# Rational approximation of holomorphic semigroups revisited

Charles Batty  St. John’s College, University of Oxford, Oxford OX1 3JP,
England<EMAIL_ADDRESS>,  Alexander Gomilko  Faculty of
Mathematics and Computer Science
Nicolas Copernicus University
Chopin Street 12/18
87-100 Toruń, Poland<EMAIL_ADDRESS>and  Yuri Tomilov  Institute of
Mathematics
Polish Academy of Sciences
Śniadeckich 8
00-656 Warsaw, Poland
and
Faculty of Mathematics and Computer Science
Nicolas Copernicus University
Chopin Street 12/18
87-100 Toruń, Poland<EMAIL_ADDRESS>
###### Abstract.

Using a recently developed $\mathcal{H}$-calculus we propose a unified
approach to the study of rational approximations of holomorphic semigroups on
Banach spaces. We provide unified and simple proofs to a number of basic
results on semigroup approximations and substantially improve some of them. We
show that many of our estimates are essentially optimal, thus complementing
the existing literature.

###### Key words and phrases:

Rational approximation, holomorphic semigroup, functional calculus, Banach
space

###### 2010 Mathematics Subject Classification:

Primary 47A60, 41A20, 41A25; Secondary 47D03, 30E10, 65J08

This work was partially supported by an NCN grant UMO- 2023/49/B/ST1/01961.

##  1\. Introduction

The theory of rational appoximations of operator semigroups is an established
area intimately related to the numerical analysis of PDEs, and also of value
in probability theory and function theory. The paper offers a new, functional
calculus approach to the study of rational approximations arising from
discretisation of an abstract Cauchy problem

(1.1) $x^{\prime}(s)=-Ax(s),\qquad s\in[0,t],\quad x(0)=x_{0},$

on a Banach space $X$, where $A$ belongs to the set of sectorial operators
${\rm Sect}(\theta)$ on $X$ of angle $\theta\in[0,\pi/2)$. Since $-A$
generates a bounded holomorphic $C_{0}$-semigroup $(e^{-tA})_{t\geq 0}$, mild
solutions to (1.1) are given by $x(s)=e^{-sA}x_{0},x_{0}\in X$. When
considering a division of $[0,t]$ into equal time steps $t/n$ and discretising
(1.1) accordingly, one approximates a bounded holomorphic $C_{0}$-semigroup
$(e^{-tA})_{t\geq 0}$ by $\\{r^{n}(tA/n):n\in\mathbb{N},t\geq 0\\}$ as
propagator, where $r$ is an appropriate rational function.

There are two main problems arising in the study of rational approximations
$r^{n}(tA/n)$ and their generalizations. The first is to find the best
possible operator norm bounds for $r^{n}(tA/n)$ and, ideally, to show that
$\sup_{n\geq 1,t\geq 0}\|r^{n}(tA/n)\|<\infty$, i.e. to prove stability of
$r^{n}(tA/n)$. If the stability does not hold, then the convergence of
$r^{n}(tA/n)x$ to $e^{-tA}x$ as $n\to\infty$ for all $t\geq 0$ may not take
place for all $x\in X$. However sharp bounds for $\|r^{n}(tA/n)\|$ help to
deal with the second problem of obtaining the convergence of
$e^{-tA}x-r^{n}(tA/n)x$ for, at least, sufficiently regular initial data $x$,
and to equip this convergence with optimal approximation rates.

The two problems above lead to the two corresponding classes of rational
functions which are basic in numerical analysis. To study stability of
rational approximations of $e^{-tA}$ it is natural and necessary (even if
$X=\mathbb{C}$) to restrict attention to $\mathcal{A}(\psi)$-stable rational
functions $r$, that is $r$ satisfying $|r(\lambda)|\leq 1$ for $\lambda$ from
$\Sigma_{\psi}:=\\{\lambda\in\mathbb{C}:|{\rm arg}\,\lambda|<\psi\\}$ with
$\psi\in(0,\pi/2]$. If $\psi=\pi/2$, then $r$ is said to be
$\mathcal{A}$-stable. The study of the approximation rates requires dealing
with rational functions which approximate $e^{-tz}$ at zero well enough: a
rational function $r$ is said to be an approximation of order $q\in\mathbb{N}$
to the exponential $e^{-z}$ if

(1.2) $e^{-z}-r(z)=\mbox{O}(|z|^{q+1}),\qquad z\to 0,$

or, in other words, $r^{(k)}(0)=(-1)^{k}$ for all $0\leq k\leq q$.

Note that the approximation problems for (1.1) are often considered with
variable step sizes $\mathcal{K}_{n}=\\{k_{j}\\}_{j=1}^{n}\subset(0,\infty)$,
$k_{1}+\cdots+k_{n}=t$, and it is of interest to obtain norm bounds for
$r(k_{1}A)\cdots r(k_{n}A)$ independent of $t$. While approximation rates can
be studied in such a setting as well, and results similar to the case of
constant steps can also be obtained (see [27, p. 95, Eq. (7)]), most of the
literature has concentrated on approximation rates for constant step sizes,
and we also follow this line of research.

Rational approximations of $(e^{-tA})_{t\geq 0}$ can be treated by variety of
means. One efficient approach, not requiring precise information on the
structure of $A$, relies on employing functional calculi for $A$. In [21]
Hersh and Kato proposed an approach to the study of rational approximations of
bounded $C_{0}$-semigroups on Banach spaces based on the Hille-Phillips
calculus, and thus on function estimates in the algebra of Laplace transforms
of bounded measures on $\mathbb{R}_{+}$. This direction was further developed
by Brenner and Thomee in [8] and a number of subsequent papers, see [5], [11],
[15], [16],[22], and [23] as examples, and many relevant approximation bounds
got their optimal form. However, in the setting of (sectorially) bounded
holomorphic semigroups the estimates offered by the Hille-Phillips calculus
appear to be rather crude. Thus, the existing arguments were mostly based on
case by case studies via the holomorphic Riesz-Dunford calculus. The arguments
relied on appropriate choices of integration contours and estimates for Riesz-
Dunford integrals. They were rather technical, and had to be adjusted to each
problem accordingly.

The aim of the paper is twofold. First, using a new functional calculus, we
provide a unified approach to a variety of norm estimates for rational
approximations of holomorphic semigroups obtained in the literature, and as a
byproduct, improve some of them. Given $A\in{\rm
Sect}(\theta),\theta\in[0,\pi/2)$, we use a bounded homomorphism from an
appropriate algebra of holomorphic functions $\mathcal{H}_{\psi}$ on a sector
$\Sigma_{\psi}$, $\psi\in(\theta,\pi/2]$, constructed in [6] and called
$\mathcal{H}$-calculus, see Section 2 for more details. The
$\mathcal{H}$-calculus is compatible with the standard holomorphic functional
calculus, and since all the relevant rational functions arising in our
considerations belong to $\mathcal{H}_{\psi}$, it reduces the study of
rational approximations to suitable $\mathcal{H}_{\psi}$-norm estimates. In
this way we develop a technique for the $\mathcal{H}_{\psi}$-norm estimates of
rational functions based on fine control of their oscillations along the rays
and sharp asymptotic estimates of approximation errors in a sector around
zero.

In particular, using the $\mathcal{H}$-calculus, we prove two stability
estimates for $\mathcal{A}(\psi)$-stable rational approximations $r$ and
variable step sizes $\mathcal{K}_{n}=(k_{j})_{j=1}^{n}$. The first one,
originally due to Bakaev [3, Theorem 2.1] (see also [26, Corollary 1.2]),
provides a general operator norm estimate for
$P_{{\mathcal{K}}_{n},r}(A)=r(k_{1}A)\cdots r(k_{n}A)$ in terms of the maximum
ratio of $k_{j}$. If $\max_{j}k_{j}\leq C\min_{j}k_{j}$ for some $C>0$, in
particular if all $k_{j}$ are the same, then the estimate shows stability of
$P_{{\mathcal{K}_{n}},r}(A)$, and, if $r$ approximate $e^{-z}$ with certain
order, this leads to convergence of $P_{{\mathcal{K}}_{n},r}$ on $X$ as
remarked in [27, p. 95]. The result was preceded by important stability
estimates in [9] and [26] yielding stability for the family
$\\{r^{n}(tA/n):n\in\mathbb{N},t\geq 0\\}$. Our second estimate, which is in a
weaker form due to Bakaev [2] and Palencia [26, Theorem 1] (see also [4] where
the result is reproved), shows stability of $P_{{\mathcal{K}}_{n},r}(A)$ if
$|r(\infty)|<1$. Both authors assume that $r$ is a first order approximation
to the exponential, their proofs are rather involved and they rely essentially
on this assumption. The $\mathcal{H}$-calculus approach is direct, and it
allows us to show that the approximation order of $r$ is irrelevant, and thus
the result holds in greater generality. These estimates serve as illustrations
of our technique, and other applications such as the study of multi-step
approximations as in [27] could probably be handled similarly. We leave this
to the interested reader. For an abstract approach to variable stepsize
approximation and other related results one may consult [1, Section II.8].

As the second aim of the paper (related to the first one), we present a new,
$\mathcal{H}$-calculus approach for obtaining rational approximation rates for
holomorphic semigroups and strengthen several well-known statements in the
literature. Let us first recall several basic bounds on approximation rates in
the setting of holomorphic semigroups.

######  Theorem 1.1.

Let $r$ be an $\mathcal{A}(\psi)$-stable rational approximation of order $q$
to the exponential, and let $A\in\operatorname{Sect}(\varphi)$ for some
$\varphi\in[0,\theta)$ and $\theta\in(0,\psi)$. Then there exists a constant
$C=C(\theta)$ such that the following hold for all $s\in[0,q]$,
$n\in\mathbb{N}$ and $t>0$:

  1. (i)

If $|r(\infty)|=1$, then

(1.3) $\|e^{-tA}x-r^{n}(tA/n)x\|\leq
CM_{\theta}(A)t^{s}n^{-s}\|A^{s}x\|,\qquad x\in\operatorname{dom}(A^{s}).$

  2. (ii)

If $|r(\infty)|<1$, then

(1.4) $\|e^{-tA}-r^{n}(tA/n)\|\leq CM_{\theta}(A)/n^{q}.$

Theorem 1.1(i) was apparently first remarked in [8, p. 694] for $s=q$ as a
consequence of arguments given by Le Roux [25], and it was repeated in [9,
Theorems 3 and 4] again for $s=q$ (mentioning also [14] and [28]). See also
[29, Theorem 9.2 and Theorem 9.3], where Theorem 1.1 with $s=q$ in (i) was
given with a full proof. Both parts (i) and (ii) of Theorem 1.1, for integer
$s$ and invertible $A$ were proved in [24, Theorem 4.2] and [24, Theorem 4.4],
while in [27, p. 95] it was noted that that the argument in [24] can be
adjusted to get rid of invertibility of $A$ and to consider even variable step
approximations, see also [20, Theorem 2.1]. (Concerning lifting the
invertibility of $A$ see the remark following [9, Theorem 5]). Generalisations
of Theorem 1.1 for invertible $A$, integer $s$ and variable time steps were
obtained in [31, Theorem 2.1 and Theorem 2.6]. (See also [31] and [32] for
other related results on rational approximation.) Note [13, Theorem 2.2 and
Theorem 2.5], where Theorem 1.1 for integer $s$ was obtained assuming that $A$
admits a bounded $H^{\infty}$-calculus. Finally, Theorem 1.1 in the present
general formulation was proved in [17, Theorem 9.2.3 and Corollary 9.2.9].

In the form of Theorem 1.1 (with varying generality) the result on rational
approximation of holomorphic semigroups has been functioning in the numerical
analysis community since the $1980$s, and it became standard (e.g. in [22]).
In this paper, we show that Theorem 1.1(i) can, in fact, be substantially
improved up to its optimal form. To this aim we introduce the notion of
$\mathcal{A}(\psi,m)$-stability of a rational function. Let $\psi\in(0,\pi/2]$
and let $r$ be a non-constant $\mathcal{A}(\psi)$-stable rational function.
Then $r(\infty)=\lim_{z\to\infty}r(z)$ exists in $\mathbb{C}$,
$|r(\infty)|\leq 1$, and there exist $a\in\mathbb{C}\setminus\\{0\\}$ and
$m\in\mathbb{N}$ such that

(1.5) $r(z)=r(\infty)-\frac{a}{z^{m}}+\mbox{O}(|z|^{-(m+1)}),\qquad
z\to\infty.$

If $r$ satisfies (1.5), then we say that $r$ is _$\mathcal{A}(\psi,m)$
-stable_. Observe that if $\epsilon>0$ and $r_{\epsilon}(z)=r(z+\epsilon)$,
then $r_{\epsilon}$ is $\mathcal{A}(\psi,m)$-stable, with the same value of
$a$ in (1.5). Moreover, if $r$ is $\mathcal{A}(\psi)$-stable, then there
exists $c_{r}>0$ such that

(1.6) $|r^{\prime}(z)|\leq\frac{c_{r}}{(1+|z|)^{2}},\qquad z\in\Sigma_{\psi}.$

Assuming that $r$ is an $\mathcal{A}(\psi,m)$-stable approximation of order
$q$ to $e^{-z}$, with $|r(\infty)|=1$, and setting for simplicity $t=1$, we
prove in Theorem 5.1 that if $s\in[0,mq/(m+1))$, then the approximation rates
$n^{-s}$ in (1.3) can be replaced by strictly sharper rates $n^{-s(1+1/m)}$.
If $s\in[mq/(m+1),q+1]$, we improve again the rates $n^{-s}$ given by (1.3)
and replace them with the rates $n^{-q}$. The latter rates are in fact the
fastest possible when $s$ varies through the whole of $[0,q+1]$ as we show in
in Theorem 7.3. Our results have an especially transparent form if $r$ is one
of the diagonal Padé approximations to $e^{-z}$. Then $m=1$, and Corollary 5.3
yields the maximal convergence rate $n^{-q}$ on quite a large interval
$[q/2,q+1]$ containing, in particular, many integers for large $q$. For
$s\in[q,q+1]$ we do not get any improvements for approximation rates thus
observing a “saturation” effect, and this interval for $s$ seems not to have
been discussed in the literature. These phenomena can be explained by the fact
that the $\mathcal{H}_{\psi}$-norm estimates for $r^{n}(\cdot/n)$ also do not
improve when $s\in[q,q+1]$.

