ˆ

so that f (z) is T -summable in the multidirection d. From the corol-

lary to Theorem 50, we then conclude that the choice of the operators

Tj is completely arbitrary, provided that they have the required or-

ˆ

ders. In particular, the sum ST ,d f only depends on the multisumma-

bility type k and the multidirection d; hence from now on we shall

ˆ ˆ

always write Sk,d f and E {z}k,d instead of ST ,d f and E {z}T ,d .

Instead of the kj , we could also use the orders κj of the integral operators

to de¬ne the type of multisummability, but in the literature the use of kj

is more common, because for formal solutions of systems (3.1) (p. 37) they

agree with the set of levels de¬ned in Section 8.4.

The above results show that in the theory of multisummability one

can arbitrarily choose the operators T1 , . . . , Tq having the given orders.

166 10. Multisummable Power Series

For practical purposes, one therefore may always take Laplace operators

Lκ1 , . . . , Lκq . This case has been studied in [18] under the name of sum-

mation by iterated Laplace integrals. For more theoretical purposes, the

original de¬nition of Ecalle in terms of acceleration operators sometimes is

more appropriate, since it has very natural properties as far as convolution

of power series is concerned.

Exercises: In view of Exercise 4 on p. 162, we will extend the de¬nition

of multisummability to power series in a root z 1/p , p ≥ 2 as follows: A

power series f in z 1/p is called k-summable in the multidirection d if and

ˆ

ˆ

only if f (z p ) is pk-summable in the multidirection d/p.

ˆ

1. Given admissible k and d, and f ∈ E {z}k,d , show for su¬ciently large

q

ˆ ˆ

natural p (depending only on k) that f = j=1 fj , with formal power

series fj in z 1/p which are kj -summable in direction dj , for 1 ¤ j ¤ q.

ˆ

2. For k1 > k2 > 0 with 1/κ = 1/k2 ’ 1/k1 ≥ 2, let |d2 ’ d1 | ¤ π/(2κ),

so that d = (d1 , d2 ) is admissible with respect to k = (k1 , k2 ). For

ˆ ˆˆ ˆ

fj ∈ E {z}kj ,dj , j = 1, 2, conclude f = f1 + f2 ∈ E {z}k,d . Show that

ˆˆ

then g = Bk1 f is κ-summable in direction d2 , hence g = Sκ,d2 g is

ˆ ˆ

holomorphic in a sector S = S(d2 , ±, r) of opening larger than 2π.

Moreover, show that ψ(z) = g(z) ’ g(ze2πi ) can be holomorphically

continued into a sector of in¬nite radius and bisecting direction d2 ’π.

ˆ

3. For k1 , k2 as in Ex. 3, show the existence of f ∈ E {z}k,d , k = (k1 , k2 ),

ˆˆ ˆˆ

d = (d1 , d2 ), which cannot be written as f = f1 + f2 , fj ∈ E {z}kj ,dj ,

j = 1, 2.

4. Under the assumptions of Theorem 50 (p. 164), show that the de-

q

ˆ ˆ

composition of f into a sum j=1 fj , in case q ≥ 2, is never unique.

10.5 Some Rules for Multisummable Power Series

The following is the analogue to some theorems in Section 6.3. To generalize

the remaining ones will be easier using a result that we shall derive in

Section 10.7.

Theorem 51 For every admissible k and d, we have the following:

ˆˆ ˆ

(a) If f , g1 , g2 ∈ E {z}k,d , then we have:

g1 + g2 ∈ E {z}k,d , Sk,d (ˆ1 + g2 ) = Sk,d g1 + Sk,d g2 ,

ˆ ˆ g ˆ ˆ ˆ

d

ˆ ˆ ˆ

f ∈ E {z}k,d , Sk,d (f ) = (Sk,d f ),

dz

z z z

ˆ ˆ ˆ

f (w)dw ∈ E {z}k,d , Sk,d f (w)dw = (Sk,d f )(w) dw.

0 0 0

10.6 Singular Multidirections 167

ˆ ˆ

(b) If f ∈ E {z}k,d and p is a natural number, then g (z) = f (z p ) ∈

ˆ

ˆ

E {z}pk,p’1 d , and (Spk,p’1 d g )(z) = (Sk,d f )(z p ).

