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Л† =Л†
Лњ Л†
only if g(z) в€јs/p g (z) in G.
Лњ
Л†
=

3. Suppose f (z) в€јs f (z) in G, and let the constant term of f (z) be zero.
=Л† Л†
Show that then z в€’1 f (z) в€јs z в€’1 f (z) in G.
Л†
=
4. Suppose f (z) в€јs f (z) in G(d, О±). Show that f (zeВ±2ПЂi ) в€јs f (z) in a
=Л† =Л†
corresponding sectorial region G(d в€“ 2ПЂ, О±).
в€ћ
Л†
5. Given s > 0 and a formal Laurent series f (z) = n=в€’m fn z n , for
some m в€€ N, show that the following two statements are equivalent:
4.6 Gevrey Asymptotics in Narrow Regions 73

fn z n в€јs
в€’1 в€ћ
(a) f (z) в€’ fn z n in G,
=
n=в€’m n=0
в€ћ
(b) z m f (z) в€јs fnв€’m z n in G.
=
n=0

Let either one serve as deп¬Ѓnition for f (z) в€јs f (z) in G.
=Л†

6. Assume that E is a Banach algebra. Let As,m (G, E ) denote the set
of functions f (z) that are meromorphic in G, so that the number
of poles of f (z) in an arbitrary closed subsector of G is п¬Ѓnite, and
such that f (z) в€јs f (z) for some formal Laurent series f . Show that
=Л† Л†
As,m (G, E ) is a diп¬Ђerential algebra. In case E = C (a п¬Ѓeld), show
that As,m (G, C ) is a diп¬Ђerential п¬Ѓeld, i.e., a diп¬Ђerential algebra in
which each element is invertible.

7. Show the existence of f в€€ A(G, E ) that is not in As (G, E ), for any
s в‰Ґ 0.

8. Let fn в€€ As (G, E ), n в‰Ґ 0. Assume that for every m в‰Ґ 0 the sequence
(m) ВЇ
(fn )n converges uniformly on every closed subsector S of G. More-
ВЇ
over, assume existence of c, K, depending upon S but independent of
(m) ВЇ
n and m, so that fn (z) в‰¤ c K m m! О“(1 + sm) for every z в€€ S.
Show f = lim fn в€€ As (G, E ). Compare this to Exercise 6 on p. 70.

4.6 Gevrey Asymptotics in Narrow Regions
The following result is a version of RittвЂ™s theorem, adapted to the theory
of Gevrey asymptotics; its proof makes use of the so-called п¬Ѓnite Laplace
operator, which is deп¬Ѓned and discussed in some detail in the following
exercises.

Proposition 10 (RittвЂ™s Theorem for Gevrey Asymptotics) For
Л†
s > 0, let f (z) в€€ C [[z]]s and a sectorial region G of opening at most sПЂ be
arbitrarily given. Then there exists a function f (z), holomorphic in G, so
that f (z) в€јs f (z) in G.
=Л†

в€ћ в€ћ
Л†
Proof: Let f (z) = 0 fn z n and deп¬Ѓne g(u) = 0 fn un /О“(1+sn), then
Л†
f (z) в€€ C [[z]]s implies holomorphy of g(u) for |u| suп¬ѓciently small. Let d
be the bisecting direction of G and deп¬Ѓne for a = ПЃeid , with suп¬ѓciently
small ПЃ > 0, and k = 1/s:
a
в€’k
g(u) exp[в€’(u/z)k ] duk .
f (z) = z (4.8)
0
74 4. Asymptotic Power Series

It follows from the exercises at the end of this section that this f (z) has
2
the desired properties.
The above result ensures that the mapping J : As (G, E ) в†’ E [[z]]s is
surjective if the opening of G is smaller than or equal to sПЂ. It is, however,
in this case not injective, as one learns from the following exercises.