If $|r(\infty)|<1$, then we show that (1.4) holds in a slightly generalised
form with the right hand side replaced by $\|A^{s}x\|/n^{q}$ for $s\in[0,q+1]$
and all $x\in\operatorname{dom}(A^{s})$. One gets (1.4) if $s=0$, but, on the
other hand, $\|A^{s}x\|$ and $\|x\|$ are, in general, not comparable for
$x\in\operatorname{dom}(A^{s})$ if $A$ is not invertible.

An important feature of our abstract approach is that we are able to show that
the obtained approximation rates are optimal in a strong sense. Namely, by an
operator-theoretical construction of independent interest, we prove that if
$r$ is an $\mathcal{A}(\psi,m)$-stable rational approximation of order $q$ to
the exponential then for all $\theta\in[0,\pi/2)$ and $s\in[0,q+1]$, there
exist a Banach space $X$, an operator $A\in\operatorname{Sect}(\theta)$ on
$X$, and a vector $x\in\operatorname{dom}(A^{s})$ such that the lower bounds
for $\|e^{-tA}x-r(A/n)^{n}x\|$ match the approximation rates obtained in
Theorem 5.1 up to a constant depending on $\theta$ and a sequence vanishing at
infinity. Optimality issues are very natural in approximation theory, and they
have been addressed thoroughly for rational approximations of bounded
$C_{0}$-semigroups, but we have not been able to find any similar results in
the literature. A typical result would obtain optimality only in a certain
(operator norm) sense, see e.g. [8, Theorem 4.4], [7, Chapter 5, Theorems 4.3
and 4.4], [23, p. 285], [15, Corollary 7.5] or [16, Prop. 4.8]. This was a
starting point of our research which led to unexpected consequences discussed
above.

### Conventions and notation

In this paper, we will consider an individual rational function $r$. It will
be implicit that some functions and constants that are introduced will depend
on $r$ and other parameters associated solely with $r$ (for example, the order
$q$). They may depend on other parameters which are independent of $r$, and we
will try to make clear which parameters are involved by writing $C=C(n,s)$ or
$C_{n,s}$, if the constants depend on $n$ and $s$.

For readers’ convenience, we list here various functions derived from $r$. The
variables are $n\in\mathbb{N}$, $s\in\mathbb{R}_{+}=[0,\infty)$,
$\gamma\in\mathbb{R}_{+}$, $z\in\mathbb{C}$. The following notation for
relevant functions will be used frequently:

  * •

$r_{n}(z)=r^{n}(z/n)$,

  * •

$r_{n,s}(z)=r_{n}(z)/z^{s}$,

  * •

$\Delta_{n}(z)=e^{-z}-r_{n}(z)$,

  * •

$\Delta_{n,s}(z)=\Delta_{n}(z)/z^{s}$.

We will also denote $\mathbb{C}_{+}:=\\{z\in\mathbb{C}:{\rm Re}\,z>0\\}$,
$\mathbb{D}:=\\{z\in\mathbb{C}:|z|<1\\}$,
$\mathbb{D}_{R}:=\\{z\in\mathbb{C}:|z|<R\\}$, and
$\Sigma_{\theta}:=\\{z\in\mathbb{C}:|{\rm arg}\,z|<\theta\\}$,
$\theta\in(0,\pi)$. The set of holomorphic functions on a domain
$\Omega\subset\mathbb{C}$ will be denoted by ${\rm Hol}(\Omega)$, and
$H^{\infty}(\Omega)$ will stand for the space of $f\in{\rm Hol}(\Omega)$ which
are bounded in $\Omega$.

If $X$ is a Banach space, then the space of bounded linear operators on $X$
will be denoted by $L(X)$, and $\operatorname{Sect}(\theta)$ will stand for
the set of sectorial operators on $X$.

##  2\. The HP-calculus and the $\mathcal{H}$-calculus

Our considerations will be based on the so-called $\mathcal{H}$-functional
calculus constructed recently in [6]. To put the $\mathcal{H}$-calculus into a
proper context, we first recall another functional caclulus, which was used in
the study of rational approximations of semigroups, see [8].

Let ${\rm M}(\mathbb{R}_{+})$ denote the Banach algebra of bounded Borel
measures on $\mathbb{R}_{+}$, and let $\mathcal{LM}(\mathbb{C}_{+})$ be the
Banach algebra of Laplace transforms $\mathcal{L}\mu$, where $\mu\in{\rm
M}(\mathbb{R}_{+})$, equipped with the norm

(2.1) $\|\mathcal{L}\mu\|_{\mathcal{LM}}:=\|\mu\|_{{\rm M}(\mathbb{R}_{+})}.$

If $-A$ generates a bounded $C_{0}$-semigroup $(e^{-tA})_{t\geq 0}$ on a
Banach space $X$, and $M:=\sup_{t\geq 0}\|e^{-tA}\|$, then the Hille-Phillips
(HP-) calculus for $A$ is defined as the bounded homomorphism
$\Psi_{A}:\mathcal{LM}\to L(X)$ given by the operator version of the Laplace
transform:

$\Psi_{A}(\mathcal{L}\mu)x=\mathcal{L}\mu(A)x=\int_{0}^{\infty}e^{-tA}x\,d\mu(t),\qquad
x\in X,\quad\mathcal{L}\mu\in\mathcal{LM},$

so that

(2.2) $\|\mathcal{L}\mu(A)\|\leq M\|\mu\|_{{\rm M}(\mathbb{R}_{+})}.$

The HP-calculus is one of the basic functional calculi used in the literature.
However, it often leads to rather crude estimates and ignores specific aspects
of important subclasses of the class of bounded $C_{0}$-semigroups, such as
sectorially bounded holomorphic semigroups. In the studies of rational
approximations for holomorphic semigroups, another (holomorphic) functional
calculus based on Riesz-Dunford integrals is used frequently. However, in this
case, the arguments depend on delicate choices of contours and involved
estimates which depend on particular contexts.

To produce sharp approximation norm bounds, finer calculi are required. One
such calculus was developed in [6], and we recall some of its features now.
Let $H^{1}(\Sigma_{\psi})$ stand for the Hardy space of functions $f\in{\rm
Hol}(\Sigma_{\psi})$ such that

$\|f\|_{H^{1}(\Sigma_{\psi})}:=\sup_{|\theta|<\psi}\,\int_{0}^{\infty}\left(|f(te^{i\theta})|+|f(te^{-i\theta})|\right)\,dt.$

It was shown in [6, Theorem 4.6] that if $f\in H^{1}(\Sigma_{\psi})$, then the
boundary values $f(te^{\pm i\psi})=\lim_{\varphi\to\pm\psi}f(te^{it\varphi})$
exist for a.e. $t$ and
$\|f\|_{H^{1}(\Sigma_{\psi})}=\|f\|_{L^{1}(\partial\Sigma_{\psi})}$.

The Hardy-Sobolev space $\mathcal{H}_{\psi}$, $\psi\in(0,\pi)$, is defined as
the space of $f\in{\rm Hol}(\Sigma_{\psi})$ such that

$\|f\|_{\mathcal{H}_{\psi,0}}:=\|f^{\prime}\|_{H^{1}(\Sigma_{\psi})}<\infty.$

By [6, Theorem 4.8], if $f\in\mathcal{H}_{\psi}$ then $f\in
H^{\infty}(\Sigma_{\psi})$, $f$ extends continuously to
$\partial\Sigma_{\psi}$, and there exists
$f(\infty)=\lim_{z\to\infty,z\in\Sigma_{\psi}}f(z)$. Note that by the above,

(2.3)
$\|f\|_{\mathcal{H}_{\psi,0}}=\|f^{\prime}\|_{L^{1}(\partial\Sigma_{\psi})}$

Moreover, by [6, Lemma 4.9], $\mathcal{H}_{\psi}$ is a Banach algebra equipped
with the norm
$\|f\|_{\mathcal{H}_{\psi}}:=\|f\|_{\infty}+\|f\|_{\mathcal{H}_{\psi,0}}$. If
$f\in{\rm Hol}(\Sigma_{\theta})$, then

(2.4) $\|f\|_{\mathcal{H}_{\psi}}\leq\|f\|_{\mathcal{H}_{\theta}},\quad
f\in\mathcal{H}_{\theta},\quad 0<\psi<\theta.$

The expression

$\|f\|^{*}_{\mathcal{H}_{\psi}}:=|f(\infty)|+\|f\|_{\mathcal{H}_{\psi,0}},\qquad
f\in\mathcal{H}_{\psi},$

defines an equivalent norm on $\mathcal{H}_{\psi}$, which is not a Banach
algebra norm. In particular, [6, Theorem 4.8(iii)] shows that, if
$f\in\mathcal{H}_{\psi}$ and $f(\infty)=0$, then

(2.5) $\|f\|_{\infty}\leq\|f\|_{\mathcal{H}_{\psi}}.$

The Banach algebras $\mathcal{H}_{\psi}$ for $\psi\in(0,\pi)$ are
isometrically isomorphic to each other, via fractional powers. Let
$\psi_{1},\psi_{2}\in(0,\pi)$ and $\gamma=\psi_{1}/\psi_{2}$. If
$f\in\mathcal{H}_{\psi_{1}}$ and $f_{\gamma}(z)=f(z^{\gamma})$, it is easy to
check that $f_{\gamma}\in\mathcal{H}_{\psi_{2}}$ and
$\|f_{\gamma}\|_{\mathcal{H}_{\psi_{2},0}}=\|f\|_{\mathcal{H}_{\psi_{1},0}}$.
See [6, Lemma 4.9].

Moreover, it is instructive to observe that if $f\in\mathcal{LM}$, then (the
restriction to $\Sigma_{\psi}$ of) $f$ belongs to $\mathcal{H}_{\psi}$ for any
$\psi\in(0,\pi/2)$. Indeed, if $f=\mathcal{L}\mu$ and $|\theta|<\psi$, then

$\displaystyle\int_{0}^{\infty}|f^{\prime}(te^{i\theta})|\,dt\leq\int_{\mathbb{R}_{+}}\int_{0}^{\infty}e^{-t\tau\cos\theta}\,dt\,\tau|\mu|(d\tau)=\frac{1}{\cos\theta}\int_{\mathbb{R}_{+}}|\mu|(d\tau),$

so that

(2.6)
$\displaystyle\|f\|^{*}_{\mathcal{H}_{\psi}}\leq|\mu(\\{0\\})|+\frac{2}{\cos\psi}\int_{(0,\infty)}|\mu|(d\tau)\leq\frac{2}{\cos\psi}\|f\|_{\mathcal{LM}}.$

As shown in [6], the algebras $\mathcal{H}_{\psi}$ allow one to define a
functional calculus for a sectorial operator $A$ on a Banach space $X$, which
takes into account the angle of sectoriality of $A$ and leads to fine
operator-norm estimates for functions of $A$. More precisely, recall that the
operator $A$ on $X$ is said to be sectorial of angle $\theta$ if
$\sigma(A)\subset\overline{\Sigma}_{\theta}$, and for every
$\psi\in(\theta,\pi)$ there exists $M_{\psi}(A)>0$ such that

$M_{\psi}(A):=\sup\\{\|\lambda
R(\lambda,A)\|:\lambda\in\Sigma_{\pi-\psi}\\}<\infty.$

As in the introduction, we let $\operatorname{Sect}(\theta)$ stand for the
class of sectorial operators of angle $\theta$, for $\psi\in[0,\pi)$ and
$\lambda\in\Sigma_{\pi-\psi}$ denote $\rho_{\lambda}(z):=(z+\lambda)^{-1}$. If
$A\in\operatorname{Sect}(\theta)$, $\psi\in(\theta,\pi)$, then by [6, Theorem
8.6] there is a unique bounded homomorphism $\Psi_{A}:\mathcal{H}_{\psi}\to
L(X)$, $\Psi_{A}(f)=f(A)$, such that
$\Psi_{A}(\rho_{\lambda})=(A+\lambda)^{-1},\lambda\in\Sigma_{\pi-\psi}$,
called the $\mathcal{H}$-calculus for $A$. The calculus is given by the
formula:

(2.7)
$f(A)=f(\infty)-\frac{1}{\pi}\int_{0}^{\infty}\left(f^{\prime}(e^{it\psi})+f^{\prime}(e^{-it\psi})\right)\operatorname{arccot}(A^{\gamma}/t^{\gamma})\,dt,$

where the operator kernel $\operatorname{arccot}(A^{\gamma}/t^{\gamma})$ is
defined by a separate formula involving the logarithmic function as explained
in [6, Section 8.2]. From (2.7) it follows that

(2.8)
$\|f(A)\|\leq|f(\infty)|+\frac{M_{\psi}(A)}{2}\|f\|_{\mathcal{H}_{\psi,0}}\leq
M_{\psi}(A)\|f\|_{\mathcal{H}_{\psi}}.$

Thus, if $f(\infty)=0$,

(2.9) $\|f(A)\|\leq\frac{M_{\psi}(A)}{2}\|f\|_{\mathcal{H}_{\psi,0}}.$

The formula (2.7) appears to depend on $\psi$, but the definitions of $f(A)$
agree provided that $\psi\in(\theta,\pi)$ and $f$ is holomorphic on
$\Sigma_{\psi}$. Recall that
$A\in\operatorname{Sect}(\theta),\theta\in[0,\pi/2)$, if and only if $-A$
generates a (sectorially bounded) holomorphic $C_{0}$-semigroup on $X$, so
that the $\mathcal{H}$-calculus applies, in fact, to the negative generators
of holomorphic semigroups.

