ON SUBDIAGONAL RATIONAL PAD´E APPROXIMATIONS AND THE BRENNER-THOM´EE APPROXIMATION THEOREM FOR OPERATOR SEMIGROUPS

. The computational powers of Mathematica are used to prove poly-nomial identities that are essential to obtain growth estimates for subdiagonal rational Pad´e approximations of the exponential function and to obtain new estimates of the constants of the Brenner-Thom´ee Approximation Theorem of Semigroup Theory.


1.
Introduction. The main result of this paper sharpens the estimate given in [13] regarding the size of the constant C m in the following key result on the approximation of strongly continuous semigroups.
This result appeared first in a 1979 paper of Reuben Hersh and Tosio Kato in the SIAM Journal of Numerical Analysis [8] (for the more general Chernoff Product formula, see [6]). In the same issue, Philip Brenner and Vidar Thomée [3] weakened the regularity assumptions and showed the theorem as stated above. Also, in [3] error estimates were given 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 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 were provided for the constant C m . This was done first in [4] and [13] for subdiagonal rational Padé-approximations r m (where [4] applies some of the estimates given in Section 1 of this paper to show that, for all x ∈ D(A α ) (α > 1 2 ), lim m→∞ r m (tA)x = T (t)x uniformly for t in compact intervals). The same estimates allow us to revisit and significantly improve the estimates concerning C m that were given in [13]. In particular, we will show that for subdiagonal Padé approximations r m with m ≥ 1 the following estimate holds.
To get started, let r m = P Q be an A -stable rational approximation to the exponential function of order m ≥ 1; i.e., P and Q are polynomials with p := deg(P ) ≤ deg(Q) =: q, and 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 m = P Q , where Moreover, for every Padé approximation r m (z) = P (z) Q(z) of the exponential of order m = p + q, (see, for example, [15], Section 75 (Die Exponentialfunktion), or [17]). As shown in [5], 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 and of odd approximation order m = 2q − 1.
for all s ∈ R. However, the last statement holds since d[2j, q] −d[2j, q] ≥ 0 for all 0 ≤ j ≤ q − 1. This proves statement (a). To prove (b), observe that (see also Theorem 3.3 in [5]). It follows from and where we setb[j, q] := 0 if j > q − 1. It can be easily seen thatẽ[n, q] = 0 if n is odd. For 0 ≤ j < q and n = 2j the identitỹ can be proven with the following Mathematica code. It follows from (4) that for all 0 ≤ j ≤ q − 1. This proves statement (b). Finally, statement (c) follows from and the fact that |r(z)| ≤ 1 for Re(z) ≤ 0 (A -stability). Proof. The first part of the proof follows exactly the proof given by Kovács [10] up to shortly after (13) below. For further reference, see also Kovács and Neubrander [12]; for the bi-continuous case, see Jara [9]. The new part of the proof starts with the Perron representation of rational Padé approximation in (14).

Remark 1. It follows from the above that |r
To begin, if r = P Q is a subdiagonal Padé approximation of the exponential, then all the distinct poles λ i of r lie in the right half-plane. By using partial fractions, for Re(z) ≤ 0, where the λ i are the roots of Q(z) and Then G 0 is a Banach algebra with norm f 0 := α var = V α (∞). Furthermore, the Hille-Phillips functional calculus (see [10]) provides the framework and justification to replace z ∈ C with Re(z) ≤ 0 by a suitable operator A in (6). Suppose A is the generator of a bounded, strongly continuous semigroup and r is an A -stable rational approximation of the exponential, then the Hille-Phillips functional calculus along with (6) implies that Since α n (0) − H t (0) = α n (∞) − H t (∞) = 0, integration by parts yields for all x ∈ X. If x ∈ D(A) then s → T (s)x is continuously differentiable with It can be shown that [10] or Lemma III.6 in [16]). Thus, for 1 ≤ k ≤ m and x ∈ D(A k+1 ), k consecutive integrations by parts yield As a consequence of (7), one obtains for t ≥ 0, n ∈ N, and x ∈ D(A k+1 ).
Working towards an estimate for h n 1 2 L 2 (R) , first recall that h n (s) = g n (s) · f n (s) which implies h n (s) = g n (s)f n (s) + f n (s)g n (s). Since |a + b| 2 ≤ 2|a| 2 + 2|b| 2 , it follows that h n