Exercises: If not otherwise speciп¬Ѓed, let g(u) be continuous for arg u = d
and 0 в‰¤ |u| в‰¤ ПЃ with п¬Ѓxed d and ПЃ > 0, and deп¬Ѓne f (z) by (4.8), for k > 0
and a = ПЃeid .
1. Show that f (z) is holomorphic on the Riemann surface of the loga-
rithm. In the literature, the mapping g в€’в†’ f is named п¬Ѓnite Laplace
operator of order k.
2. Assume that complex numbers gn and real numbers cn в‰Ґ 0 (for n в‰Ґ 0)
exist so that for every N в‰Ґ 0 and every u as above rg (u, N ) в‰¤ cN .
(a) Setting g(u) = 0 for arg u = d, |u| > ПЃ, show rg (u, N ) в‰¤ cN ,
Лњ
for every N в‰Ґ 0, every u with arg u = d, and
N в€’1
cn ПЃnв€’N }.
cN = max{cN ,
Лњ
n=0

(b) For z with cos(k(d в€’ arg z)) в‰Ґ Оµ > 0 and N в‰Ґ 0, set fn =
gn О“(1 + n/k) and show
rf (z, N )z n в‰¤ KN = Оµв€’1в€’N/k cN О“(1 + N/k).
Лњ
3. With cn , Kn as above, assume for s1 в‰Ґ 0 that cn в‰¤ c K n О“(1 + ns1 ),
n в‰Ґ 0, with suп¬ѓciently large c, K в‰Ґ 0, independent of n. Show that
ЛњЛњ
then Kn в‰¤ cK n О“(1 + ns2 ), with s2 = 1/k + s1 and suп¬ѓciently large
ЛњЛњ
c, K в‰Ґ 0 (independent of n, but depending on Оµ > 0). Use this to
prove the above version of RittвЂ™s theorem for Gevrey asymptotics.
4. For k > 0, b в€€ E , and c > 0, let f (z) = b exp[в€’cz в€’k ]. Show that
f (z) в€ј1/k Л† in S(0, ПЂ/k), with Л† being the zero power series.
0 0
=
5. Use the previous exercise to conclude that to every sectorial region
G of opening not more than ПЂ/k there exists f (z), holomorphic and
nonzero in G, with f (z) в€ј1/k Л† in G.
0
=
6. Let G be a sectorial region of arbitrary opening, and let f (z) be
holomorphic in G with f (z) в€ј1/k Л† in G. To each closed subsector S
ВЇ
0
=
of G, п¬Ѓnd c1 , c2 > 0 so that f (z) в‰¤ c1 exp[в€’c2 |z|в€’k ] in S.
ВЇ

7. Let G be a sectorial region of opening not more than sПЂ, and let
f be holomorphic in G with f (z) в€јs Л† in G. Moreover, let g(z) be
=0
holomorphic in G and so that for k < 1/s and some real c1 , c2 > 0
we have g(z) в‰¤ c1 exp[c2 |z|в€’k ] in G. Show f (z)g(z) в€јs Л† in G.
=0
4.7 Gevrey Asymptotics in Wide Regions 75

4.7 Gevrey Asymptotics in Wide Regions
We have learned from Proposition 10 that the mapping J : As (G, E ) в†’
E [[z]]s is surjective for sectorial regions of opening at most sПЂ. We now
show that for wider regions J is injective вЂ“ the fact that then it is no longer
surjective will follow from results in the next chapter. The proposition we
are going to show is due to Watson :

Proposition 11 (WatsonвЂ™s Lemma) Suppose that G is a sectorial region
of opening more than sПЂ, s > 0, and let f в€€ H(G, E ) satisfy f (z) в€јs Л† in
=0
G. Then f (z) в‰Ў 0 in G.

ВЇ ВЇ
Proof: Let S = S(d, О±, ПЃ) be any closed subsector of G of opening О± > sПЂ.
From Exercise 6 on p. 74 we conclude for suitably large constants c, K and
k = 1/s
в€’k
f (z) в‰¤ c eв€’K|z| , z в€€ S. ВЇ
ВЇ
In particular, f (z) is bounded, say, by C, on S. For Оє = ПЂ/О± (< k)
and z = z(w) = eid (ПЃв€’Оє + w)в€’1/Оє we have z в€€ S for every w in the
ВЇ
closed right half-plane. Thus, for arbitrary x > 0, the function g(w) =
exp[x w]f (z(w)) is bounded by C on the line Re w = 0 and, because of
Оє < k, bounded by some, possibly larger, constant for Re w в‰Ґ 0. PhragmВґn-
e
LindelВЁfвЂ™s principle (p. 235) then implies
o

g(w) = exp[x Re w] f (z(w)) в‰¤ C Re w в‰Ґ 0.