We emphasize two crucial properties of the $\mathcal{H}$-calculus, which are
implicit in [6] in view of a somewhat indirect approach to the
$\mathcal{H}$-calculus developed there. Note that by [6, Theorems 5.10, 11.1],
the algebras $\mathcal{H}_{\psi}$ are invariant under shifts, i.e., if
$f\in\mathcal{H}_{\psi}$, $\epsilon>0$, and $f_{\epsilon}(z)=f(z+\epsilon)$,
then $f_{\epsilon}\in\mathcal{H}_{\psi}$, and the map $f\mapsto
f_{\varepsilon}$ from $\mathcal{H}_{\psi}$ to $\mathcal{H}_{\psi}$ is bounded.
Thus, it is easy to see that if $A\in\operatorname{Sect}(\theta)$, then
$A+\varepsilon$ has a bounded $\mathcal{H}_{\psi}$-calculus given by

$\Phi_{A+\epsilon}(f)=\Phi_{A}(f_{\epsilon}).$

Since the $\mathcal{H}_{\psi}$-calculus for $A+\varepsilon$ is unique, it
follows that

(2.10) $f(A+\epsilon)=f_{\epsilon}(A).$

Since $-A$ is the generator of a $C_{0}$-semigroup $(T(t))_{t\geq 0}$ on $X$,
and $\mathcal{H}_{\psi}$ contains the functions $e^{-t\cdot}$ for $t\geq 0$,
it is natural to expect that

$e^{-t\cdot}(A)=T(t),\qquad t\geq 0.$

This follows from the fact that if $t\geq 0$ then $e^{-t\cdot}$ is the limit
in $\mathcal{H}_{\psi}$ of $\rho_{1}(t\cdot/n)^{n},\,n\to\infty$, proved in
Remark 5.2 below.

We will be concerned only with the exponential functions $(e^{-tz})_{t\geq 0}$
and rational functions $r$ which are $\mathcal{A}(\psi)$-stable, and functions
derived from these, applied in situations where $\psi\in(0,\pi/2)$, and $A$ is
sectorial of angle $\theta\in[0,\psi)$. The operators $(e^{-tA})_{t\geq 0}$
form a $C_{0}$-semigroup which extends to a bounded holomorphic semigroup on a
sector. The operators $r(A)$ have the natural definition for rational
functions.

##  3\. Estimates for rational approximations of the exponential

###  3.1. Stability estimates

In this section we extend known results about the stability of certain classes
of rational approximations to the exponential function with variable
stepsizes. In the next section we will apply these results to sectorial
operators. For $R>0$, recall the notation
$\mathbb{D}_{R}=\\{z\in\mathbb{C}:|z|<R\\}$.

######  Proposition 3.1.

Let $f$ be a holomorphic function on $\mathbb{D}_{R}$ which is non-constant,
$\theta\in(-\pi,\pi]$, and let $f_{\theta}(t):=|f(te^{i\theta})|,\,0<t<R$.
Then there exist $R_{0}\in(0,R)$ and a finite (possibly empty) set
$\Theta_{f}\subset(-\pi,\pi]$ such that for every
$\theta\in(\pi,\pi]\setminus\Theta_{f}$,

(3.1) $0<|f^{\prime}(te^{i\theta})|\leq
C_{\theta}|f^{\prime}_{\theta}(t)|,\quad t\in(0,R_{0}),\quad$

for some $C_{\theta}>0$.

###### Proof.

By adjusting $R$ if necessary, we may assume that $f^{\prime}$ has no zeroes
in $\mathbb{D}_{R}\setminus\\{0\\}$. From the Taylor series for $f$, there
exist $k\in\mathbb{N}$ and $b=|b|e^{i\varphi}\neq 0$ such that

$f(z)=f(0)+bz^{k}+\mbox{O}(z^{k+1}),\qquad z\to 0,\quad z\in\mathbb{D}_{R},$

and

(3.2) $|f^{\prime}(te^{i\theta})|=k|b|t^{k-1}+\mbox{O}(t^{k}),\qquad t\to
0,\quad\theta\in(-\pi,\pi].$

If $f(0)=0$, then

$f_{\theta}(t)=|b|t^{k}+\mbox{O}(t^{k+1}),\qquad
f^{\prime}_{\theta}(t)=k|b|t^{k-1}+\mbox{O}(t^{k}),$

uniformly in $\theta$, and in view of (3.2), the inequality (3.1) holds for
some $R_{0}>0$ and every $\theta\in(-\pi,\pi]$.

If $f(0)\not=0$, we may assume, without loss of generality, that $f(0)=1$. A
simple calculation reveals that

$f_{\theta}^{2}(t)=1+2|b|t^{k}\cos(\varphi+k\theta)+\mbox{O}(t^{k+1}),\\\ $

uniformly in $\theta$. Hence, for every $\theta\in(-\pi,\pi]$,

$\displaystyle f_{\theta}(t)$ $\displaystyle=1+{\rm O}(t^{k}),$ $\displaystyle
f_{\theta}(t)f^{\prime}_{\theta}(t)$
$\displaystyle=k|b|t^{k-1}\cos(\varphi+k\theta)+{\rm O}(t^{k}).$

Taking into account (3.2), we infer that (3.1) is satisfied for some $R_{0}>0$
and

$\Theta_{f}:=\\{\theta\in(-\pi,\pi]:\,\cos(\varphi+k\theta)=0\\}.\qed$

######  Corollary 3.2.

Let $r$ be a rational function which is non-constant, holomorphic at zero and
bounded at infinity. Then there exist a finite (possibly empty) set
$\Theta_{r}\subset(-\pi,\pi]$ and $R_{1}>R_{0}>0$ such that for every
$\theta\in(-\pi,\pi]\setminus\Theta_{r}$

(3.3) $0<|r^{\prime}(te^{i\theta})|\leq
c_{\theta}|r^{\prime}_{\theta}(t)|,\qquad t\in(0,R_{0})\cup(R_{1},\infty),$

for some $c_{\theta}>0$.

###### Proof.

Employing Proposition 3.1 we infer that there is $R_{0}>0$ such that the
inequality in (3.3) holds for $t\in(0,R_{0})$ and all except finitely many
$\theta$. Passing then to $r^{*}(z):=r(1/z)$ and employing Proposition 3.1
again we obtain $R_{1}>0$ such that the inequality in (3.3) also holds for
$t>R_{1}$ and all except finitely many $\theta$. ∎

A consequence of (3.3) is that $r_{\theta}^{\prime}$ has no zero in
$(0,R_{0})$ or $(R_{1},\infty)$. This has the consequence that if $g$ is a
non-negative measurable function on an interval $I$ contained in $(0,R_{0})$
then

(3.4)
$\int_{I}|r_{\theta}^{\prime}(t)|g(t)\,dt=\left|\int_{I}r_{\theta}^{\prime}(t)g(t)\,dt\right|.$

######  Example 3.3.

We illustrate Proposition 3.1 by considering a Cayley transform. Let
$z_{0}=\tau e^{i\varphi}\in\mathbb{C}_{+}$ and

$r(z)=\frac{z-z_{0}}{z+\overline{z}_{0}},\qquad z\in\mathbb{C}_{+}.$

We will show that for every $\theta\in(-\pi/2,\pi/2)$,

(3.5) $r_{\theta}^{\prime}(t)<0,\quad t\in(0,\tau),\qquad\text{and}\qquad
r_{\theta}^{\prime}(t)>0,\quad t\in(\tau,\infty),$

and for every $\delta\in(0,1)$ there exists $c_{\delta}>0$ such that

(3.6) $|r^{\prime}(te^{i\theta})|\leq
c_{\delta}\frac{|r_{\theta}^{\prime}(t)|}{\cos\theta},\quad
t\in(0,(1-\delta)\tau)\cup((1+\delta)\tau,\infty).$

Indeed, note that

$|r^{\prime}(te^{i\theta})|=\frac{2\tau\cos\varphi}{t^{2}+2t\tau\cos(\theta+\varphi)+\tau^{2}},$

and

$r^{2}_{\theta}(t)=\frac{t^{2}-2t\tau\cos(\theta-\varphi)+\tau^{2}}{t^{2}+2t\tau\cos(\theta+\varphi)+\tau^{2}}.$

Differentiation gives

$r_{\theta}(t)r_{\theta}^{\prime}(t)=\frac{2\tau(t^{2}-\tau^{2})\cos\theta\cos\varphi}{(t^{2}+2t\tau\cos(\theta+\varphi)+\tau^{2})^{2}}=\frac{(t^{2}-\tau^{2})\cos\theta}{t^{2}+2t\tau\cos(\theta+\varphi)+\tau^{2}}|r^{\prime}(te^{i\theta}|.$

Since $r_{\theta}(t)>0$ and $r_{\theta}(t)\leq 1$, this shows that (3.5)
holds. Moreover,

$\displaystyle|r^{\prime}(te^{i\theta})|$
$\displaystyle=\frac{t^{2}+2t\tau\cos(\theta+\varphi)+\tau^{2}}{|t^{2}-\tau^{2}|\cos\theta}|r_{\theta}(t)r^{\prime}_{\theta}(t)|$
$\displaystyle\leq\frac{(t+\tau)^{2}}{|t^{2}-\tau^{2}|\cos\theta}|r^{\prime}_{\theta}(t)|=\frac{t+\tau}{|t-\tau|\cos\theta}|r^{\prime}_{\theta}(t)|,$

so (3.6) holds with $c_{\delta}=2\delta^{-1}-1$.

In Example 3.3, $r\in H^{\infty}(\mathbb{C}_{+})$ and (3.3) holds for all
$\theta\in(-\pi/2,\pi/2)$. In the general case, when $r\in
H^{\infty}(\Sigma_{\psi})$ and $\psi\in(0,\pi/2]$, there may be exceptional
values of $\theta$ for which (3.3) does not hold as the following two examples
show.

######  Example 3.4.

Fix $\varphi\in(0,\pi/2)$ and consider the rational function

$r(z):=\frac{z-e^{-i\varphi}}{z+e^{-i\varphi}}.$

Note that $r\in H^{\infty}(\mathbb{C}_{+})$. If
$z=ie^{-i\varphi}t=te^{i(\pi/2-\varphi)},t>0,$ then $z\in\mathbb{C}_{+}$ and

$|r(ie^{-i\varphi}t)|=\left|\frac{it-1}{it+1}\right|=1,\qquad t>0.$

Hence

$\frac{d}{dt}|r(ie^{-i\varphi}t)|\equiv 0,\quad t>0,$

and so (3.3) does not hold for $\theta=\pi/2-\varphi$.

######  Example 3.5.

For a fixed $\varphi\in(-\pi/2,\pi/2)$ define (a “shifted Cayley transform”)

$r(z):=\frac{z+1-e^{-i\varphi}}{z+1+e^{i\varphi}}$

and note that $\|r\|_{{H}^{\infty}(\mathbb{C}_{+})}=1$ and $r(\infty)=1$.
Moreover, for every $t>0$,

$|r(t)|^{2}=\frac{(t+1)^{2}-2(t+1)\cos\varphi+1}{(t+1)^{2}+2(t+1)\cos\varphi+1},$

By simple calculations,

$\left.\frac{d}{dt}|r(t)|\right|_{t=0}=0,\qquad|r^{\prime}(0)|=\frac{2}{|1+e^{i\varphi}|^{2}}\not=0.$

This implies that (3.3) does not hold for $\theta=0$ and small $t>0$.

Now we turn our attention towards rational approximations schemes involving
products of a single function over variable stepsizes. Let $r$ be an
$\mathcal{A}(\psi)$-stable rational function, and let $\theta\in(0,\psi)$. Let
$\mathcal{K}_{n}=(k_{j})_{j=1}^{n}$ be a sequence in $(0,\infty)$ of length
$n$, $K_{0}=\min_{j}\,\\{k_{j}\\}$ and $K_{1}=\max_{j}\,\\{k_{j}\\}$. Let

(3.7) $P_{\mathcal{K}_{n},r}(z):=\prod_{j=1}^{n}r(k_{j}z),\qquad
z\in\Sigma_{\psi}.$

By (1.6) and (2.3), we have $P_{{\mathcal{K}}_{n},r}\in\mathcal{H}_{\theta,0}$
and

(3.8)
$\|P_{{\mathcal{K}}_{n},r}\|_{\mathcal{H}_{\theta,0}}=\|P_{{\mathcal{K}}_{n},r}^{\prime}\|_{L^{1}(\partial\Sigma_{\theta})}.$

We will give two estimates for the right-hand side of (3.8), firstly an
estimate which is valid for all $\mathcal{A}(\psi)$-stable rational functions,
but the estimate depends on $\mathcal{K}_{n}$ as well as $r$ and $\theta$, and
secondly an estimate which is independent of $\mathcal{K}_{n}$ provided that
$|r(\infty)|<1$.

