¯

phic in the interior of Sm , continuous up to its boundary, and Xm (z) ¤

4’m c |z|’2 , which for m = 0 is correct. Then for any path of integration

¯

in Sm , the above integral converges compactly for z not on the path. Con-

¯—

sequently, Um (z) is holomorphic in interior points of Sm and tends to 0 as

¯— ¯—

z ’ ∞ in Sm+1 . Moreover, for z ∈ Sm+1 we can always integrate along

the boundary of Sm , in which case |u ’ z| ≥ δm+1 ’ δm = 2’m’1 δ. There-

¯

fore, Um (z) ¤ 2’m Lc [π δ]’1 , z ∈ Sm+1 , where L is an upper bound

¯—

for the integral of |u|’2 d|z| along the corresponding path. By choosing ρ ˜

’m’3

large enough, we can ensure that 8 L c ¤ δ π, so that Um (z) ¤ 2

¯— ¯—

in Sm+1 . Consequently, I + Um (z) is invertible in Sm+1 , and according to

Exercise 1 on p. 104, its inverse can be bounded by 2. Therefore, we obtain

|z|2 Xm+1 (z) ¤ 2’3m’2 c ¤ 4’m’1 c in Sm+1 . Altogether, this shows that

¯

the above iterative de¬nition works, producing sequences Um (z), Xm (z)

— —

which are holomorphic in S∞ , resp. S∞ , being the intersection of Sm , resp.

Sm . From the de¬nition of Um (z), and using Cauchy™s formula, we ob-

tain Um (ze2πi ) ’ Um (z) = Xm (z), z ∈ S∞ , m ∈ N0 . Using this and

the de¬nition of the Xm (z), one then ¬nds [I + Um (z)] [I + Xm (z)] =

[I + Xm+1 (z)] [I + Um (ze2πi )]. We now de¬ne matrices Tm (z) by

I + Tm (z) = [I + Um (z)] · . . . · [I + U0 (z)], m ≥ 0, z ∈ S∞ .

—

Owing to Exercise 4, the sequence Tm (z) converges uniformly on S∞ , de¬n-

ing a matrix function T (z) that is holomorphic there and vanishes for

z ’ ∞, since all the Tm (z) do the same. According to the above relations

9.7 The Freedom of the Highest-Level Invariants 157

for Um (z), we ¬nd [I + Tm (z)] [I + X(z)] = [I + Xm+1 (z)] [I + Tm (ze2πi )].

Since Xm (z) ’ 0 for m ’ ∞, uniformly on S∞ , this then implies [I +

T (z)] [I + X(z)] = I + T (ze2πi ) in S∞ , which completes the proof. 2

In order to formulate in which sense the Stokes multipliers are free, con-

ˆ

sider any system (3.1) (p. 37). This system has an HLFFS (F (z), Y (z)), and

corresponding Stokes™ multipliers Vj , which are restricted by (9.4) and the

support condition stated in Theorem 45. Now, consider any set of constant

matrices Vj , j ∈ Z, with Vj+j0 = e’2πiL Vj e2πiL , and (n, m) ∈ Suppj ’

˜ ˜ ˜

˜ (j)

Vn,m = 0. Using the above lemma, it is easy to prove existence of another

system having an HLFFS with the same Y (z) and these Stokes multipliers:

Theorem 49 Under the assumptions stated above, there exists a system

˜ ˆ ˆ

z x = A(z) x having an HLFFS (I + T (z) F (z), Y (z)), with a formal ana-

˜ ˜

ˆ

lytic transformation I + T (z) of Gevrey order k, and corresponding Stokes™

˜

multipliers Vj , j ∈ Z.

˜

Proof: For the proof, it su¬ces to consider the case of Vj = Vj for all but

one index j between 0 and j0 ’1; applying this result repeatedly, the general

˜

case follows immediately. For which index j we allow Vj to di¬er from Vj is

also inessential, because a change of variable z ’ zeid , for suitable d, may

be used to shift the enumeration of the Stokes multipliers by any amount.

Thus, for the proof of the above result we restrict ourselves to the case of

˜ ˜

Vj = Vj , 1 ¤ j ¤ j0 ’ 1, and we de¬ne W by I + V0 = (I + W ) (I + V0 ).

According to Exercise 4 on p. 142, this matrix W also satis¬es the support

condition for j = 0, so that the X(z) = X0 (z) W X0 (z) ∼1/k ˆ in S0 ©

’1

0

=

S’1 , whose boundary rays are d’1 ± π/(2k). Hence, Lemma 17 may be

¯

applied to this matrix X(z) and any closed subsector S(ρ) of S0 © S’1 .

