174 10. Multisummable Power Series

but ¬nitely many exceptional ones, while given dj , . . . , dq , the value

dj’1 can be any number with 2κj |dj ’ dj’1 | ¤ π but ¬nitely many,

for 2 ¤ j ¤ q. Using this terminology, show the following analogue to

the Main Decomposition Theorem:

ˆ

Theorem: Let f be multisummable. Then there exist k1 > . . . >

kq > 0, so that for almost all multidirections d = (d1 , . . . , dq ) we have

ˆˆ ˆ ˆ

f = f1 + . . . + fq , with fj ∈ E {z}kj ,dj .

Discuss in which cases we can take k = (k1 , . . . , kq ) to be the optimal

ˆ

type of f .

11

Ecalle™s Acceleration Operators

Although Ecalle™s acceleration operators are in no way special in our pre-

sentation of multisummability theory, they are nonetheless important in

many applications, owing to their close relation with Laplace transform

and convolution of functions. Therefore we now will introduce them and

show their main properties.

For real ± > 1 and complex z, let

1

u1/±’1 exp[u ’ z u1/± ] du,

C± (z) =

2πi γ

with a path of integration γ as in Hankel™s integral for the inverse Gamma

function: from ∞ along arg u = ’π to some u0 < 0, then on the circle

|u| = |u0 | to arg u = π, and back to ∞ along this ray. Because of ± > 1,

this integral represents an entire function of z. By termwise integration of

∞

exp[’z u1/± ] = 0 (’z)n un/± /n! and use of Hankel™s formula (p. 228), we

¬nd the power series expansion

∞

(’z)n

C± (z) = .

n! “(1 ’ (n + 1)/±)

n=0

Using (B.14) (p. 232) and Theorem 69 (p. 233), one can show that C±

is of exponential order β = ±/(± ’ 1) and ¬nite type. For ± = 2, one

can express C2 in terms of a con¬‚uent hypergeometric function, while for

± ’ ∞, C± (z) ’ e’z . By a change of variable zu1/± = w’1 , and then

substituting z = t’1 , we see that t’1 C± (t’1 ) is the Borel transform of

index ± of z ’1 e’1/z . From Theorem 32 (p. 95) we see that e(t) = t C± (t)

176 11. Ecalle™s Acceleration Operators

therefore is a kernel function of order β = ±/(± ’ 1), corresponding to the

moment function m(u) = “(1 + u)/“(1 + u/±).

11.1 De¬nition of the Acceleration Operators

˜ ˜

For real numbers d and k > k > 0, set ± = k/k. Then the function

ek,k (t) = k tk C± (tk ) is a kernel function of order κ = k β = (1/k ’ 1/k)’1 .

˜

˜

The corresponding integral operator, denoted by Ak,k , shall be named the

˜

˜ ˜

acceleration operator with indices k and k, bearing in mind that k > k is

always implicitly assumed. One veri¬es easily that

∞(d)

’k

f (t) C± (t/z)k d tk ,

(Ak,k f )(z) = z

˜

0

for f ∈ A(κ) (S, E ). Note that our de¬nition of acceleration operators di¬ers

slightly from the one used by other authors; this is so to make them match

with our de¬nition of Borel and Laplace transforms.

It follows right from the de¬nition that the moment function correspond-

˜

ing to Ak,k is “(1 + u/k)/“(1 + u/k). This motivates to de¬ne the formal

˜

∞

ˆ fn z n by

acceleration operator of a formal power series f (z) = 0

∞

“(1 + n/k)

ˆ˜ ˆ

(Ak,k f )(z) = z n fn . (11.1)

˜

“(1 + n/k)

0

The following theorem is a result of our general discussion of integral

operators, especially of Theorem 27 (p. 91):

Theorem 55 For k > k > 0 and 1/κ = 1/k ’ 1/k, let f ∈ A(κ) (S, E )

˜ ˜

for a sector S = S(d, ±), and let g = Ak,k f , de¬ned in a corresponding

˜

sectorial region G = G(d, ± + π/k). For s1 ≥ 0, assume f (z) ∼s1 f (z) in

=ˆ

ˆˆ

S, take s2 = 1/κ + s1 , and let g = A˜ f . Then

ˆ k,k

g(z) ∼s2 g (z)

=ˆ in G.

Considering the corresponding moment functions, we obtain from Theo-

rem 31 (p. 94) that Lk = Lk — Ak,k . Therefore, Lemma 18 (p. 160) shows

˜ ˜

that Lk = Lk —¦ Ak,k on A(k) (S, E ):

˜ ˜

Theorem 56 In addition to the assumptions of Theorem 55 (p. 176), let f

be of exponential growth not more than k. Then g = Ak,k f is holomorphic

˜

˜ ˜

and of exponential growth not more than k in S = S(d, ± + π/κ), and

Lk g = Lk f .

˜

The inverse of the acceleration operators are brie¬‚y discussed in the

following exercises.

11.2 Ecalle™s De¬nition of Multisummability 177

˜ ˜

Exercises: In the following exercises, ¬x k > k > 0, and set ± = k/k,

∞

˜

1/κ = 1/k ’ 1/k. Moreover, let D± (z) = z n “(1 + n/±)/n! , for

n=0

z ∈ C.

1. Show that D± , de¬ned above, is an entire function of exponential

order β = ±/(± ’ 1) and ¬nite type. Moreover, prove the integral

∞(d)

exp[zx1/± ’ x] dx, for every z, and

representation D± (z) = 0

d ∈ (’π/2, π/2).