In the statements and proofs of these two theorems, the notation $C$ denotes a
constant which may depend on $r$, $\psi$ and $\theta$, but does not depend on
$\mathcal{K}_{n}$. The value of $C$ may change from place to place.

######  Theorem 3.6.

Let $r$ be an $\mathcal{A}(\psi)$-stable rational function, where
$\psi\in(0,\pi/2]$, and let $\theta\in(0,\psi)$. There is a constant $C$ such
that

(3.9) $\|P_{\mathcal{K}_{n},r}\|_{\mathcal{H}_{\theta,0}}\leq
C\left(1+\log\left(\frac{K_{1}}{K_{0}}\right)\right)$

for all finite sequences $\mathcal{K}_{n}$ in $(0,\infty)$.

###### Proof.

Let $R_{0}$, $R_{1}$ and $\Theta_{r}$ be as in Corollary 3.2, and

$a:=\frac{R_{0}}{K_{1}},\qquad b:=\frac{R_{1}}{K_{0}}.$

By the maximum principle, it suffices to consider $\theta\in(0,\psi)$,
sufficiently close to $\psi$ that (3.3) holds for $\pm\theta$.

Let $\mathcal{K}_{n}$ and $P_{\mathcal{K}_{n},r}$ be as above. Since
$\|P_{\mathcal{K}_{n},r}\|_{H^{\infty}(\Sigma_{\psi})}\leq 1$, there is a
constant $C$ (depending only on $\theta$ and $\psi$) such that

(3.10) $|P_{{\mathcal{K}}_{n},r}^{\prime}(z)|\leq\frac{C}{|z|},\qquad
z\in\overline{\Sigma}_{\theta},\,z\neq 0.$

Using (3.3), (3.10) and (3.4), we have

$\displaystyle\hskip-15.0pt\int_{0}^{\infty}\left|P_{{\mathcal{K}}_{n},r}^{\prime}(te^{i\theta})\right|\,dt$
$\displaystyle\leq\int_{0}^{a}\sum_{j=1}^{n}k_{j}|r^{\prime}(k_{j}te^{i\theta})|\prod_{l=1,\;l\not=j}^{n}|r(k_{l}te^{i\theta})|\,dt$
$\displaystyle\hskip
15.0pt\hbox{}+\int_{a}^{b}\left|P_{{\mathcal{K}}_{n},r}^{\prime}(te^{i\theta})\right|\,dt+\int_{b}^{\infty}\sum_{j=1}^{n}k_{j}|r^{\prime}(k_{j}te^{i\theta})|\prod_{l=1,\;l\not=j}^{n}|r(k_{l}te^{i\theta})|\,dt$
$\displaystyle\leq
C\Bigg{|}\int_{0}^{a}\sum_{j=1}^{n}\frac{d}{dt}|r(k_{j}te^{i\theta})|\prod_{l=1,\;l\not=j}^{n}|r(k_{l}te^{i\theta})|\,dt\Bigg{|}$
$\displaystyle\hskip
15.0pt\hbox{}+C\int_{a}^{b}\frac{dt}{t}+C\Bigg{|}\int_{b}^{\infty}\sum_{j=1}^{n}\frac{d}{dt}|r(k_{j}te^{i\theta})|\prod_{l=1,\;l\not=j}^{n}|r(k_{l}te^{i\theta})|\,dt\Bigg{|}$
$\displaystyle=C\Bigg{|}\int_{0}^{a}\frac{d}{dt}\Big{|}\prod_{j=1}^{n}r(k_{j}te^{i\theta})\Big{|}\,dt\Bigg{|}+C\log\left(\frac{b}{a}\right)+C\Bigg{|}\int_{b}^{\infty}\frac{d}{dt}\Big{|}\prod_{j=1}^{n}r(k_{j}te^{i\theta})\Big{|}\,dt\Bigg{|}$
$\displaystyle\leq C+C\log\left(\frac{R_{1}K_{1}}{R_{0}K_{0}}\right)\leq
C\left(1+\log\left(\frac{K_{1}}{K_{0}}\right)\right).$

The equality sign is valid by the sign argument in (3.4). The same argument
also works for $-\theta$, and hence (3.9) follows. ∎

######  Theorem 3.7.

Let $r$ be an $\mathcal{A}(\psi)$-stable rational function with
$|r(\infty)|<1$, and let $\theta\in(0,\psi)$. There exists $C$ such that

(3.11) $\|P_{\mathcal{K}_{n},r}\|_{{\mathcal{H}}_{\theta,0}}\leq C$

for every finite sequence $\mathcal{K}_{n}\subseteq(0,\infty)$.

###### Proof.

Without loss of generality, we may assume that $R_{0}=1$; otherwise consider
$r(zR_{0})$ in place of $r(z)$, noting that this change of variable does not
change $\|P_{\mathcal{K}_{n},r}\|_{\mathcal{H}_{\theta,0}}$. Thus

(3.12) $0<|r^{\prime}(te^{i\theta})|\leq C|r^{\prime}_{\theta}(t)|,\qquad
t\in(0,1).$

By the maximum principle there exists $\kappa_{r}\in(0,1)$ such that

(3.13) $|r(z)|\leq\kappa_{r},\qquad z\in\Sigma_{\theta},\;|z|\geq 1.$

Let $\mathcal{K}_{n}$ be a sequence of length $n$ in $(0,\infty)$. We have
shown in Theorem 3.6 that $P_{\mathcal{K}_{n},r}\in{\mathcal{H}}_{\theta,0}$.
To obtain (3.11), we will use (3.13) to improve the estimate of the
$L^{1}$-norm of $P_{\mathcal{K}_{n},r}$ on the boundary of $\Sigma_{\theta}$,
where $\theta\in(0,\psi)$ and $(\ref{CP1D})$ holds for $\pm\theta$.

Let $\mathcal{K}_{n}=(k_{j})_{j=1}^{n}$, where $0<k_{1}\leq k_{2}\leq\dots\leq
k_{n}$ without loss of generality, and let

(3.14) $t_{0}=0,\;\quad
t_{j}=\frac{1}{k_{n-j+1}},\;j=1,\dots,n,\quad\;t_{n+1}=\infty.$

Let

$S_{r,m}(\mathcal{K}_{n};\theta):=\int_{t_{m-1}}^{t_{m}}\big{|}P^{\prime}_{\mathcal{K}_{n},r}(te^{i\theta})\big{|}\,dt,\qquad
m=1,\dots,n+1.$

Then

(3.15)
$\int_{0}^{\infty}\big{|}P^{\prime}_{\mathcal{K}_{n},r}(te^{i\theta})\big{|}\,dt=\sum_{m=1}^{n+1}S_{r,m}(\mathcal{K}_{n};\theta).$

Taking into account that

$k_{j}t\leq 1,\quad j=1,\dots,n,\quad t\in(0,1/k_{n}),$

and applying (3.14) and (3.4), we have

$\displaystyle S_{r,1}(\mathcal{K}_{n};\theta)$
$\displaystyle\leq\sum_{j=1}^{n}k_{j}\int_{0}^{1/k_{n}}|r^{\prime}(k_{j}e^{i\theta}t)|\prod_{l=1,\;l\not=j}^{n}|r_{\theta}(k_{l}t)|\,dt$
$\displaystyle\leq
C\sum_{j=1}^{n}k_{j}\int_{0}^{1/k_{n}}\left|\frac{d}{dt}r_{\theta}(k_{j}t)\right|\prod_{l=1,\;l\not=j}^{n}|r_{\theta}(k_{l}t)|\,dt$
$\displaystyle=C\left|\int_{0}^{1/k_{n}}\frac{d}{dt}|P_{\mathcal{K}_{n},r}(te^{i\theta})|\,dt\right|\leq
C.$

For $2\leq m\leq n$, we have

$\displaystyle k_{j}t$ $\displaystyle\leq 1,\quad j=1,\dots,n-m+1,\quad
t\leq\frac{1}{k_{n-m+1}},$ $\displaystyle k_{l}t$ $\displaystyle\geq 1,\quad
l=n-m+2,\dots,n,\quad t\geq\frac{1}{k_{n-m+2}}.$

Using (3.12), (1.6), (3.13) and (3.4), we obtain

$\displaystyle S_{r,m}(\mathcal{K}_{n};\theta)$
$\displaystyle\leq\sum_{j=1}^{n}k_{j}\int_{1/k_{n-m+2}}^{1/k_{n-m+1}}|r^{\prime}(k_{j}e^{i\theta}t)|\prod_{l=1,\;l\not=j}^{n}|r_{\theta}(k_{l}t)|\,dt$
$\displaystyle\leq
C\kappa_{r}^{m-1}\sum_{j=1}^{n-m+1}k_{j}\int_{1/k_{n-m+2}}^{1/k_{n-m+1}}\left|\frac{d}{dt}r_{\theta}(k_{j}t)\right|\prod_{l=1,\;l\not=j}^{n-m+1}|r_{\theta}(k_{l}t)|\,dt$
$\displaystyle\hskip
15.0pt\hbox{}+\sum_{j=n-m+2}^{n}k_{j}\int_{1/k_{n-m+2}}^{1/k_{n-m+1}}|r^{\prime}(k_{j}e^{i\theta}t)|\prod_{l=1,\;l\not=j}^{n}|r_{\theta}(k_{l}t)|\,dt$
$\displaystyle=C\kappa_{r}^{m-1}\left|\int_{1/k_{n-m+2}}^{1/k_{n-m+1}}\frac{d}{dt}|P_{n-m+2}(te^{i\theta})|\,dt\right|$
$\displaystyle\hskip
12.0pt\hbox{}+c_{r}\kappa_{r}^{m-2}\sum_{j=n-m+2}^{n}\int_{1/k_{n-m+2}}^{1/k_{n-m+1}}\frac{k_{j}}{(1+k_{j}t)^{2}}\,dt$
$\displaystyle\leq C\kappa_{r}^{m-1}+c_{r}(m-1)\kappa_{r}^{m-2}.$

For $m=n+1$, we have

$\displaystyle S_{r,n+1}(\mathcal{K}_{n};\theta)$
$\displaystyle\leq\sum_{j=1}^{n}k_{j}\int_{1/k_{1}}^{\infty}|r^{\prime}(k_{j}e^{i\theta}t)|\prod_{l=1,\;l\not=j}^{n}|r_{\theta}(k_{l}t)|\,dt$
$\displaystyle\leq
c_{r}\kappa_{r}^{n-1}\sum_{j=1}^{n}\int_{1/k_{1}}^{\infty}\frac{k_{j}}{(1+k_{j}t)^{2}}\,dt\leq
c_{r}n\kappa_{r}^{n-1}.$

Thus,

$\displaystyle\int_{0}^{\infty}\big{|}P^{\prime}_{\mathcal{K}_{n},r}(te^{i\theta})\big{|}\,dt$
$\displaystyle\leq
C+\sum_{m=2}^{n}(C\kappa_{r}^{m-1}+c_{r}(m-1)\kappa_{r}^{m-2})+c_{r}n\kappa_{r}^{n-1}$
$\displaystyle=C\sum_{m=1}^{n}\kappa_{r}^{m-1}+c_{r}\sum_{m=1}^{n}m\kappa_{r}^{m-1}\leq\frac{C}{1-\kappa_{r}}+\frac{c_{r}}{(1-\kappa_{r})^{2}}.$

and completely similarly

$\displaystyle\int_{0}^{\infty}\big{|}P^{\prime}_{\mathcal{K}_{n},r}(te^{-i\theta})\big{|}\,dt\leq\frac{C}{1-\kappa_{r}}+\frac{c_{r}}{(1-\kappa_{r})^{2}}.$

Now (3.11) is immediate. ∎

###  3.2. Approximation rate estimates

In this section, we prove several auxiliary lemmas about
$\mathcal{A}(\psi,m)$-stable rational approximations of the exponential
function, which lead to the improved version of Theorem 1.1 by means of the
$\mathcal{H}$-calculus. Recalling the definition of $A(\psi,m)$-stable
rational approximations from the Introduction, we start with a simple example
of an $\mathcal{A}(\pi/6,2)$-stable function which approximates $e^{-z}$ with
$q=1$ and $|r(\infty)|=1$, illustrating this new sharper notion of stability.