Letting x в†’ в€ћ completes the proof. 2
Given a sectorial region G, we shall say that a subspace B of A(G, E )
is an asymptotic space, if the mapping J : B в†’ E [[z]]s is injective. In this
terminology we may express the above proposition as saying that As (G, E )
is an asymptotic space if the opening of G is larger than sПЂ. This result, in
a way, is the key to what will follow in the next chapters: Given a formal
Л†
power series f (z) and a sectorial region G of opening larger than sПЂ, if we
succeed in п¬Ѓnding a holomorphic function f with f (z) в€јs f (z) in G, then
=Л†
it is unique, and it is justiп¬Ѓed, if not to say very natural, to consider this
Л†
f as a sum of some sort for f . While this abstract deп¬Ѓnition of a sum will
be seen to have very natural properties, one certainly would like to have
a way of somehow calculating f , and we shall see that this indeed can be
done, at least in principle.

Exercises:
Л†
1. Let f be holomorphic for |z| < ПЃ, ПЃ > 0, and let f be its power series
expansion. Conclude that then for every sectorial region G and every
s > 0 we have f (z) в€јs f (z) in G.
=Л†
76 4. Asymptotic Power Series

2. Show the following converse of the previous exercise: Let G be a
sectorial region of opening more than (2 + s)ПЂ, for some s > 0, and
assume f (z) в€јs f (z) in G. Show that then f в€€ C {z}.
=Л† Л†
5
Integral Operators

In this chapter we introduce a certain type of integral operators that shall
play an important role later on. The simplest ones in this class are Laplace
operators, while some of the others will be EcalleвЂ™s acceleration operators,
that will be studied in detail in Chapter 11. All of them will be used in the
next chapter to deп¬Ѓne some summability methods that, in the terminology
common in this п¬Ѓeld, are called moment methods. As we shall prove, most
of these methods are equivalent in the sense that they sum the same formal
power series to the same holomorphic functions. The reason for investigat-
ing all these equivalent methods is that for particular formal power series it
will be easier to check applicability of a particular method. Thus having all
of them at our disposal gives a great deal of п¬‚exibility. Moreover, it also is
of theoretical interest to know what properties of the methods are needed
to sum a certain class of formal power series. It should, however, be noted
that statements on methods being equivalent are here to be understood
for summation of power series in interior points of certain regions, while
the methods may be inequivalent when studying the same series at some
boundary point.
Laplace and Borel operators are used in several diп¬Ђerent areas of math-
ematics, and are therefore deп¬Ѓned and studied in many textbooks, such
as  and others. They will also be the main tool in our theory of multi-
summability. Here, we only apply them to functions that are holomorphic
in sectorial regions and continuous at the origin, which simpliп¬Ѓes most of
the proofs for their properties. For example, the identity theorem for holo-
morphic functions implies that formulas proven for some вЂњsmallвЂќ set of
complex numbers, e.g., an inп¬Ѓnite set accumulating at some interior point
78 5. Integral Operators

of a region, immediately extend to the full region where all terms involved
are holomorphic. Moreover, it will be convenient to slightly adjust the def-
inition of Laplace operators, so that a power of the independent variable is
mapped to the same power times a constant. For these reasons, we choose
to include all the proofs for the properties of these operators.
A very recent paper dealing with Borel transform is by Fruchard and
SchВЁfke . For a discussion of related but more general integral operators
a
see, e.g., Braaksma , Schuitman , or their joint article .

5.1 Laplace Operators
Let S = S(d, О±) be a sector of inп¬Ѓnite radius, and let f в€€ A(k) (S, E ),
which we deп¬Ѓned to mean that f is holomorphic and of exponential growth
at most k > 0 in S and continuous at the origin. For П„ with |d в€’ П„ | < О±/2,
в€ћ(П„ )
f (u) exp[в€’(u/z)k ] duk , with integration along arg u = П„ ,
the integral 0
converges absolutely and compactly in the open set cos(k[П„ в€’arg z]) > c|z|k ,
if c is taken suп¬ѓciently large, depending upon f and П„ . On the Riemann
surface of the logarithm, the region described by this inequality in general
has inп¬Ѓnitely many connected components, one of which is speciп¬Ѓed by the
inequalities (4.1) (p. 61). These describe a sectorial region G = G(П„, ПЂ/k)
of opening ПЂ/k and bisecting direction П„ , which has been discussed in Ex-
ercise 2 on p. 61. In G, the function
в€ћ(П„ )
в€’k
f (u) exp[в€’(u/z)k ] duk
g(z) = z (5.1)
0

is holomorphic. According to the deп¬Ѓnition of exponential growth, the
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