2. Prove1

∞

“(±(n + 1))

D± (’1/z) ∼1 ± (’1)n z ±(n+1)

=

n!

n=0

in S(0, π(1 + 1/±)).

3. For a sectorial region G = G(d, φ) of opening more than π/κ, and

f ∈ H(G, E ) continuous at the origin, de¬ne the deceleration operator

˜

Dk,k with indices k and k by

˜

1

z k f (z)D± (u/z)k d z ’k ,

(Dk,k f )(u) =

˜

2πi γκ („ )

with γκ („ ) as in Section 5.2. Show that Dk,k and Ak,k are inverse to

˜ ˜

one another in the sense of Section 5.7.

11.2 Ecalle™s De¬nition of Multisummability

Let any summability type k = (k1 , . . . , kq ) be given. By de¬nition, the

corresponding κj satisfy κ1 = k1 , 1/κj = 1/kj ’1/kj’1 , 2 ¤ j ¤ q. Since for

k-summability we may choose any integral operators Tj having the required

orders κj , we may take T1 = Lk1 , Tj = Akj’1 ,kj , 2 ¤ j ¤ q. These operators

were used by Ecalle in his original de¬nition of multisummability. It is a

very natural choice for the following reason: By looking at the corresponding

moment functions, we ¬nd

Lk1 — Ak1 ,k2 — . . . — Akq’1 ,kq = Lkq .

Therefore, we may say that the iterated operator Lk1 —¦Ak1 ,k2 —¦. . .—¦Akq’1 ,kq is

a natural extension of the operator Lkq to a wider class of functions. Hence,

1 Here we use a more general de¬nition of Gevrey asymptotics: For ±, k > 0, we

∞ ∞

say that f (z) ∼s f z ±n in S(d, β, r) if and only if f (z 1/±) ∼s± f z n in

= =

n=0 n n=0 n

± ). Compare this to Exercise 2 on p. 72 to see that this is correct in case of

S(±d, ±β, r

± being a natural number.

178 11. Ecalle™s Acceleration Operators

k-summability di¬ers from kq -summability in using this extended Laplace

operator instead of the usual one. Another good reason for prefering this

choice of operators over others lies in their strong relation with convolution

of functions. This will be discussed next.

Exercises: Let a ¬xed summability type k = (k1 , . . . , kq ) be given, and

let the corresponding operators be chosen as above.

ˆ ˆ

1. Show that for f ∈ E [[z]], the series fj de¬ned in Exercise 2 on p. 162

fn z n /“(1 + n/kj ).

are equal to

ˆ

2. Let f ∈ E {z}k,d , for a multidirection d = (d1 , . . . , dq ) with d1 =

. . . = dq . For s ≥ 1/kq , de¬ne f (z; s) as the sum of the convergent

ˆˆ ˆ

series B1/s f . Interpret k-summability of f in this multidirection as a

continuation of f (z; s) with respect to s to the interval s ≥ 1/k1 .

11.3 Convolutions

The main advantage of acceleration operators over general ones of the same

order lies in the fact that they behave well with respect to convolution of

functions, which we shall introduce and discuss now. To do so, we shall for

simplicity of notation assume that E is a Banach algebra, although similar

results hold in all cases where the products of the functions resp. power

series occuring are de¬ned, e.g., in the situation of Theorem 14 (p. 67).

ˆˆ

For arbitrary k > 0, let s = 1/k. Given any f , g ∈ E [[z]], consider

ˆ ˆ ˆ ˆ ˆˆ

h = Bk [(Lk f ) (Lk g )]. The coe¬cients hn of this series are given by

n

1

fn’m “(1 + s(n ’ m)) gm “(1 + sm), n ≥ 0.

hn =

“(1 + sn) m=0

Using the Beta Integral (p. 229) and integrating termwise, this identity can

zˆ

ˆ ˆ

be written as h(z s ) = (d/dz) [ 0 f ((z ’ t)s ) g (ts )dt]. The series h will be

ˆ

ˆ ˆ ˆˆ

called formal convolution of index k of f and g , and we write h = f —k g .

ˆ

Now, let G be a sectorial region and consider two functions f, g ∈ H(G, E )

which are continuous at the origin. We de¬ne a function h(z), holomorphic

in G, by

z

d

f ((z ’ t)s ) g(ts )dt , z s ∈ G,

s

h(z ) = (11.2)

dz 0

and call h the convolution of f and g (with index k). In shorthand we shall

write f —k g for this function h.

The convolution operator is well behaved with respect to Gevrey asymp-

totics, as is shown in the following lemma:

11.3 Convolutions 179

Lemma 22 For k, f , g, and G as above, assume f (z) ∼s1 f (z), g(z) ∼s1

=ˆ =

g (z) in G, for some s1 > 0. Then we have (f —k g)(z) ∼s1 (f —k g )(z) in G.

ˆˆ

ˆ =

In case G is a sector of in¬nite radius and f , g are of exponential growth

not more than κ, with some κ > 0, then so is f —k g.

¯

Proof: By assumption, for every closed subsector S of G there exist c, K >

0 so that |rf (z, N )|, |rg (z, N )| ¤ c K N “(1 + s1 N ) for every N ≥ 0 and

¯ ¯

z ∈ S. Using the Beta Integral, one can show for z s ∈ S and h = f —k g,

ˆ ˆ— ˆ