######  Example 3.8.

Let

$r(z)=\frac{1-4z^{3}}{1+z+4z^{3}},\quad z\in\mathbb{C}_{+}.$

Then $r(\infty)=-1$, and

$r(z)+1=\frac{2+z}{1+z+4z^{3}}=\frac{1}{4z^{2}}+{\rm O}(|z|^{-3}),\qquad
z\to\infty.$

Moreover,

$r(0)=1,\qquad r^{\prime}(0)=-1,\qquad r^{\prime\prime}(0)=2,$

so $r$ approximates $e^{-z}$ with exact order $q=1$. The poles of $r$ are at
$-1/2$ and at

$\frac{1\pm i\sqrt{7}}{4}=\frac{e^{\pm
i\varphi}}{\sqrt{2}},\quad\varphi=\arctan\sqrt{7}\in(\pi/3,\pi/2).$

Thus $r$ is holomorphic on $\Sigma_{\pi/6}$. If $t>0$ and
$z=te^{i\pi/6}=(\sqrt{3}+i)t/2$, then $z^{3}=it^{3}$ and

$\displaystyle|r(te^{\pm i\pi/6})|^{2}$
$\displaystyle=\frac{16t^{6}+1}{(4t^{3}+t/2)^{2}+(\sqrt{3}t/2+1)^{2}}$
$\displaystyle=\frac{16t^{6}+4}{16t^{6}+4t^{4}+t^{2}+\sqrt{3}t+4}<1.$

Thus $r$ is $\mathcal{A}(\pi/6,2)$-stable.

For the study of approximation rates for rational approximations
$r(t\cdot/n)^{n}$ to the exponential, it is crucial to have sharp estimates
for the size of the rational function $r$ near infinity. The following lemma
provides such an estimate in terms the Taylor expansion of $r$ at infinity.

######  Lemma 3.9.

Let $r$ be an $\mathcal{A}(\psi,m)$-stable rational function with
$|r(\infty)|=1$. Then for every $\theta\in(0,\psi)$ there exist
$b_{1},b_{2}>0$ and $R\geq 1$, such that

(3.16) $e^{-b_{2}/|z|^{m}}\leq|r(z)|\leq e^{-b_{1}/|z|^{m}}$

for all $z\in\Sigma_{\theta}$ with $|z|\geq R$.

###### Proof.

Without loss of generality, we may assume that $r(\infty)=1$. If
$a=|a|e^{i\beta}$, $\beta\in(-\pi,\pi]$, then it follows from
$\mathcal{A}(\psi,m)$-stability that

$|r(te^{i\varphi})|^{2}=1-2|a|t^{-m}\cos(\beta-m\varphi)+\mbox{O}(t^{-m-1}),\qquad
t\to\infty,\quad|\varphi|\leq\psi.$

It follows that

$\cos(\beta\pm m\varphi)\geq 0,\qquad|\varphi|\leq\psi,$

and then

$|\beta\pm m\psi|\leq\pi/2,$

and finally

$|\beta|+m\psi=\max\\{|\beta-m\psi|,|\beta+m\psi|\\}\leq\pi/2.$

Let $z=te^{i\varphi}$, with $t>0$, $|\varphi|\leq\theta$, and let

$\omega:=\max\,\\{|\beta-m\theta|,|\beta+m\theta|\\}<\pi/2.$

We infer from (1.5) that there exists $c>0$ such that

(3.17)
$|r(te^{i\varphi})|\leq\left|1-\frac{|a|e^{i\beta}e^{-im\varphi}}{t^{m}}\right|+\frac{c}{t^{m+1}}\leq\left|1-\frac{|a|e^{i\omega}}{t^{m}}\right|+\frac{c}{t^{m+1}}.$

If

(3.18) $t\geq(|a|/\cos\omega)^{1/m},$

we may apply an elementary inequality

$|1-z|\leq 1-\frac{{\rm Re}\,z}{2}\qquad\mbox{if}\qquad|z|^{2}\leq{\rm
Re}\,z,$

and conclude that

(3.19) $|r(te^{i\varphi}|\leq
1-\frac{|a|\cos\omega}{2t^{m}}+\frac{c}{t^{m+1}}.$

If in addition to (3.18),

(3.20) $t\geq\left(\frac{4c}{|a|\cos\omega}\right)^{1/(m+1)},$

then it follows from (3.19) and the inequality $1-\tau\leq e^{-\tau}$ that

(3.21) $|r(te^{i\varphi})|\leq 1-\frac{|a|\cos\omega}{4t^{m}}\leq
e^{-|a|\cos\omega/(4t^{m})}.$

Thus the second inequality in (3.16) holds with $b_{1}=(|a|\cos\omega)/4$ and

$t\geq
R_{1}:=\max\,\left\\{\left(\frac{|a|}{\cos\omega}\right)^{1/m},\left(\frac{4c}{|a|\cos\omega}\right)^{1/(m+1)}\right\\}.$

For the first inequality, there exists $c^{\prime}>0$ such that

(3.22) $|r(te^{i\varphi})|\geq
1-\frac{|a|}{t^{m}}-\frac{c^{\prime}}{t^{m+1}},\qquad t>0.$

Let

(3.23) $R_{2}:=\max\,\\{c^{\prime}/|a|,(8|a|)^{1/m}\\}.$

If $t\geq R_{2}$, then the elementary bound $1-\tau/2\geq e^{-\tau}$,
$\tau\in[0,1]$, shows that

$|r(te^{i\varphi})|\geq 1-\frac{|a|}{t^{m}}-\frac{c^{\prime}}{t^{m+1}}\geq
1-\frac{2|a|}{t^{m}}\geq e^{-4|a|/t^{m}}.$

Hence the first inequality in (3.16) is true with with $b_{2}=4|a|$ and $t\geq
R_{2}$ as in (3.23). So (3.16) holds with $R=\max(R_{1},R_{2},1)$. ∎

Given an $\mathcal{A}(\psi)$-stable rational function $r$ and
$n\in\mathbb{N}$, define

$r_{n}(z):=r^{n}(z/n).$

For $\varphi\in[-\psi,\psi]$, $n\in\mathbb{N}$, $s\geq 0$, $\epsilon\geq 0$
and $R>0$, let

(3.24) $\displaystyle
Q^{\epsilon}_{R}(\varphi,n,s):=\int_{Rn}^{\infty}\left|\frac{d}{dt}\left(\frac{r_{n}(te^{i\varphi})}{(te^{i\varphi}+\epsilon)^{s}}\right)\right|\,dt$
$\displaystyle=\frac{1}{n^{s}}\int_{R}^{\infty}\left|n(ue^{i\varphi}+\epsilon)r^{\prime}(ue^{i\varphi})-sr(ue^{i\varphi})\right|\,|r^{n-1}(ue^{i\varphi})|\,\frac{du}{|ue^{i\varphi}+\epsilon|^{s+1}},$

by a change of variable $t=ne^{i\varphi}u$.

The next rather technical lemma is basic for obtaining our improvements of
approximation rates for rational approximations.

######  Lemma 3.10.

Let $r$ be an $\mathcal{A}(\psi,m)$-stable rational function such that
$|r(\infty)|=1$, and let $\theta\in(0,\psi)$ and $T>0$. Then there exist
$R\geq 1$ and $C_{1}=C_{1}(\theta,T)>0$ and $C_{2}=C_{2}(\theta,\epsilon,T)>0$
such that for all $\varphi\in[-\theta,\theta]$, $s\in[0,T]$, and
$n\in\mathbb{N}$,

(3.25) $Q^{0}_{R}(\varphi,n,s)\leq\frac{C_{1}}{n^{(1+m^{-1})s}},$

and, for every $\epsilon\geq 0$,

(3.26) $Q^{\epsilon}_{R}(\varphi,n,s)\geq\frac{C_{2}}{n^{(1+m^{-1})s}}.$

###### Proof.

Let $b_{1}$, $b_{2}$ and $R\geq 1$ be as in Lemma 3.9. Let
$\varphi\in[-\theta,\theta]$, and $s\geq 0$. Note that $|r(te^{i\varphi})|\leq
1$ and, from (1.5), that

(3.27) $|r^{\prime}(z)|\leq\frac{c_{\theta}}{|z|^{m+1}},\qquad
z\in\Sigma_{\theta}.$

With the changes of variables $\tau=t^{-m}$ and $v=(n-1)\tau$ for $n\geq 2$ in
(3.24), we can estimate as follows for $\epsilon=0$:

$\displaystyle Q^{0}_{R}(\varphi,n,s)$
$\displaystyle\leq\frac{1}{n^{s}}\int_{R}^{\infty}\,(c_{\theta}nt^{-m}+s)|r^{n-1}(te^{i\varphi})|\,\frac{dt}{t^{s+1}}$
$\displaystyle\leq\frac{1}{mn^{s}}\int_{0}^{1/R^{m}}(c_{\theta}n\tau+s)\,e^{-b_{1}(n-1)\tau}\tau^{s/m-1}\,d\tau$
$\displaystyle\leq\frac{C(\theta,m,b_{1})}{n^{s}}\int_{0}^{\infty}(n\tau+s)\tau^{s/m-1}\,e^{-b_{1}(n-1)\tau}\,d\tau$
$\displaystyle\leq\frac{C(\theta,m,b_{1})}{(n-1)^{s/m}n^{s}}\int_{0}^{\infty}(2v+s)v^{s/m-1}\,e^{-b_{1}v}\,dv$
$\displaystyle=\frac{C(\theta,m,b_{1})(2+mb_{1})}{b_{1}}\frac{\Gamma(1+s/m)}{b_{1}^{s/m}{(n-1)^{s/m}n^{s}}},$

where $\Gamma$ stands for the gamma function. For $T>0$, this estimate is
bounded by an expression of the form on the right-hand side of (3.25), where
$C_{1}$ is independent of $s\in[0,T]$ and $n\geq 2$.

For $n=1$, we have

$Q^{0}_{R}(\varphi,1,s)\leq\frac{1}{m}\int_{0}^{1/R^{m}}(c_{\theta}\tau+s)\tau^{s/m-1}\,d\tau,$

which is easily seen to be finite and bounded for $s\in[0,T]$, so there is an
upper bound as in (3.25), valid for all $\varphi\in[-\theta,\theta]$,
$s\in[0,T]$ and $n\in\mathbb{N}$.

Now we will justify the lower bound in (3.25). Let $\epsilon\geq 0$ be fixed.
There exists $\tilde{c}_{\theta}>0$ such that

(3.28) $|r^{\prime}(z)|\geq\frac{\tilde{c}_{\theta}}{|z|^{m+1}},\qquad
z\in\Sigma_{\theta},\quad|z|\geq R.$

Let $T>0$, and

$R_{T}=\frac{1}{2}\left(\frac{\tilde{c}_{\theta}}{2T}\right)^{1/m}.$

Let $s\in[0,T]$. Let $n\geq(R/R_{T})^{m}$ and $z=te^{i\varphi}$, where
$|\varphi|<\theta$ and $R_{T}n^{1/m}\leq t\leq 2R_{T}n^{1/m}$. Then

$n|z+\epsilon|\geq n|z|\,|r^{\prime}(z)|\geq 2T\geq 2s.$

Using (3.28), (3.16) the inequalities

$e^{-b_{2}(n-1)/t^{m}}\geq e^{-b_{2}/R_{T}^{m}},\qquad t\geq R_{T}n^{1/m},$

and

$|te^{i\varphi}+\epsilon|\leq(1+\epsilon)t,\qquad t\geq 1,$

we infer that

$\displaystyle(1+\epsilon)^{s+1}Q^{\epsilon}_{R}(\varphi,n,s)$
$\displaystyle\geq\frac{1}{n^{s}}\int_{R_{T}n^{1/m}}^{2R_{T}n^{1/m}}\,(nt|r^{\prime}(te^{i\varphi})|-s)|r^{n-1}(te^{i\varphi})|\,\frac{dt}{t^{s+1}}$
$\displaystyle\geq\frac{n}{2n^{s}}\int_{R_{T}n^{1/m}}^{2R_{T}n^{1/m}}\,|r^{\prime}(te^{i\varphi})||r^{n-1}(te^{i\varphi})|\,\frac{dt}{t^{s}}$
$\displaystyle\geq\frac{n}{2n^{s}}\int_{R_{T}n^{1/m}}^{2R_{T}n^{1/m}}e^{-b_{2}(n-1)/t^{m}}\,\frac{dt}{t^{s+m+1}}$
$\displaystyle\geq\frac{n}{2n^{s}}\int_{R_{T}n^{1/m}}^{2R_{T}n^{1/m}}e^{-b_{2}/R_{T}^{m}}\,\frac{dt}{t^{s+m+1}}$
$\displaystyle=\frac{\left(1-2^{-(s+m)}\right)e^{-b_{2}/R_{T}^{m}}}{2(s+m)R_{T}^{s+m}}\,n^{-(1+m^{-1})s}.$

By taking the minimum value of the term for $s\in[0,T]$, we obtain a lower
bound of the form (3.26) for all but a finite set of $n$. For each of the
remaining values of $n$, the function

$(\varphi,s)\mapsto n^{(1+m^{-1})s}Q^{\epsilon}_{R}(\varphi,n,s)$

is continuous on $[-\theta,\theta]\times[0,T]$, so it attains a positive
minimum value. Thus a lower bound as in (3.25) can be obtained for all
$\varphi\in[-\theta,\theta]$, $s\in[0,T]$, and $n\in\mathbb{N}$. ∎

For an $\mathcal{A}(\psi)$-stable rational function $r$, $n\in\mathbb{N}$ and
$s\geq 0$, we define

$\Delta_{n,s}(z)=\frac{e^{-z}-r_{n}(z)}{z^{s}},\qquad z\in\Sigma_{\psi},\quad
n\in\mathbb{N}.$

A somewhat tedious proof of the next auxiliary estimate is postponed to the
Appendix.