ˆ

(c) If f ∈ E {z}k,d and p is a natural number for which g (z) = f (z 1/p ) is

ˆ ˆ

ˆ

again a power series, then g (z) ∈ E {z}p’1 k,pd , and (Sp’1 k,pd g )(z) =

ˆ ˆ

1/p

ˆ

(Sk,d f )(z ).

Proof: Statements (b) and (c) follow directly from the de¬nition and

Exercise 2 on p. 88. For (a), we may assume that the numbers κj all are

larger than 1/2, because if not we may use (b), (c) with su¬ciently large p.

In this case, (a) follows from Theorem 50 (p. 164) and the corresponding

2

results in Section 6.3.

While some of the rules for k-summability have been generalized to mul-

tisummability, we did not yet do so for those concerning products of series.

This shall be done later with help of a result that characterizes those func-

tions arising as sums of multisummable series.

Exercises: Let admissible k and d be given.

1. Show that the exercises in Section 6.3 generalize to multisummability.

∞

ˆ

2. Let f (z) = n=’p fn z n be a formal Laurent series, with p ∈ N and

ˆ

fn ∈ E . There are two ways of de¬ning multisummability of f :

ˆ

(a) We say that f is k-summable in the multidirection d if g (z) =

ˆ

∞ ˆ

∈ E {z}k,d , and we then de¬ne Sk,d f = Sk,d g +

n

n=0 fn z ˆ

’1 n

n=’p fn z .

ˆ

(b) We say that f is k-summable in the multidirection d if g (z) =ˆ

z p f (z) ∈ E {z}k,d , and we then de¬ne Sk,d f = z ’p Sk,d g .

ˆ ˆ ˆ

Show that both de¬nitions are equivalent, and in case f’p = . . . =

f’1 = 0 coincide with the original de¬nition for power series.

10.6 Singular Multidirections

Let a multisummability type k = (k1 , . . . , kq ) and a formal power series

ˆ

f (z) ∈ E [[z]] be given. A multidirection d, admissible with respect to k,

ˆ ˆ

will be called singular for f , if and only if f ∈ E {z}k,d ; otherwise, we

say that d is nonsingular. It is possible that all d are singular, e.g., when

ˆ ˆˆ

f ∈ E [[z]]1/kq , or when g = S (Bkq f ) cannot be holomorphically continued

across the boundary of some bounded region containing the origin. The

168 10. Multisummable Power Series

ˆ

set of all singular multidirections will be called the singular set of f (with

respect to k).

ˆ

Inspecting the de¬nition of E {z}k,d , one sees that the reason for f ∈

E {z}k,d , i.e., d singular, will be connected to the “level” j, since it may be

so that the functions fj , . . . , fq , de¬ned in Exercise 2 on p. 162, all exist, but

fj cannot be holomorphically continued into any sector of in¬nite radius

and bisecting direction dj , or in every such sector has exponential growth

larger than κj , so that application of the next operator fails. If this occurs,

we shall say that d is singular of level j.

ˆ

Let d = (d1 , . . . , dq ) be singular of level j for f . Then the above discussion

˜ ˜ ˜ ˜

implies that all multidirections d = (d1 , . . . , dq ) with dν = dν for j ¤ ν ¤ q

ˆ

are automatically singular of level j for f , and we shall identify all these

singular multidirections. Moreover, in view of Lemma 19 (p. 163) we may

˜ ˜

also identify admissible multidirections d and d with dj ’ dj = 2µπ, for

ˆ

some µ ∈ Z. After doing so, the singular set of f may or may not contain

ˆ

¬nitely many elements. If it does, we shall say that f is k-summable. As

for q = 1 we shall write E {z}k for the set of all k-summable formal power

series with coe¬cients in E .

For q = 1 we have seen that absence of singular directions implies con-

ˆ

vergence of the series f . This generalizes to arbitrary q as follows:

Proposition 22 Let k = (k1 , . . . , kq ), q ≥ 2, be admissible, and assume

ˆ

that f ∈ E {z}k has no singular multidirections of level j, for some ¬xed j,

ˆ ˜

1 ¤ j ¤ q. Then f ∈ E {z}k , with k = πj (k) = (k1 , . . . , kj’1 , kj+1 , . . . , kq ).

