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h = f ˆk g :
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.

Using the above estimates, we ¬nd
f ((z ’ t)s ) tsN rg (ts , N )dt ¤ c2 K N “(1 + N s1 )|z|1+sN /(1 + s1 N ),

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

This shows altogether for su¬ciently large c, K > 0, independent of N , z
as above, and N ≥ 0
wsN rh (ws , N )dw ¤ cK N |z|1+sN “(1 + N s1 ).

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

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-
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

uk wk
(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
(f±,k —k g)(z) = g(t) dtk , ± > 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
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
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 .
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
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

formally hold when we replace aj by its power series, fj by fj , and xj by
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
2. Formally, show (aj —kj xj )(z) = xj (z) ’
ˆ ˆ b (z, t) xj (t) dt,
ˆ with

an (z kj ’ tkj )n/kj ’1 /“(n/kj ).
bj (z, t) =

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.
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-
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
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


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