######  Lemma 3.11.

Let $r$ be an $A(\psi)$-stable rational approximation of order $q$ to the
exponential, and let $\theta\in(0,\psi)$ and $R>0$. Then there exist $C>0$ and
$\alpha>0$ depending on $r,\theta$ and $R$ such that

(3.29) $\left|\Delta_{n,s}^{\prime}(z)\right|\leq
C\left(q+1-s+|z|\right)\frac{|z|^{q-s}}{n^{q}}e^{-\alpha|z|}$

for all $n\in\mathbb{N}$, $z\in\Sigma_{\theta}$, $|z|<Rn$, and $s\in[0,q+1]$.

We also need the following estimate, under the assumptions of Lemma 3.11 and
the additional assumption that $R\geq 1$:

(3.30)
$\displaystyle\int_{Rn}^{\infty}\left|\frac{d}{dt}\left(\frac{e^{-te^{i\varphi}}}{(te^{i\varphi}+\epsilon)^{s}}\right)\right|\,dt$
$\displaystyle\leq\int_{Rn}^{\infty}\left(\frac{s}{t^{s+1}}+\frac{1}{t^{s}}\right)e^{-t\cos\varphi}\,dt$
$\displaystyle\leq\frac{q+2}{\cos\varphi}e^{-Rn\cos\varphi}.$

######  Remark 3.12.

It should be noted that Lemmas 3.9 and 3.10 as well as the estimate in (3.30)
apply in exactly the same form for the shifts of $r$, i.e., the functions
$r_{\epsilon}(z)=r(z+\epsilon)$, with possibly different values for
$C=C(\epsilon)$ depending on $\epsilon$.

Lemmas 3.10 and 3.11 lead to the following function-theoretical result on
rational approximations, which after applying the $\mathcal{H}$-calculus, will
transform into a statement on rational approximation rates for holomorphic
semigroups.

######  Theorem 3.13.

Let $r$ be an $\mathcal{A}(\psi,m)$-stable rational approximation of order $q$
to the exponential, and let $\theta\in(0,\psi)$. Then the following hold.

  1. (i)

If $|r(\infty)|=1$, then there exists $C=C(\theta)$ such that

(3.31)
$\|\Delta_{n,s}\|_{{\mathcal{H}}_{\theta,0}}\leq\frac{C}{n^{\delta_{s}}}$

for all $s\in[0,q+1]$ and $n\in\mathbb{N}$, where
$\delta_{s}:=\min\,\\{(1+m^{-1})s,q\\}$.

  2. (ii)

If $|r(\infty)|<1$, then there exists $C=C(\theta)$ such that

(3.32) $\|\Delta_{n,s}\|_{{\mathcal{H}}_{\theta,0}}\leq\frac{C}{n^{q}}$

for all $s\in[0,q+1]$ and $n\in\mathbb{N}$.

###### Proof.

We start with a proof of (i). Let $s\in[0,q+1]$ be fixed. Let $R\geq 1$ be
given by Lemma 3.10 with $T=q+1$, and fix $C$ and $\alpha$ given by Lemma 3.11
for that value of $R$. Using (3.29), (3.30) and Lemma 3.10, we obtain that

(3.33) $\displaystyle\|\Delta_{n,s}\|_{{\mathcal{H}}_{\theta,0}}$
$\displaystyle\leq\frac{2C}{n^{q}}\int_{0}^{Rn}(q+1-s+t)t^{q-s}e^{-\alpha
t}\,dt$ $\displaystyle\hskip
15.0pt\hbox{}+2\int_{Rn}^{\infty}\left|\Delta^{\prime}_{n,s}(te^{i\theta})-r^{\prime}_{n,s}(te^{i\theta})\right|\,dt+2\int_{Rn}^{\infty}\,|r_{n,s}^{\prime}(te^{i\theta})|\,dt$
$\displaystyle\leq\frac{2C}{n^{q}}\int_{0}^{\infty}(q+1-s+t)t^{q-s}e^{-\alpha
t}\,dt$ $\displaystyle\hskip
15.0pt\hbox{}+\frac{2(q+2)}{\cos\theta}e^{-Rn\cos\theta}+2Q^{0}_{R}(\theta,n,s)$
$\displaystyle\leq
C_{1}\left(\frac{1}{n^{q}}+\frac{1}{n^{(1+m^{-1})s}}\right)\leq\frac{C_{2}}{n^{\delta_{s}}},$

for appropriate constants $C_{1}$ and $C_{2}$, so (i) is proved.

Now assume that $|r(\infty)|<1$. We use the same approach as for (i), taking
$R=1$ in Lemma 3.11. We follow the argument for (i) but Lemma 3.10 is
inapplicable, and we need to estimate $Q_{1}^{0}(\varphi,n,s)$ in a different
way. By the assumptions and the maximum principle, we observe that, for
$z\in\Sigma_{\theta}$ with $|z|>1$,

(3.34) $|r(z)|\leq\kappa,\qquad|r^{\prime}(z)|\leq\frac{c_{0}}{|z|^{2}},$

for some $\kappa\in(|r(\infty)|,1)$, $c_{0}=c_{0}(\theta)$. Setting
$z=te^{i\varphi}$ and using (3.34) in (3.24), we infer that there exists
$C=C(\theta)$ such that for every $\varphi\in(0,\theta)$,

$Q_{1}^{0}(\varphi,n,s)\leq\frac{\kappa^{n-1}}{n^{s}}\int_{n}^{\infty}\left(\frac{c_{0}n^{2}}{t^{s+2}}+\frac{s}{t^{s+1}}\right)\,dt\leq\frac{C}{n^{q}}.$

Applying this estimate to replace Lemma 3.10 and Lemma 3.11 with $R=1$, and
arguing as in (3.33), we obtain (3.32). ∎

##  4\. Stability of operator approximations

Now we are ready to formulate our stability estimates for rational
approximations of (sectorially) bounded holomorphic semigroups. Since the
negative generators of such semigroups coincide with sectorial operators of
angle less than $\pi/2,$ we employ the language of sectorial operators, and
deduce our results as direct corollaries of the $\mathcal{H}$-calculus, using
(2.8) along with Theorems 3.6 and 3.7. Firstly, we obtain the following
general statement on stability of variable stepsize rational approximations
first proved in [3] and containing a well-known, constant-step stability
result from [9], [24] (see also [29] and [17]). Recall that for a finite
sequence $\mathcal{K}_{n}:=(k_{j})_{j=1}^{n}\subset(0,\infty)$, we let
$K_{0}=\min\\{k_{j}\\}$ and $K_{1}=\max\\{k_{j}\\}$. The proof is an immediate
consequence of Theorem 3.6 and (2.8).

######  Theorem 4.1.

Let $r$ be an $\mathcal{A}(\psi)$-stable rational function, where
$\psi\in(0,\pi/2]$, and let $\theta\in(0,\psi)$. Then there exists
$C=C(\theta)$ such that, for every operator $A\in\operatorname{Sect}(\varphi)$
for some $\varphi\in(0,\theta)$ and every finite sequence
$\mathcal{K}_{n}\subset(0,\infty)$,

$\bigg{\|}\prod_{j=1}^{n}r(k_{j}A)\bigg{\|}\leq
CM_{\theta}(A)\left(1+\log\left(\frac{K_{1}}{K_{0}}\right)\right).$

In particular, for any $n\in\mathbb{N}$,

$\left\|r^{n}(A/n)\right\|\leq CM_{\theta}(A).$

Secondly, our methods allow us to strengthen substantially a well-known result
on variable stepsize approximation due to Palencia [27] and Bakaev [2]
(reproved in [4]) and to obtain the next statement. It is an immediate
consequence of Theorem 3.7 and (2.8).

######  Theorem 4.2.

Let $r$ be an $\mathcal{A}(\psi)$-stable rational function with
$|r(\infty)|<1$, where $\psi\in(0,\pi/2]$, and let $\theta\in(0,\psi)$. Then
there exists $C=C(\theta)$ such that, for every operator
$A\in\operatorname{Sect}(\varphi)$ for some $\varphi\in(0,\theta)$ and every
finite sequence $\mathcal{K}_{n}\subset(0,\infty)$,

$\bigg{\|}\prod_{j=1}^{n}r(k_{j}A)\bigg{\|}\leq CM_{\theta}(A).$

The assumptions in [2] and [27] (and in [4]) included not only that
$|r(\infty)|<1$, but also that $r$ approximates $e^{-z}$ with order $1$. This
assumption is indispensable in their proofs which are rather technical. For
results related to Theorem 4.2 see also [5] where the first order of
approximation also appears to be crucial.

##  5\. Rates for operator approximations

The $\mathcal{H}$-calculus also leads to sharp estimates of approximation
rates, in particular to a refined version of Theorem 1.1. It might appear to
be a mere improvement of Theorem 1.1, taking into account fine structure of
$r$. However Theorem 5.1(i) gives a substantially better estimate than Theorem
1.1(i) when $s\in(0,q)$ (and it is essentially vacuous when $s=0$). Theorem
5.1(ii) for $s=0$ gives the result in Theorem 1.1(ii), and extends it slightly
for smooth vectors $x$. Moreover, the bounds given by Theorem 5.1 are optimal
as we prove in Section 7. The proofs are of a common form, based on the
$\mathcal{H}_{\psi}$-calculus and its compatibility with the extended
holomorphic calculus.

######  Theorem 5.1.

Let $r$ be an $\mathcal{A}(\psi,m)$-stable rational approximation of order
$q$, and let $\varphi\in(0,\psi)$ and $\theta\in(\varphi,\psi)$. Then there
exists a constant $C=C(\theta)$ such that the following holds for all
operators $A\in\operatorname{Sect}(\varphi)$ for some $\varphi\in[0,\theta)$:

  1. (i)

If $|r(\infty)|=1$, $s\in[0,q+1]$, and $\delta_{s}:=\min\\{(1+m^{-1})s,q\\}$,
then for all $x\in{\rm dom}(A^{s})$, $t>0$ and $n\in\mathbb{N}$,

(5.1)
$\|e^{-tA}x-r^{n}(tA/n)x\|\leq\frac{CM_{\theta}(A)t^{s}}{n^{\delta_{s}}}\|A^{s}x\|.$

  2. (ii)

If $|r(\infty)|<1$ and $s\in[0,q+1]$, then for all $x\in{\rm dom}(A^{s})$,
$t>0$ and $n\in\mathbb{N}$,

(5.2)
$\|e^{-tA}x-r^{n}(tA/n)x\|\leq\frac{CM_{\theta}(A)t^{s}}{n^{q}}\|A^{s}x\|.$

###### Proof.

Replacing $A$ by $tA$ for $t>0$, we may assume that $t=1$. For $s\in[0,q+1]$,
let $g_{s}(z)=z^{-s}$ and

$\Delta_{n}(z):=e^{-z}-r_{n}(z),\qquad z\in\Sigma_{\psi}.$

Then $g_{s}\Delta_{n}=\Delta_{n,s}\in\mathcal{H}_{\psi}$.

For $\varepsilon>0$, the extended holomorphic calculus can be applied to the
invertible operator $A+\varepsilon$. In that calculus,
$g_{s}(A+\varepsilon)=(A+\varepsilon)^{-s}$ in the standard theory of
fractional powers. Moreover, $g_{s}$ is in the domain of the extended
$\mathcal{H}_{\psi}$-calculus for $A+\varepsilon$, and
$\Delta_{n}(A+\varepsilon)=e^{-(A+\varepsilon)}-r_{n}(A+\varepsilon)$ in both
calculi. Then

$\Delta_{n,s}(A+\varepsilon)=(A+\varepsilon)^{-s}\big{(}e^{-(A+\varepsilon)}-r_{n}(A+\varepsilon)\big{)}$

in the extended holomorphic calculus. Since the two calculi are compatible
(see [6, Proposition 8.1(iv)]) this also holds in the
$\mathcal{H}_{\psi}$-calculus. Hence, for
$x\in\operatorname{dom}{(A^{s})}=\operatorname{dom}((A+\varepsilon)^{s})$,

$\displaystyle\big{\|}\big{(}e^{-(A+\varepsilon)}-r_{n}(A+\varepsilon)\big{)}x\big{\|}$
$\displaystyle=\left\|\Delta_{n,s}(A+\varepsilon)(A+\varepsilon)^{s}x\right\|$
$\displaystyle\leq\|\Delta_{n,s}(A+\varepsilon)\|\,\|(A+\varepsilon)^{s}x\|.$

If $s>0$, then we can use the estimates from Theorem 3.13 and (2.8) to obtain
(5.1) or (5.2) with $A$ replaced by $A+\varepsilon$. Letting $\varepsilon\to
0$, and using the relation that $\lim_{\varepsilon\to
0}M_{\theta}(A+\varepsilon)=M_{\theta}(A)$ and [17, Proposition 3.1.9 c)], we
obtain (5.1) or (5.2) for $A$. For $s=0$, $\Delta_{n,0}(\infty)=-r(\infty)$
and there is an additional term $|r(\infty)|^{n}$ in the estimate for
$\|\Delta_{n,0}(A+\varepsilon)\|$. This can be covered by changing the
constant $C$. ∎

######  Remark 5.2.

The case $s=0$ of Theorem 5.1(ii) shows that for every $t\geq 0$,

$\lim_{n\to\infty}\|e^{-tA}-r^{n}(tA/n)\|=0.$

Let $\rho_{1}(z)=(z+1)^{-1}$. Then $\rho_{1}$ is $\mathcal{A}(\pi/2,1)$-stable
rational approximation of order $1$, with $\rho_{1}(\infty)=0$. If $A\in{\rm
Sect(\pi/2)}$ then $-A$ generates a $C_{0}$-semigroup $(T(t))_{t\geq 0}$, and
it is known that $\rho_{1}^{n}(tA/n)$ converges strongly to $T(t)$ for each
$t\geq 0$ ([12, Corollary III.5.5]). Hence $T(t)=e^{-tA}$.