˜

Moreover, for every nonsingular multidirection d = (d1 , . . . , dq ) correspond-

˜

ing to k, the multidirection d = (d1 , . . . , dj’1 , dj+1 , . . . , dq ) is nonsingular

˜ ˆ ˆ

with respect to k, and Sk,d f = Sk,d f .

˜˜

Proof: Without loss of generality, we assume that the operators Tj used

are all Laplace operators of orders κj . Absence of singular multidirections

ˆ’1 ˆ

ˆ’1

of level j = q implies that fq = S (Tq —¦ . . . —¦ T1 f ) is entire and of

exponential growth κq in arbitrary sectors of in¬nite radius. This shows

that fq’1 = Tq fq is holomorphic at the origin, completing the proof in

this case. For j ¤ q ’ 1, consider any nonsingular multidirection d and the

corresponding functions f0 , . . . , fq de¬ned in Exercise 2 on p. 162. Absence

of singular multidirections of this level shows existence of a sector S of

in¬nite radius, bisecting direction dj+1 and opening more than π/κj+1

in which fj is holomorphic and of exponential growth at most κj . For

’1

fj+1 = Tj+1 fj , we obtain from Exercise 2 on p. 821 that fj+1 (z) is of

exponential growth not larger than k, with 1/k = 1/κj + 1/κj+1 , in a small

sector with bisecting direction dj+1 . Consequently, the operator Tj — Tj+1

can be applied to connect fj+1 with fj’1 . This is equivalent to saying that

’1

1 By choice of the operators, Tj+1 is the Borel operator of order κj+1 .

10.7 Applications of Cauchy-Heine Transforms 169

ˆ˜ ˜

f is k-summable in the multidirection d = (d1 , . . . , dj’1 , dj+1 , . . . , dq ), and

ˆ

f0 = Sk,d f . 2

˜˜

This result has the following converse:

˜

Proposition 23 Let k = (k1 , . . . , kq ), q ≥ 2, be admissible, and set k =

πj (k) = (k1 , . . . , kj’1 , kj+1 , . . . , kq ) for some ¬xed j, 1 ¤ j ¤ q. Then

ˆ ˆ

every f ∈ E {z}k is also in E {z}k , and therefore f has no singular mul-

˜

tidirections of level j. In other words, a multidirection d = (d1 , . . . , dq )

ˆ

is nonsingular for f , regarded as an element of E {z}k , if and only if

˜ ˆ

d = (d1 , . . . , dj’1 , dj+1 , . . . , dq ) is nonsingular for f in E {z}k .

˜

Proof: The proof is obvious for j = q, so assume otherwise. For operators

T1 , . . . , Tq of respective orders κ1 , . . . , κq , the collection

T1 , . . . , Tj’1 , Tj — Tj+1 , Tj+2 , . . . , Tq

˜

can serve as operators for k-summability. Using Lemma 18 (p. 160), one can

then replace Tj — Tj+1 by the iteration Tj —¦ Tj+1 , choosing any admissible

2

direction to integrate along. This then completes the proof.

Exercises: In the following exercises, consider some admissible k =

(k1 , . . . , qq ) with q ≥ 2.

ˆ ˆ ˜

1. Let f ∈ E {z}k © E [[z]]1/κ , for κ > kq . Show f ∈ E {z}k , k = πq (k).

˜

ˆ

2. Let f ∈ E {z}k have no singular multidirections of whatever level.

ˆ

Show that then f converges.

10.7 Applications of Cauchy-Heine Transforms

The following two propositions characterize functions f that are the sum of

multisummable power series. To do so, we de¬ne for every admissible k =

(k1 , . . . , kq ) and d = (d1 , . . . , dq ) closed intervals I1 , . . . , Iq corresponding to

d by Ij = [dj ’ π/(2kj ), dj + π/(2kj )], 1 ¤ j ¤ q. Admissibility of d with

respect to k then is equivalent to the inclusions I1 ‚ I2 ‚ . . . ‚ Iq . We also

set k0 = ∞ and recall that f (z) ∼0 f (z) in S implies that f will converge

=ˆ ˆ

and be the power series expansion of f about the origin. In particular, if

ˆ 0,

f = ˆ then f vanishes identically.

Proposition 24 Let k = (k1 , . . . , kq ) and d = (d1 , . . . , dq ) be admissible,

ˆ