¯

By choosing S(ρ) large, resp. δ > 0 small enough, the matrix T (z) so

obtained is holomorphic in a corresponding sector S — with boundary rays

d’1 ’π/(2k)+µ, resp. d’1 +π(2+1/(2k))’µ, where µ > 0 is so small that no

singular direction lies within (d’1 ’ µ, d’1 + µ). Moreover, we may assume

that I + T (z) is invertible there; otherwise, the radius of S — can be made

larger. For X(z) = [T + T (z)] X0 (z), one can verify X(ze2πi ) = X(z) [I +

˜ ˜ ˜

W ] exp[2πi M0 ], with M0 as in (9.5). De¬ning A(z) = z X (z) X ’1 (z), we

˜ ˜ ˜

˜

have a system for which X(z) is a fundamental solution.

Since the matrix X(z) is asymptotically zero of Gevrey order 1/k, we

conclude from Proposition 18 (p. 121) that T (z) ∼1/k T (z) in S — . There-

ˆ

=

ˆ ˆ ˜

fore, [I + T (z)] F (z), Y (z)) is an HLFFS of z x = A(z) x, and its data pairs

˜ ˜

ˆ (z), Y (z)). According to the Main Theorem in

are the same as those of (F

ˆ

Section 8.3, T (z) is k-summable, with singular directions among those for

the system (3.1). Owing to the size of S — , we conclude that the k-sum of

ˆ

T (z) equals T (z), for all directions d between d’1 ’ µ and d’1 + 2π + µ. Be-

cause of the smallness of µ, this shows that d1 is the only singular direction,

ˆ ˜

modulo 2π, of T (z). Consequently, for 0 ¤ j ¤ j0 , the matrices Fj (z) =

158 9. Stokes™ Phenomenon

ˆ ˆ

[I +T (z)] Fj (z) are the HLNS corresponding to [I + T (z)] F (z), Y (z)). From

˜

this we obtain that the corresponding Stokes™ multipliers Vj equal Vj , for

1 ¤ j ¤ j0 . The matrix F’1 (z) = [I + T (ze2πi )] F’1 (z) has the correct

˜

asymptotic in S’1 , and hence is the corresponding HLNS. This implies

˜ 2

I + V0 = [I + W ] (I + V0 ), completing the proof.

Exercises:

1. For x ≥ 0, show log(1 + x) ¤ x.

∞

xm < ∞, show that the

2. For nonnegative real numbers xm with 0

n

sequence pn = m=0 (1 + xm ) converges.

3. For arbitrary ν — ν-matrices An , show (I + Am ) · . . . · (I + A0 ) ’ I ¤

m

n=0 (1 + An ) ’ 1.

4. Show that the sequence Tm (z), de¬ned in the proof of Lemma 17,

converges uniformly on S∞ .

10

Multisummable Power Series

In previous chapters we have shown that HLFFS are very natural to con-

sider when discussing Stokes™ phenomenon, since they are k-summable,

for some suitable k > 0. Unfortunately, their computation is not so easy,

because of one step requiring use of Banach™s ¬xed point theorem. So in

applications one may prefer to work with formal fundamental solutions in

the classical sense. They can be computed relatively easily, using computer

algebra tools which will brie¬‚y be discussed in Section 13.5. In Section 8.4

we have shown these FFS to be a product of ¬nitely many matrix power

series, each of which is k-summable with a value of k depending on the

factor. However, this factorization is neither unique nor fully constructive,

so the problem of summation of FFS remains. In this chapter we are now

presenting a summability method that is stronger than k-summability for

every k > 0, enabling us to sum FFS as a whole. This method, named multi-

summability, was ¬rst introduced in somewhat di¬erent form by Ecalle [94],

using what he called acceleration operators. Here, we present an equivalent

de¬nition, based on the more general integral operators introduced in Sec-

tions 5.5 and 5.6. We also show that it would be su¬cient to work with

Laplace operators only, but it can be convenient in applications to have

the more general integral operators at hand: Sometimes one will meet for-

mal power series for which it will be simpler to show applicability of some

particular integral operator, but more complicated to do so for a Laplace

operator, although theoretically they are equivalent.

As we stated above, multisummability will turn out to be stronger than

k-summability, for every k > 0. On the other hand, it will be not too

much stronger, as we are going to show that every multisummable formal

160 10. Multisummable Power Series

power series can be written as a sum of ¬nitely many formal power series,

each of which is k-summable, in some direction, for an individual k > 0,

depending on the series. Thus, roughly speaking, the set of multisummable

power series is the linear hull of the union of the sets of k-summable ones.

This description, although quite suggestive, is not entirely correct, as one

also has to consider the corresponding directions, but that shall be made

clearer later on.

Several authors have presented and/or applied Ecalle™s theory of mul-

tisummability, e.g., Martinet and Ramis [186, 187], Malgrange and Ramis

[185], Loday-Richaud [169, 170], Thomann [260“262], Balser and Tovbis

[43], Malgrange [184], and Jung, Naegele, and Thomann [142].