The following corollary provides a natural example of a function $r$
satisfying the assumptions of Theorem 5.1(i).

######  Corollary 5.3.

Let $r=P_{k}/Q_{k}$, $k\in\mathbb{N}$, be the diagonal Padé approximation of
order $2k$ to $e^{-z}$, and let
$A\in\operatorname{Sect}(\varphi),\varphi\in[0,\pi/2)$ and
$\theta\in(\varphi,\pi/2)$. Then there exists $C=C(\theta)$ such that, for all
$s\in[0,q+1]$, $t\geq 0$, $n\in\mathbb{N}$ and $x\in{\rm dom}(A^{s})$,

(5.3)
$\|e^{-tA}x-r^{n}(tA/n)x\|\leq\frac{CM_{\theta}(A)t^{s}}{n^{\gamma_{s}}}\|A^{s}x\|,$

where $\gamma_{s}=2\min\\{s,k\\}$.

###### Proof.

By [19, Theorem 4.12] (or [30, Theorem 7]), the diagonal Padé approximations
of $e^{-z}$ are $\mathcal{A}(\pi/2)$-stable. By [19, Theorem 3.12], $P_{k}$
and $Q_{k}$ have the same degree, $P_{k}(z)=Q_{k}(-z)$ for all
$z\in\mathbb{C},$ and $P_{k}$ contain the term $z^{k-1}$ with a non-zero
coefficient. Thus $r$ satisfies (1.5) with $m=1$, that is, they are
$\mathcal{A}(\pi/2,1)$-stable. Now (5.3) follows directly from (5.1). ∎

##  6\. Optimality in terms of the algebra $\mathcal{H}_{\psi}$

In this and the next section we address the optimality of Theorem 5.1, and
show that the approximation rates provided by this theorem are sharp. We
define

(6.1) $\Delta_{n,s,1}(z):=\Delta_{n,s}(z+1),\qquad z\in\mathbb{C}_{+}.$

######  Theorem 6.1.

Let $r$ be an $\mathcal{A}(\psi,m)$-stable rational approximation. Then the
following hold.

  * (i)

If $r$ is of exact order $q$, then there exists $C>0$ such that, for all
$\theta\in(0,\psi)$, $s\in[0,q+1]$ and $n\in\mathbb{N}$,

(6.2)
$\|\Delta_{n,s}\|_{\mathcal{H}_{\theta,0}}\geq|\Delta_{n,s}(1)|\geq\frac{C}{n^{q}}.$

  * (ii)

If $r$ is of order $q$ and $|r(\infty)|=1$, then there exists $C=C(\theta)$
such that, for all $\theta\in(0,\psi)$, $s\in[0,q+1]$ and $n\in\mathbb{N}$,

(6.3)
$\|\Delta_{n,s,1}\|_{{\mathcal{H}}_{\theta,0}}\geq\frac{C}{n^{(1+m^{-1})s}}.$

###### Proof.

The first inequality in (6.2) is immediate from (2.5), so we only need to
prove the second inequality. Since $r$ is of exact order $q$, there exists
$a\neq 0$ such that

$r(z)=e^{-z}+az^{q+1}+\mbox{O}(|z|^{q+2}),\qquad z\to 0.$

Let

$\displaystyle g(z)$
$\displaystyle:=r(z)e^{z}-1-az^{q+1}=\mbox{O}(|z|^{q+2}),$ $\displaystyle
h(z)$ $\displaystyle:=r(z)e^{z}-1=g(z)+az^{q+1}=\mbox{O}(|z|^{q+1}).$

There exist $R>0$ and $c$ such that $g$ and $h$ are holomorphic in
$\mathbb{D}_{R}$ and

(6.4) $|g(z)|\leq
c|z|^{q+2},\qquad|h(z)|\leq\min\\{\tfrac{1}{2},c|z|^{q+1}\\},\qquad
z\in\mathbb{D}_{R}.$

Let $z\in\mathbb{D}_{R}$ and

$\eta(z)=\log(1+h(z))-az^{q+1},\quad z\in\mathbb{D}_{R}.$

Using the elementary inequality

$|\log(1+\zeta)-\zeta|\leq|\zeta|^{2},\qquad|\zeta|\leq 1/2,$

for $\zeta=h(z)$, we obtain

$|\eta(z)|\leq|h(z)|^{2}+|g(z)|\leq c_{1}|z|^{q+2},$

and

$r(z)=e^{-z}(1+h(z))=e^{-z}e^{az^{q+1}}e^{\eta(z)}.$

Now let $z\in\mathbb{C}$ and $n\geq|z|/R$, so that $z/n\in\mathbb{D}_{R}$.
Then

(6.5)
$\Delta_{n,s}(z)=\frac{e^{-z}-r_{n}(z)}{z^{s}}=\frac{e^{-z}(1-e^{az^{q+1}/n^{q}}e^{\eta_{n}(z)})}{z^{s}},$

where

$|\eta_{n}(z)|=n|\eta(z/n)|\leq c_{1}\frac{|z|^{q+2}}{n^{q+1}},\qquad
z\in\mathbb{D}_{Rn}.$

Setting $z=1$ in (6.5) and letting $n\to\infty$, we see that
$\eta_{n}(1)=\mbox{O}(n^{-(q+1)})$, and we infer that

$\displaystyle|\Delta_{n,s}(1)|=e^{-1}|1-\exp\left(a/n^{q}+\eta_{n}(1)\right)|=\frac{|a|}{en^{q}}+\mbox{O}(n^{-(q+1)}),\quad
n\to\infty,$

uniformly for $s\in[0,q+1]$. Thus (6.2) holds.

Now we will prove (6.3) under the additional assumption that $|r(\infty)|=1$.
Let $s\in[0,q+1]$ and $n\in\mathbb{N}$. Using (3.30) with $\epsilon=1$, we
note that

$\|\Delta_{n,s,1}\|_{{\mathcal{H}}_{\theta,0}}\geq
Q_{1}^{1}(\theta,n,s)-\frac{q+2}{\cos\theta}e^{-Rn\cos\theta}.$

Taking into account Remark 3.12 and applying the bound (3.26) of Lemma 3.10
with $T=q+1$ and $r_{1}(z):=r(z+1)$ in place of $r$, we infer that there
exists $C=C(\theta)>0$ such that

$\|\Delta_{n,s,1}\|_{{\mathcal{H}}_{\theta,0}}\geq\frac{C}{n^{(1+m^{-1})s}}.$

Thus (6.3) holds. ∎

For $\gamma\in(0,1)$, $s\geq 0$, and $n\in\mathbb{N}$, let

${\Delta}^{[\gamma]}_{n,s,1}(z):={\Delta}_{n,s,1}(z^{\gamma}),\qquad
z\in\mathbb{C}_{+}.$

######  Corollary 6.2.

Let $r$ be an $\mathcal{A}(\psi,m)$-stable rational approximation of order
$q$, with $|r(\infty)|=1$, and let $\psi\in(0,\pi/2]$, $\theta\in(0,\psi)$,
$\gamma:={2\theta}/{\pi}\in(0,1)$, $s\in[0,q+1]$ and $n\in\mathbb{N}$. Then
$\Delta^{[\gamma]}_{n,s,1}\in\mathcal{LM}$. Moreover, there is a constant
$C=C(\theta)>0$ such that, for all $s\in[0,q+1]$ and $n\in\mathbb{N}$,

(6.6)
$\|\Delta^{[\gamma]}_{n,s,1}\|_{\mathcal{LM}}\geq\frac{C}{n^{\delta_{s}}},$

where $\delta_{s}=\min\,\\{(1+m^{-1})s,q\\}$.

###### Proof.

Let $n\in\mathbb{N}$ and $s\in[0,q+1]$. By direct verification,
$\Delta_{n,s,1}\in\mathcal{H}_{\theta}$. Then, by [6, Lemma 4.9]
$\Delta^{[\gamma]}_{n,s,1}\in\mathcal{H}_{\pi/2}$, and by [6, Theorem 4.12] we
have $\Delta^{[\gamma]}_{n,s,1}\in\mathcal{LM}$.

Let $\varphi\in(0,\pi/2)$ and $\tilde{\theta}=\gamma\varphi\in(0,\theta)$.
Then, as noted in Section 2, $\Delta^{[\gamma]}_{n,s,1}$ belongs to
$\mathcal{H}_{\varphi}$ and (2.6) shows that

$\|\Delta^{[\gamma]}_{n,s,1}\|_{\mathcal{LM}}\geq\frac{\cos\varphi}{2}\|\Delta^{[\gamma]}_{n,s,1}\|_{\mathcal{H}_{\varphi,0}}=\frac{\cos\varphi}{2}\|\Delta_{n,s,1}\|_{\mathcal{H}_{\tilde{\theta},0}}.$

The assertion (6.6) now follows from Theorem 6.1. ∎

######  Remark 6.3.

Let $\mathcal{L}L^{1}$ denote the space of Laplace transforms $\mathcal{L}f$
of functions $f$ in $L^{1}(\mathbb{R}_{+})$. This is a non-unital closed
subalgebra of the Hille-Phillips algebra $\mathcal{LM}$. Theorem 4.12 in [6]
actually shows that $\Delta_{n,s,1}^{[\gamma]}\in\mathcal{L}L^{1}+\mathbb{C}$,
where $\mathbb{C}$ represents the constant functions on $\mathbb{C}_{+}$. The
space $\mathcal{L}L^{1}$ arises also in Section 7.

##  7\. Optimality for semigroup generators

The estimates in Theorem 5.1 for semigroup generators are optimal, as we will
show in this section. We start by showing that Theorem 5.1(ii) is optimal.
Given the estimate in Theorem 6.1(ii), the optimality of (5.2) is shown in the
following trivial example.

######  Example 7.1.

Let $r$ be an $\mathcal{A}(\psi)$-stable rational approximation of exact order
$q$, and let $A$ be the identity operator $I$ on any Banach space $X$, so $A$
is sectorial of angle $0$. Then $A^{s}$ is the identity operator and
$\Delta_{n,s}(A)$ is the operator of multiplication by the scalar
$\Delta_{n,s}(1)$ on $X$. Thus it is immediate from Theorem 6.1(i) that there
exists $C>0$ such that

$\|(e^{-A}-r^{n}(A/n))x\|=|\Delta_{n,s}(1)|\,\|A^{s}x\|\geq\frac{C}{n^{q}}\|x\|$

for all $s\in[0,q+1]$, $n\in\mathbb{N}$ and $x\in X$. Thus the estimate (5.2)
in Theorem 5.1 is optimal for semigroup generators (up to a multiplicative
constant).

Next we will show that Theorem 5.1(i) is optimal, by first considering a
specific example of a semigroup of multiplication operators.

######  Example 7.2.

For $f\in L^{1}(\mathbb{R}_{+})$, let $\mathcal{L}f$ be the Laplace transform
of $f$. Consider the Banach space

$X=\mathcal{L}L_{1}:=\big{\\{}\mathcal{L}f:f\in
L^{1}(\mathbb{R}_{+})\big{\\}},$

with the norm given by

(7.1)
$\|\mathcal{L}f\|_{X}:=\|f\|_{L^{1}(\mathbb{R}_{+})},\qquad\mathcal{L}f\in X,$

cf. (2.1). For $t\geq 0$, let

$(T(t)\mathcal{L}f)(z)=e^{-tz}(\mathcal{L}f)(z),\qquad\mathcal{L}f\in
X,\;z\in\mathbb{C}_{+}.$

The map $f\mapsto\mathcal{L}f$ is a surjective isometry from
$L^{1}(\mathbb{R}_{+})$ to $X$ which transforms the $C_{0}$-semigroup of right
shifts on $L^{1}(\mathbb{R}_{+})$ into the multiplication operators on $X$.
Let $-D$ be the generator of $(T(t))_{t\geq 0}$. By the HP-calculus, for all
$\mathcal{L}f\in X$ and $\mathcal{L}\mu\in\mathcal{LM}$,

$\displaystyle(\mathcal{L}\mu(D)\mathcal{L}f)(z)$
$\displaystyle=\int_{0}^{\infty}(T(t)\mathcal{L}f)(z)\,d\mu(t)$
$\displaystyle=\int_{0}^{\infty}e^{-zt}\mathcal{L}f(z)\,\,d\mu(t)=\mathcal{L}\mu(z)\mathcal{L}f(z)=\mathcal{L}(f*\mu)(z).$

Then, by [10, Proposition 4.7.43(i)] (cf. (2.2)),

(7.2) $\|\mathcal{L}\mu(D)\|=\|\mathcal{L}\mu\|_{\mathcal{LM}}.$

For $\mathcal{L}f\in X$ and $z,\lambda\in\mathbb{C}_{+}$,

$\big{(}(\lambda+D)^{-1}\mathcal{L}f\,\big{)}(z)=\int_{0}^{\infty}e^{-\lambda
t}e^{-zt}(\mathcal{L}f)(z)\,dt=\frac{(\mathcal{L}f)(z)}{\lambda+z}.$

Hence $D$ is of the form

(7.3) $\displaystyle{\rm dom}(D)$ $\displaystyle=\big{\\{}\mathcal{L}f\in
X:\,z(\mathcal{L})f(z)\in X\big{\\}},$ $\displaystyle(D\mathcal{L}f)(z)$
$\displaystyle=z(\mathcal{L}f)(z),\qquad z\in\mathbb{C}_{+}.$

The considerations above are variations of the elementary transference
principle considered in [18, Example 5.2], for example.

