˜ ˆ

only if g(z) ∼s/p g (z) in G.

˜

ˆ

=

3. Suppose f (z) ∼s f (z) in G, and let the constant term of f (z) be zero.

=ˆ ˆ

Show that then z ’1 f (z) ∼s z ’1 f (z) in G.

ˆ

=

4. Suppose f (z) ∼s f (z) in G(d, ±). Show that f (ze±2πi ) ∼s f (z) in a

=ˆ =ˆ

corresponding sectorial region G(d “ 2π, ±).

∞

ˆ

5. Given s > 0 and a formal Laurent series f (z) = n=’m fn z n , for

some m ∈ N, show that the following two statements are equivalent:

4.6 Gevrey Asymptotics in Narrow Regions 73

fn z n ∼s

’1 ∞

(a) f (z) ’ fn z n in G,

=

n=’m n=0

∞

(b) z m f (z) ∼s fn’m z n in G.

=

n=0

Let either one serve as de¬nition for f (z) ∼s f (z) in G.

=ˆ

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 ¬nite, and

such that f (z) ∼s f (z) for some formal Laurent series f . Show that

=ˆ ˆ

As,m (G, E ) is a di¬erential algebra. In case E = C (a ¬eld), show

that As,m (G, C ) is a di¬erential ¬eld, i.e., a di¬erential algebra in

which each element is invertible.

7. Show the existence of f ∈ A(G, E ) that is not in As (G, E ), for any

s ≥ 0.

8. Let fn ∈ As (G, E ), n ≥ 0. Assume that for every m ≥ 0 the sequence

(m) ¯

(fn )n converges uniformly on every closed subsector S of G. More-

¯

over, assume existence of c, K, depending upon S but independent of

(m) ¯

n and m, so that fn (z) ¤ c K m m! “(1 + sm) for every z ∈ S.

Show f = lim fn ∈ 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 Ritt™s theorem, adapted to the theory

of Gevrey asymptotics; its proof makes use of the so-called ¬nite Laplace

operator, which is de¬ned and discussed in some detail in the following

exercises.

Proposition 10 (Ritt™s Theorem for Gevrey Asymptotics) For

ˆ

s > 0, let f (z) ∈ C [[z]]s and a sectorial region G of opening at most sπ be

arbitrarily given. Then there exists a function f (z), holomorphic in G, so

that f (z) ∼s f (z) in G.

=ˆ

∞ ∞

ˆ

Proof: Let f (z) = 0 fn z n and de¬ne g(u) = 0 fn un /“(1+sn), then

ˆ

f (z) ∈ C [[z]]s implies holomorphy of g(u) for |u| su¬ciently small. Let d

be the bisecting direction of G and de¬ne for a = ρeid , with su¬ciently

small ρ > 0, and k = 1/s:

a

’k

g(u) exp[’(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 ) ’ E [[z]]s is

surjective if the opening of G is smaller than or equal to sπ. It is, however,

in this case not injective, as one learns from the following exercises.

Exercises: If not otherwise speci¬ed, let g(u) be continuous for arg u = d

and 0 ¤ |u| ¤ ρ with ¬xed d and ρ > 0, and de¬ne f (z) by (4.8), for k > 0

and a = ρeid .

1. Show that f (z) is holomorphic on the Riemann surface of the loga-

rithm. In the literature, the mapping g ’’ f is named ¬nite Laplace

operator of order k.

2. Assume that complex numbers gn and real numbers cn ≥ 0 (for n ≥ 0)

exist so that for every N ≥ 0 and every u as above rg (u, N ) ¤ cN .

(a) Setting g(u) = 0 for arg u = d, |u| > ρ, show rg (u, N ) ¤ cN ,

˜

for every N ≥ 0, every u with arg u = d, and

N ’1

cn ρn’N }.

cN = max{cN ,

˜

n=0

(b) For z with cos(k(d ’ arg z)) ≥ µ > 0 and N ≥ 0, set fn =

gn “(1 + n/k) and show

rf (z, N )z n ¤ KN = µ’1’N/k cN “(1 + N/k).

˜

3. With cn , Kn as above, assume for s1 ≥ 0 that cn ¤ c K n “(1 + ns1 ),

n ≥ 0, with su¬ciently large c, K ≥ 0, independent of n. Show that

˜˜

then Kn ¤ cK n “(1 + ns2 ), with s2 = 1/k + s1 and su¬ciently large

˜˜

c, K ≥ 0 (independent of n, but depending on µ > 0). Use this to

prove the above version of Ritt™s theorem for Gevrey asymptotics.

