z

d

f ((z ’ t)s ) tsN rg (ts , N )

sN s

z rh (z , N ) =

dz 0

N ’1

gm (z ’ t)sm ts(N ’m) rf (ts , N ’ m) dt.

+

m=0

Using the above estimates, we ¬nd

z

f ((z ’ t)s ) tsN rg (ts , N )dt ¤ c2 K N “(1 + N s1 )|z|1+sN /(1 + s1 N ),

0

whereas, using Exercise 1 on p. 41,

z N ’1

gm tsm (z ’ t)s(N ’m) rf ((z ’ t)s , N ’ m) dt

0 m=0

N ’1

c2 K N |z|1+sN

¤ “(1 + ms1 ) “(1 + (N ’ m)s1 ).

1 + sN m=0

ˆˆ

This shows altogether for su¬ciently large c, K > 0, independent of N , z

as above, and N ≥ 0

z

wsN rh (ws , N )dw ¤ cK N |z|1+sN “(1 + N s1 ).

ˆˆ

0

Using Cauchy™s formula for the ¬rst derivative, one can see that this implies

h(z) ∼s1 h(z) in G. An elementary estimate can be used to show that h is

=ˆ

2

of the desired exponential growth, in case f and g are.

As we indicated above, the acceleration operators are well behaved with

respect to convolutions. To prove this, we ¬rst show that the Laplace op-

erator of order k maps convolutions of the same index onto products:

Theorem 57 Assume that E is a Banach algebra. Let S be a sector of

in¬nite radius, and let k > 0 be arbitrarily given. Moreover, let f, g be

E -valued functions, holomorphic in S, continuous at the origin, and of

exponential growth not more than k. Then

Lk (f —k g) = (Lk f )(Lk g).

180 11. Ecalle™s Acceleration Operators

˜

Proof: Setting h = f —k g, h = Lk h, s = 1/k, and choosing d appro-

∞(d)

˜

priately, we have h(z s ) = 0 exp[’w/z] h(ws ) dw. Inserting for h(ws ),

integrating by parts, and then interchanging the order of integration shows

˜ 2

h = (Lk f )(Lk g).

We now deal with the acceleration operators:

Theorem 58 Assume that E is a Banach algebra. Let S be a sector of

˜ ˜

in¬nite radius, let k > k > 0 be arbitrarily given, and take 1/κ = 1/k ’1/k.

Moreover, let f, g be E -valued functions, holomorphic in S, continuous at

the origin, and of exponential growth not more than κ. Then

Ak,k (f —k g) = (Ak,k f ) —k (Ak,k g).

˜ ˜ ˜ ˜

˜ ˜ ˜

Proof: With h = (Ak,k f ) —k (Ak,k g), ± = k/k, s = 1/k, and suitable d,

˜

˜ ˜ ˜

we ¬nd

∞(d) ∞(d)

˜˜ g(w)k(z 1/± , w, u) dwk duk ,

h(z s ) = f (u)

0 0

z

uk wk

d

1/±

(z ’ t)’1/± C± t’1/± C±

k(z , w, u) = dt.

(z ’ t)1/± t1/±

dz 0

For ¬xed w and u, k(z, w, u) is the convolution of index ± of the functions

z ’1 C± (uk /z) and z ’1 C± (wk /z), which in turn are the Borel transform

k k

of order ± of z ’1 e’u /z and z ’1 e’w /z . From the previous theorem we

therefore ¬nd that k(z, w, u) is the Borel transform of the same order of the

k k k k

product z ’2 e’(u +w )/z = ’(‚/‚uk )z ’1 e’(u +w )/z . This in turn shows

k(z, w, u) = ’(‚/‚uk )z ’1 C± ((uk + wk )/z). So we obtain, replacing z by

˜

z ± , hence z s by z s with s = 1/k, and making corresponding changes of

variables in the integrals:

∞(kd) ∞(kd)

’1 ‚

˜

h(z s ) = f (us ) C± ((u + w)/z) du g(ws ) dw.

z ‚u

0 0

Setting u = t ’ w, hence (‚/‚u) = (‚/‚t), and interchanging the order of

integration, followed by integration by parts then gives

∞(kd) t

1 d

˜ f ((t ’ w)s )g(ws ) dw dt.

h(z s ) = C± (t/z)

z dt

0 0

˜

This, however, is equivalent to h = Ak,k (f —k g). 2

˜

Exercises: As above, assume that E is a Banach algebra. Let S be a

sectorial region, and let f, g be E -valued, holomorphic in G, and continuous

at the origin.

11.4 Convolution Equations 181

1. Show that limk’∞ (f —k g)(z) = f (z) g(z), z ∈ G.

2. Let E have unit element e, and set f±,k (z) = z ± e/“(1+±/k), ± ∈ R0 .

Show that f0,k —k g = g, while

(z k ’ tk )±/k’1

z

(f±,k —k g)(z) = g(t) dtk , ± > 0.

“(±/k)

0

11.4 Convolution Equations

We brie¬‚y mention the following application of the previous results: Let E

be a Banach algebra with unit element e, let an ∈ E be so that a(z) =

∞ ˆ

e ’ 1 an z n converges for |z| < ρ, with ρ > 0. Moreover, let f ∈ E {z}k,d ,

for some admissible k = (k1 , . . . , kq ) and d = (d1 , . . . , dq ). For a given

natural number r, the inhomogeneous di¬erential equation

ˆ

z r+1 x + a(z) x = f (z) (11.3)

∞ n

then has a unique formal solution x(z) = ˆ 0 xn z . Its coe¬cients are

n’1

given by the recursion xn = fn ’ (n ’ r) xn’r + m=0 an’m xm , n ≥ 0,

with x’r = . . . = x’1 = 0. We wish to show that this solution again

is multisummable. For simplicity of notation, let us assume that for some,

necessarily unique, j we have kj = r; if this were not so, we had to introduce

˜

a summability type k corresponding to the set { r, k1 , . . . , kq }. To deal

with the above question, it is convenient to make the canonical choice

ˆ ˆˆ

of operators Lk1 , Ak1 ,k2 , . . . , Akq’1 ,kq , so that fj = Bkj f . For notational

convenience, we here set k0 = ∞ and Ak0 ,k1 = Lk1 .