Let $\theta\in[0,\pi/2)$, $\gamma=2\theta/\pi$ and $A:=D^{\gamma}$, where $D$
is given by (7.3). By the extended HP (or holomorphic) functional calculus,
$A\in\operatorname{Sect}(\theta)$, and

$(Af)(z)=z^{\gamma}f(z),\qquad f\in X,\quad z\in\mathbb{C}_{+}.$

Let $r$ be an $\mathcal{A}(\psi,m)$-stable rational approximation of order $q$
with $|r(\infty)|=1$, and let $s\in[0,q+1]$. By the composition rule for the
extended holomorphic calculus [18, Theorem 2.4.2],

$\Delta_{n,s,1}(A)={\Delta}^{[\gamma]}_{n,s,1}(D).$

It follows from Corollary 6.2 that

(7.4)
$\|\Delta_{n,s,1}(A)\|=\|\Delta^{[\gamma]}_{n,s,1}(D)\|=\|\Delta^{[\gamma]}_{n,s,1}\|_{\mathcal{LM}}\geq\frac{C}{n^{\delta_{s}}}$

for some $C=C(\theta)>0$ and all $n\in\mathbb{N}$ and $s\in[0,q+1]$, where
$\mathbf{\delta_{s}=}\min\,\\{(1+m^{-1})s,q\\}$.

Next we show that (5.1) is optimal when $|r(\infty)|=1$ and $(1+m^{-1})s<q$.
The case when $s\in[mq/(m+1),q+1)$ has already been covered in Example 7.1.

######  Theorem 7.3.

Let $r$ be an $A(\psi,m)$-stable rational approximation of order $q$ with
$|r(\infty)|=1$. Then for every $(\alpha_{n})_{n\geq 1}\subset(0,\infty)$ with
$\lim_{n\to\infty}\alpha_{n}=0$ and all $\theta\in[0,\pi/2)$ and
$s\in[0,mq/(m+1))$, there exist a Banach space $\mathcal{X}$, an operator
$\mathcal{A}\in\operatorname{Sect}(\theta)$ on $\mathcal{X}$, a vector
$x\in{\rm dom}(\mathcal{A}^{s})$ and a constant $C=C(\theta)>0$ such that

(7.5) $\|e^{-\mathcal{A}}x-r^{n}(\mathcal{A}/n)x\|\geq
C\alpha_{n}n^{-(1+m^{-1})s},\qquad n\in\mathbb{N}.$

###### Proof.

Without loss of generality, we can assume that $(\alpha_{n})_{n\geq
1}\subset(0,1/2)$. Let $\theta\in[0,\pi/2)$ be fixed, and let $X$ and
$A\in\operatorname{Sect}(\theta)$ be as in Example 7.2.

Define a Banach space $\mathcal{X}=c_{0}(X)$ by

$\displaystyle\mathcal{X}:=$ $\displaystyle\\{(x_{n})_{n\geq 1}\subset
X:\,\lim_{n\to\infty}\,\|x_{n}\|=0\\},$ $\displaystyle\|x\|_{\mathcal{X}}:=$
$\displaystyle\sup_{n\geq 1}\,\|x_{n}\|,\qquad x=(x_{n})_{n\geq
1}\in\mathcal{X},$

and consider the linear operator $\mathcal{A}$ on $\mathcal{X}$ given by

$\displaystyle{\rm dom}(\mathcal{A})$ $\displaystyle:=\\{(x_{n})_{n\geq
1}\in\mathcal{X}:x_{n}\in{\rm dom}(A),\,\,(Ax_{n})_{n\geq
1}\in\mathcal{X}\\},$ $\displaystyle\mathcal{A}x$
$\displaystyle:=((A+I)x_{n})_{n\geq 1},\qquad x=(x_{n})_{n\geq
1}\in\rm{dom}(\mathcal{A}).$

It is straightforward to show that
$\mathcal{A}\in\operatorname{Sect}(\theta)$.

Let $s\in[0,mq/(m+1))$. By Example 7.2 there exists $C=C(\theta)>0$ such that
for all $n\in\mathbb{N}$,

$\|\Delta_{n,s,1}(A)\|\geq{C}{n^{-(1+m^{-1})s}}.$

Hence there exist $y_{n}\in X$, satisfying

$\|y_{n}\|={\alpha_{n}},\qquad\|\Delta_{n,s,1}(A)y_{n}\|\geq
2^{-1}C{\alpha_{n}}n^{-(1+m^{-1})s}.$

Observe that by the spectral mapping theorem for fractional powers [17,
Theorem 3.1.1 j)], the operators $(A+I)^{s}$ and $\mathcal{A}^{s}$ are
boundedly invertible. Moreover, the operators $\mathcal{A}^{-s}$ and
$\Delta_{n}(\mathcal{A})$ act on $\mathcal{X}$ as diagonal copies of
$(A+I)^{-s}$ and $\Delta_{n}(A+I)=\Delta_{n,1}(A)$ (see (2.10)).

Let $y=(y_{n})_{n\geq 1}\in\mathcal{X}$, $x_{n}=(A+I)^{-s}y_{n}$ and

$x=\mathcal{A}^{-s}y=((A+I)^{-s}y_{n})_{n\geq 1}.$

Thus $x\in{\rm dom}(\mathcal{A}^{s})$. Using the product rule for the
(extended) holomorphic functional calculus, we infer that, for every
$n\in\mathbb{N}$,

$\|\Delta_{n}(\mathcal{A})x\|_{\mathcal{X}}\geq\|\Delta_{n}(A+I)x_{n}\|=\|\Delta_{n,1}(A)x_{n}\|\\\
=\|\Delta_{n,s,1}(A)y_{n}\|\geq 2^{-1}C\alpha_{n}n^{-(1+m^{-1})s}.\qed$

######  Remark 7.4.

The Banach space $\mathcal{X}$ in Theorem 7.3 is the same space, independent
of $r$, $(\alpha_{n})$, $\theta$ and $s$, and the operator $\mathcal{A}$
depends only on $\theta$.

##  8\. Appendix: Proof of Lemma 3.11

We recall Lemma 3.11 and give a proof. For $\psi\in(0,\pi/2]$ and $R>0$, let

$\Sigma_{\psi}(R):=\\{z\in\Sigma_{\psi}:|z|<R\\}.$

We show that if $r$ is an $\mathcal{A}(\psi)$-stable rational approximation of
order $q$, and $\theta\in(0,\psi)$, $R>0$, then there exist $\alpha>0$ and
$C=C(\theta,R)$ such that

(8.1) $\left|\Delta_{n,s}^{\prime}(z)\right|\leq
C(q+1-s+|z|)\frac{|z|^{q-s}}{n^{q}}e^{-\alpha|z|},$

for all $z\in\Sigma_{\theta}(Rn)$, $s\in[0,q+1]$ and $n\in\mathbb{N}$.

To this aim, let $\theta\in(0,\psi)$ and $R>0$ be fixed. Constants that appear
may depend on $r$, $\theta$ and $R$ but not on $s$ or $n$.

For $z\in\Sigma_{\psi}$ and $n\in\mathbb{N}$, define

$F_{n,s}(z):=\frac{e^{-nz}-r^{n}(z)}{z^{s}},$

and note that

$\Delta^{\prime}_{n,s}(z)=\frac{1}{n^{s+1}}F^{\prime}_{n,s}(z/n).$

Thus the estimate (8.1) is equivalent to the estimate

(8.2) $\left|F^{\prime}_{n,s}(z)\right|\leq
Cn((q+1-s)+n|z|)|z|^{q-s}e^{-\alpha n|z|}$

for some $C,\alpha>0$ (with the same values as in (8.1)) and all
$z\in\Sigma_{\theta}(R)$, $s\in[0,q+1]$ and $n\in\mathbb{N}$.

To establish (8.2) we need several auxiliary bounds. By (1.2), there exists
$a\in\mathbb{C}$ such that

(8.3) $r(z)-e^{-z}=az^{q+1}+g_{1}(z),\qquad z\in\Sigma_{\psi},$

where

$|g_{1}(z)|\leq M_{0}|z|^{q+2}\qquad\text{and}\qquad|g_{1}^{\prime}(z)|\leq
M_{1}|z|^{q+1}$

for some $M_{0},M_{1}>0$. In particular,

(8.4) $r^{\prime}(z)+r(z)=a(q+1)z^{q}+h(z),\qquad|h(z)|\leq M_{2}|z|^{q+1}$

for some $M_{2}>0$.

By [29, Lemmas 9.2 and 9.4], for every $R>0$ there exist $\alpha>0$ and
$C_{1}>0$ (depending on $\theta$ and $R$) such that

(8.5) $\displaystyle|r(z)|$ $\displaystyle\leq e^{-\alpha|z|},$ (8.6)
$\displaystyle|r^{k}(z)-e^{-kz}|$ $\displaystyle\leq C_{1}n|z|^{q+1}e^{-\alpha
k|z|},$

for all $z\in\Sigma_{\theta}(R)$ and $k\in\mathbb{N}$. To deduce (8.2), we
will derive a strengthened version of (8.6). Namely, we will show that, for
all $z\in\Sigma_{\theta}(R)$ and $n\in\mathbb{N}$,

(8.7) $r^{n}(z)-e^{-nz}=anz^{q+1}r^{n-1}(z)+g_{n}(z),$

with

(8.8) $|g_{n}(z)|\leq M_{3}n^{2}|z|^{q+2}e^{-\alpha n|z|}$

for some $M_{3}>0$.

If $z\in\Sigma_{\theta}(R)$, then

$\hbox{}\hskip 30.0pt(r^{n}(z)-e^{-nz})-n(r(z)-e^{-z})r^{n-1}(z)\\\
=(r(z)-e^{-z})\sum_{j=0}^{n-1}r^{j}(z)(e^{-(n-1-j)z}-r^{n-j-1}(z)).\hskip
30.0pt$

Hence

$g_{n}(z)=(r(z)-e^{-z})\sum_{j=0}^{n-1}r^{j}(z)(e^{-(n-1-j)z}-r^{n-j-1}(z))+ng_{1}(z)r^{n-1}(z).$

Using (8.5), (8.6) (for $k=1,2,\dots,n-1$) and (8.3), we obtain

$\displaystyle|g_{n}(z)|$ $\displaystyle\leq
C_{1}|z|^{q+1}e^{-\alpha|z|}\sum_{j=0}^{n-1}e^{-\alpha
j|z|}C_{1}(n-j-1)|z|^{q+1}e^{-\alpha(n-j-1)|z|}$ $\displaystyle\hbox{}\hskip
30.0pt+nM_{0}|z|^{q+2}e^{-\alpha(n-1)|z|}$
$\displaystyle\leq\left(C_{1}^{2}|z|^{q}+M_{0}e^{\alpha|z|}\right)n^{2}|z|^{q+2}e^{-\alpha
n|z|}\leq M_{3}n^{2}|z|^{q+2}e^{-\alpha n|z|},$

where $M_{3}=C_{1}^{2}R^{q}+M_{0}e^{\alpha R}$.

Now, observing that

$(e^{-nz}-r^{n}(z))^{\prime}=n(r^{n}(z)-e^{-nz})-n\left(r^{\prime}(z)+r(z)\right)r^{n-1}(z),$

and using (8.4) and (8.7), we write

$\displaystyle F^{\prime}_{n,s}(z)$
$\displaystyle=\frac{n(r^{n}(z)-e^{-nz})}{z^{s}}-\frac{nr^{n-1}(z)(r^{\prime}(z)+r(z))}{z^{s}}+\frac{s(r^{n}(z)-e^{-nz})}{z^{s+1}}$
$\displaystyle=\frac{n(r^{n}(z)-e^{-nz})}{z^{s}}-na(q+1-s)z^{q-s}r^{n-1}(z)$
$\displaystyle\hskip
30.0pt\hbox{}-\frac{sg_{n}(z)}{z^{s+1}}-\frac{nr^{n-1}(z)h(z)}{z^{s}}.$

Using (8.4), (8.5), (8.6) (for $k=n$), (8.7) and (8.8), we obtain that

$\displaystyle|F^{\prime}_{n,s}(z)|$ $\displaystyle\leq
C_{1}n^{2}|z|^{q+1-s}e^{-\alpha n|z|}+n|a|(q+1-s)|z|^{q-s}e^{-\alpha(n-1)|z|}$
$\displaystyle\hskip 20.0pt\hbox{}+M_{3}(q+1)n^{2}|z|^{q+1-s}e^{-\alpha
n|z|}+M_{2}|z|^{q+1-s}e^{-\alpha(n-1)|z|}$ $\displaystyle\leq
Cn(q+1-s+n|z|)|z|^{q-s}e^{-\alpha n|z|},$

where

$C=\max\left(|a|,C_{1}+M_{3}(q+1)R+M_{2}Re^{\alpha R}\right).$

Thus (8.2) holds.

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