4. For k > 0, b ∈ E , and c > 0, let f (z) = b exp[’cz ’k ]. Show that

f (z) ∼1/k ˆ in S(0, π/k), with ˆ 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 π/k there exists f (z), holomorphic and

nonzero in G, with f (z) ∼1/k ˆ in G.

0

=

6. Let G be a sectorial region of arbitrary opening, and let f (z) be

holomorphic in G with f (z) ∼1/k ˆ in G. To each closed subsector S

¯

0

=

of G, ¬nd c1 , c2 > 0 so that f (z) ¤ c1 exp[’c2 |z|’k ] in S.

¯

7. Let G be a sectorial region of opening not more than sπ, and let

f be holomorphic in G with f (z) ∼s ˆ 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) ¤ c1 exp[c2 |z|’k ] in G. Show f (z)g(z) ∼s ˆ 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 ) ’

E [[z]]s is surjective for sectorial regions of opening at most sπ. We now

show that for wider regions J is injective “ 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 [282]:

Proposition 11 (Watson™s Lemma) Suppose that G is a sectorial region

of opening more than sπ, s > 0, and let f ∈ H(G, E ) satisfy f (z) ∼s ˆ in

=0

G. Then f (z) ≡ 0 in G.

¯ ¯

Proof: Let S = S(d, ±, ρ) be any closed subsector of G of opening ± > sπ.

From Exercise 6 on p. 74 we conclude for suitably large constants c, K and

k = 1/s

’k

f (z) ¤ c e’K|z| , z ∈ S. ¯

¯

In particular, f (z) is bounded, say, by C, on S. For κ = π/± (< k)

and z = z(w) = eid (ρ’κ + w)’1/κ we have z ∈ S for every w in the

¯

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

κ < k, bounded by some, possibly larger, constant for Re w ≥ 0. Phragm´n-

e

Lindel¨f™s principle (p. 235) then implies

o

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

Letting x ’ ∞ 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 ’ 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 sπ. This result, in

a way, is the key to what will follow in the next chapters: Given a formal

ˆ

power series f (z) and a sectorial region G of opening larger than sπ, if we

succeed in ¬nding a holomorphic function f with f (z) ∼s f (z) in G, then

=ˆ

it is unique, and it is justi¬ed, if not to say very natural, to consider this

ˆ

f as a sum of some sort for f . While this abstract de¬nition 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:

ˆ

1. Let f be holomorphic for |z| < ρ, ρ > 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) ∼s f (z) in G.

=ˆ

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)π, for some s > 0, and

assume f (z) ∼s f (z) in G. Show that then f ∈ C {z}.

=ˆ ˆ

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 Ecalle™s acceleration operators,

that will be studied in detail in Chapter 11. All of them will be used in the

next chapter to de¬ne some summability methods that, in the terminology

common in this ¬eld, 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 ¬‚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 di¬erent areas of math-

ematics, and are therefore de¬ned and studied in many textbooks, such

as [284] 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 simpli¬es most of

the proofs for their properties. For example, the identity theorem for holo-

morphic functions implies that formulas proven for some “small” set of

complex numbers, e.g., an in¬nite 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

Sch¨fke [104]. For a discussion of related but more general integral operators

a

see, e.g., Braaksma [65], Schuitman [246], or their joint article [73].

5.1 Laplace Operators

Let S = S(d, ±) be a sector of in¬nite radius, and let f ∈ A(k) (S, E ),

which we de¬ned to mean that f is holomorphic and of exponential growth

at most k > 0 in S and continuous at the origin. For „ with |d ’ „ | < ±/2,

∞(„ )

f (u) exp[’(u/z)k ] duk , with integration along arg u = „ ,

the integral 0

converges absolutely and compactly in the open set cos(k[„ ’arg z]) > c|z|k ,

if c is taken su¬ciently large, depending upon f and „ . On the Riemann

surface of the logarithm, the region described by this inequality in general

has in¬nitely many connected components, one of which is speci¬ed by the

inequalities (4.1) (p. 61). These describe a sectorial region G = G(„, π/k)

of opening π/k and bisecting direction „ , which has been discussed in Ex-

ercise 2 on p. 61. In G, the function

∞(„ )

’k

f (u) exp[’(u/z)k ] duk

g(z) = z (5.1)

0

is holomorphic. According to the de¬nition of exponential growth, the