10.1 Convolution Versus Iteration of Operators

In Sections 5.5 and 5.6 we introduced general kernel functions e(z) and cor-

responding integral operators T . The most important examples of kernels

of order k > 0 are e(z) = k z k exp[z k ]; the corresponding operators be-

ing Laplace resp. Borel operators. Other examples are Ecalle™s acceleration

operators, which shall be investigated in the following chapter.

Given two kernels e1 (z), e2 (z) of orders k1 , k2 > 0 with corresponding

operators T1 , T2 and moment functions m1 (u), m2 (u), we have de¬ned

in Theorem 31 (p. 94) a new kernel e(z) of order k = (1/k1 + 1/k2 )’1

with corresponding moment function m1 (u) m2 (u). This kernel e(z) shall

be called the convolution e1 — e2 of e1 and e2 . It de¬nes an integral operator

T1 — T2 , for which we now investigate its relation with T1 —¦ T2 .

Lemma 18 Under the above assumptions, let f ∈ A(k) (S(d, µ), E ), for

some d ∈ R, µ > 0, so that T1 —T2 f is de¬ned. Then g = T2 f is holomorphic

in the sector S(d, µ + π/k2 ), and is of exponential growth not more than k1

there; consequently, T1 g is de¬ned, integrating along any direction d1 with

2|d ’ d1 | < µ + π/k2 , and T1 g = T1 — T2 f .

Proof: Use the de¬nition of e1 — e2 and justify interchanging the order of

integration in the formula representing T1 — T2 f . 2

The above lemma shows that T1 — T2 equals T1 —¦ T2 whenever the ¬rst one

is de¬ned. However, examples show that T1 —¦ T2 can be applied to a wider

set of functions. In a sense, this observation is the key to multisummability.

Exercises: In the following exercises, let kernels ej (z) of arbitrary orders

κj > 0 (1 ¤ j ¤ q) be given.

1. Show e1 — e2 = e2 — e1 , (e1 — e2 ) — e3 = e1 — (e2 — e3 ).

2. Let f ∈ A(k) (S(d, µ), E ), for k ’1 = 1 κ’1 . Show T1 — . . . — Tq f =

q

j

T1 —¦ . . . —¦ Tq f and discuss the choice of directions of integration in

10.2 Multisummability in Directions 161

the multiple integral on the right-hand side. Furthermore, show that

for f as above, T1 —¦ . . . —¦ Tq does not depend on the enumeration of

the Tj .

10.2 Multisummability in Directions

In what follows, we shall consider a ¬xed tuple T = (T1 , . . . , Tq ) of inte-

gral operators Tj of respective orders κj > 0. Furthermore, we consider a

likewise ¬xed multidirection d = (d1 , . . . , dq ), which we call admissible with

respect to T , provided that

2κj |dj ’ dj’1 | ¤ π, 2 ¤ j ¤ q. (10.1)

Given T and d, we are now going to de¬ne T -summability in the multidi-

ˆ

rection d. To do so, we consider the inverse of the formal operators Tj , i.e.,

(Tj’1 f )(z) = fn z n /mj (n), for f (z) = fn z n ∈ E [[z]].

ˆˆ ˆ

ˆ

• For q = 1, we say that a formal power series f ∈ E [[z]] is T -summable

ˆ’1 ˆ

in the multidirection d, provided that g = T1 f ∈ E {z}, and that

ˆ

its sum g(z) is in A(κ1 ) (S(d1 , µ), E ), for some µ > 0.

˜

˜

• For q ≥ 2, set T = (T2 , . . . , Tq ), d = (d2 , . . . , dq ). Then we say that

ˆ

a formal power series f ∈ E [[z]] is T -summable in the multidirection

ˆ’1 ˆ ˜

˜

d, provided that g = T1 f is T -summable in the multidirection d,

ˆ

and that its T -sum g(z) is in A(κ1 ) (S(d1 , µ), E ), for some µ > 0.

˜

• In both cases, note that the operator T1 can be applied to the function

g(z), integrating along a ray inside the sector S(d1 , µ). We then say

ˆ

that f = T1 g is the T -sum of f in the multidirection d, and write

ˆ

f = ST ,d f .

We shall write E {z}T ,d for the set of all formal power series that are

T -summable in the multidirection d. Observe for q = 1, that Theorem 38

(p. 108) implies E {z}T ,d = E {z}κ1 ,d .

Exercises: In what follows, let T and d, as above, be given.

1. Show that the above inductive de¬nition of T -summability in a mul-

ˆ

tidirection d is equivalent to the following: Given f (z), we ¬rst divide

the coe¬cients fn by m1 (n) · . . . · mq (n) to obtain a convergent series

ˆ

“ hence f (z) has to be in E [[z]]s , with s = 1/κ1 + . . . + 1/κq , for this