ˆ

Corresponding to f , there exist functions f0 , . . . , fq and formal power

ˆ ˆ

series f0 , . . . , fq as in Exercise 2 on p. 162. Each fj is holomorphic in a

sectorial region G(dj+1 , ±j ), with ±j > π/κj+1 (in case j ¤ q ’1), resp. in a

disc about the origin (in case j = q). Moreover, for j ≥ 1, the function fj (z)

admits holomorphic continuation into a sector S(dj , µ) and is of exponential

growth at most κj there. Any two consecutive functions fj , fj+1 are related

via the operator Akj ,kj+1 . Our goal is to prove the existence of functions xj ,

corresponding to x, and having the same features. To do so, we investigate

ˆ

the identities

(δ ’ r) (fr,kj —kj xj )(z) + (aj —kj xj )(z) = fj (z), 1 ¤ j ¤ q, (11.4)

where aj denotes the Borel transform of order kj of a, δ stands for the

operator z (d/dz), and fr,kj is as in Exercise 2 on p. 181. These equations

182 11. Ecalle™s Acceleration Operators

ˆ

formally hold when we replace aj by its power series, fj by fj , and xj by

ˆˆ

Bkj x. Without going into detail, we state the following:

For j with kj ¤ r, this identity is nothing but a Volterra-type integral

equation for the function xj . The usual iteration technique shows that every

such equation has a unique solution that has the same features as fj : It is

holomorphic in the region G(dj+1 , ±j ) (in case j ¤ q ’ 1), resp. in a disc

about the origin (in case j = q). Moreover, for j ≥ 1, the function xj ad-

mits holomorphic continuation into the sector S(dj , µ) and is of exponential

growth at most κj there. It then follows from uniqueness of the solutions

that, for every such j, we have xj = Akj ,kj+1 xj+1 , in case j < q.

For j with kj > r, identity (11.4) is of a di¬erent nature: It can be inter-

preted as a singular integral equation for xj , hence the standard iteration

method fails. However, once a solution xj is known, say, for z in a sector of

¬nite radius, then (11.4), for j ≥ 1, can still serve to show continuation of

xj into S(dj , µ), and get a growth estimate allowing application of Akj’1 ,kj .

This operator then de¬nes the next function xj’1 , and in this fashion all

x0 , . . . , xq are obtained.

While in the general theory of multisummability the choice of operators

was of no real importance, here it is essential to choose the acceleration

operators: For others, there is no notion comparing to the convolution of

functions, making it impossible to study the equations that correspond to

(11.3) by formal application of the inverse operators Tj’1 .

Exercises: Use the notation introduced above.

˜

1. For k > k > 0, show δ(Lk f ) = Lk (δf ), δ(Ak,k f ) = Ak,k (δf ), when-

˜ ˜

ever f is holomorphic and of appropriate growth in a sector of in¬nite

radius.

z

2. Formally, show (aj —kj xj )(z) = xj (z) ’

ˆ ˆ b (z, t) xj (t) dt,

ˆ with

0j

∞

an (z kj ’ tkj )n/kj ’1 /“(n/kj ).

bj (z, t) =

1

u

3. Show (δ ’ r) (fr,r —r x)(u) = r [ur x(u) ’ 0 x(t) dtr ], and write (11.4),

ˆ ˆ ˆ

for kj = r, as a Volterra integral equation.

12

Other Related Questions

In this chapter, we mention some additional results related to the theory

of either multisummability or ODE in the complex plane:

In the ¬rst section we shall give necessary and su¬cient conditions for so-

called matrix summability methods to be stronger than multisummability.

These conditions are very much analogous to the classical characterization

of regular summability methods. They may also be viewed as analogues

of conditions characterizing what is called power series regularity, meaning

that a summation method sums convergent power series, inside their disc of

convergence, to the correct sum. So far, only one matrix method is known

to be stronger than multisummability. On the other hand, for the subclass

of power series methods it has been shown in [32] that none of them can

have this property.

In the second section, we show in which sense a system of ODE of

Poincar´ rank r ≥ 2 is equivalent to one of rank 1, but higher dimen-

e

sion. This sometimes can be useful in generalizing results for systems of

rank r = 1 to the general case.

Sections 12.3 and 12.4 deal with two di¬erent but, as far as the methods

used are concerned, intimately related problems that both can roughly be

characterized as follows: Given a certain class of systems of ODE, together

with a number of parameters that, at least theoretically, can be computed

from every such system, then are these parameters free in the sense that

they can independently take on every value? For the Riemann-Hilbert prob-

lem, the class of systems under consideration are the Fuchsian systems, and

the parameters are their monodromy data. For the problem of Birkho¬ re-

duction the systems are those with polynomial coe¬cient matrix, and the

184 12. Other Related Questions

parameters are the so-called invariants associated to the singular point at

in¬nity which is assumed to have Poincar´ rank r ≥ 1. Both problems have

e

been believed to be entirely solved for quite some time, but we shall explain

that the answer to the ¬rst one is, in fact negative, while the second one is

still partially open.

The ¬nal section then deals with the central connection problem: Think

of a function given by a convergent power series; then what can be said

about its behavior on the boundary of the disc of convergence? If the func-

tion satis¬es a system of ODE, then its behavior is essentially determined