or not there is a sequence r m of rational Padé approximations r m such that r m (tA)x → T (t)x as m → ∞ uniformly on compacts for all x ∈ D(A).
In order to translate Theorem 3.1 into the language of Laplace transforms, the following remark is useful.
for Re(λ) > 0, and let A be the generator of a bounded, strongly continuous semigroup T (t) on some Banach space Z with resolvent R(λ, A) for Re(λ) > 0. Then the following three problems are equivalent.
Proof. The implication (I) =⇒ (II) follows from the fact that t → T (t)z ∈ C b ([0, ∞), Z) and R(λ, A)z = ∞ 0 e −λt T (t)z dt for all z ∈ Z; i.e., the resolvent of the operator A is the Laplace transform of the operator semigroup T (t) (see [2], [5]). The implication (II) =⇒ (III) holds because (a) the shift semigroup is bi-continuous with generator A = d ds on F = C b ([0, ∞), X), and (b) all approximation results for strongly continuous semigroups extend to bi-continuous ones (The shift semigroup is not strongly continuous in C b ([0, ∞), X) with respect to the norm topology but only with respect to the topology of uniform convergence on compact subsets of [0, ∞). For a discussion and further references on "bi-continuous semigroups," see [8]). Finally, the implication (III) =⇒ (I) is due to the observation that the Corollary 1. Let P, Q be given by (1) and let r(z) = P (z) Q(z) = b1 λ1−z + b2 λ2−z +· · ·+ bq λq−z be the subdiagonal Padé approximation scheme of the exponential of order m = 2q − 1, where λ i (1 ≤ i ≤ q) are the q distinct roots of the polynomial Q and where the constants C r are as in Theorem 3.1.
As immediate consequence of the Hille-Phillips functional calculus in combination with (12) - (16) and Proposition 1 one obtains the following approximation results.