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T = (T1 , . . . , Tq ) of integral operators of respective orders Оє1 , . . . , Оєq
Л†
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-
Л†
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
Л† Л†
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 Оєj of the integral operators
to deп¬Ѓne 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 deп¬Ѓned 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
LОє1 , . . . , LОєq . This case has been studied in  under the name of sum-
mation by iterated Laplace integrals. For more theoretical purposes, the
original deп¬Ѓnition 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 deп¬Ѓnition
of multisummability to power series in a root z 1/p , p в‰Ґ 2 as follows: A
power series f in z 1/p is called k-summable in the multidirection d if and
Л†
Л†
only if f (z p ) is pk-summable in the multidirection d/p.
Л†
1. Given admissible k and d, and f в€€ E {z}k,d , show for suп¬ѓciently large
q
Л† Л†
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 в‰¤ j в‰¤ q.
Л†

2. For k1 > k2 > 0 with 1/Оє = 1/k2 в€’ 1/k1 в‰Ґ 2, let |d2 в€’ d1 | в‰¤ ПЂ/(2Оє),
so that d = (d1 , d2 ) is admissible with respect to k = (k1 , k2 ). For
Л† Л†Л† Л†
fj в€€ E {z}kj ,dj , j = 1, 2, conclude f = f1 + f2 в€€ E {z}k,d . Show that
Л†Л†
then g = Bk1 f is Оє-summable in direction d2 , hence g = SОє,d2 g is
Л† Л†
holomorphic in a sector S = S(d2 , О±, r) of opening larger than 2ПЂ.
Moreover, show that П€(z) = g(z) в€’ g(ze2ПЂi ) can be holomorphically
continued into a sector of inп¬Ѓnite radius and bisecting direction d2 в€’ПЂ.
Л†
3. For k1 , k2 as in Ex. 3, show the existence of f в€€ E {z}k,d , k = (k1 , k2 ),
Л†Л† Л†Л†
d = (d1 , d2 ), which cannot be written as f = f1 + f2 , fj в€€ E {z}kj ,dj ,
j = 1, 2.
4. Under the assumptions of Theorem 50 (p. 164), show that the de-
q
Л† Л†
composition of f into a sum j=1 fj , in case q в‰Ґ 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:
Л†Л† Л†
(a) If f , g1 , g2 в€€ E {z}k,d , then we have:
g1 + g2 в€€ E {z}k,d , Sk,d (Л†1 + g2 ) = Sk,d g1 + Sk,d g2 ,
Л† Л† g Л† Л† Л†
d
Л† Л† Л†
f в€€ E {z}k,d , Sk,d (f ) = (Sk,d f ),
dz
z z z
Л† Л† Л†
f (w)dw в€€ E {z}k,d , Sk,d f (w)dw = (Sk,d f )(w) dw.
0 0 0
10.6 Singular Multidirections 167

Л† Л†
(b) If f в€€ E {z}k,d and p is a natural number, then g (z) = f (z p ) в€€
Л†
Л†
E {z}pk,pв€’1 d , and (Spk,pв€’1 d g )(z) = (Sk,d f )(z p ).
Л†

(c) If f в€€ E {z}k,d and p is a natural number for which g (z) = f (z 1/p ) is
Л† Л†
Л†
again a power series, then g (z) в€€ E {z}pв€’1 k,pd , and (Spв€’1 k,pd g )(z) =
Л† Л†
1/p
Л†
(Sk,d f )(z ).

Proof: Statements (b) and (c) follow directly from the deп¬Ѓnition and
Exercise 2 on p. 88. For (a), we may assume that the numbers Оєj all are
larger than 1/2, because if not we may use (b), (c) with suп¬ѓciently 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.
в€ћ
Л†
2. Let f (z) = n=в€’p fn z n be a formal Laurent series, with p в€€ N and
Л†
fn в€€ E . There are two ways of deп¬Ѓning multisummability of f :
Л†
(a) We say that f is k-summable in the multidirection d if g (z) =
Л†
в€ћ Л†
в€€ E {z}k,d , and we then deп¬Ѓne Sk,d f = Sk,d g +
n
n=0 fn z Л†
в€’1 n
n=в€’p fn z .
Л†
(b) We say that f is k-summable in the multidirection d if g (z) =Л†
z p f (z) в€€ E {z}k,d , and we then deп¬Ѓne Sk,d f = z в€’p Sk,d g .
Л† Л† Л†

Show that both deп¬Ѓnitions are equivalent, and in case fв€’p = . . . =
fв€’1 = 0 coincide with the original deп¬Ѓnition for power series.

10.6 Singular Multidirections
Let a multisummability type k = (k1 , . . . , kq ) and a formal power series
Л†
f (z) в€€ E [[z]] be given. A multidirection d, admissible with respect to k,
Л† Л†
will be called singular for f , if and only if f в€€ E {z}k,d ; otherwise, we
say that d is nonsingular. It is possible that all d are singular, e.g., when
Л† Л†Л†
f в€€ 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

Л†
set of all singular multidirections will be called the singular set of f (with
respect to k).
Л†
Inspecting the deп¬Ѓnition of E {z}k,d , one sees that the reason for f в€€
E {z}k,d , i.e., d singular, will be connected to the вЂњlevelвЂќ j, since it may be
so that the functions fj , . . . , fq , deп¬Ѓned in Exercise 2 on p. 162, all exist, but
fj cannot be holomorphically continued into any sector of inп¬Ѓnite radius
and bisecting direction dj , or in every such sector has exponential growth
larger than Оєj , so that application of the next operator fails. If this occurs,
we shall say that d is singular of level j.
Л†
Let d = (d1 , . . . , dq ) be singular of level j for f . Then the above discussion
Лњ Лњ Лњ Лњ
implies that all multidirections d = (d1 , . . . , dq ) with dОЅ = dОЅ for j в‰¤ ОЅ в‰¤ q
Л†
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
Лњ Лњ
also identify admissible multidirections d and d with dj в€’ dj = 2ВµПЂ, for
Л†
some Вµ в€€ Z. After doing so, the singular set of f may or may not contain
Л†
п¬Ѓnitely 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 coeп¬ѓcients in E .
For q = 1 we have seen that absence of singular directions implies con-
Л†
vergence of the series f . This generalizes to arbitrary q as follows:
Proposition 22 Let k = (k1 , . . . , kq ), q в‰Ґ 2, be admissible, and assume
Л†
that f в€€ E {z}k has no singular multidirections of level j, for some п¬Ѓxed j,
Л† Лњ
1 в‰¤ j в‰¤ q. Then f в€€ E {z}k , with k = ПЂj (k) = (k1 , . . . , kjв€’1 , kj+1 , . . . , kq ).
Лњ
Moreover, for every nonsingular multidirection d = (d1 , . . . , dq ) correspond-
Лњ
ing to k, the multidirection d = (d1 , . . . , djв€’1 , dj+1 , . . . , dq ) is nonsingular
Лњ Л† Л†
with respect to k, and Sk,d f = Sk,d f .
ЛњЛњ

Proof: Without loss of generality, we assume that the operators Tj used
are all Laplace operators of orders Оєj . Absence of singular multidirections
Л†в€’1 Л†
Л†в€’1
of level j = q implies that fq = S (Tq в—¦ . . . в—¦ T1 f ) is entire and of
exponential growth Оєq in arbitrary sectors of inп¬Ѓnite radius. This shows
that fqв€’1 = Tq fq is holomorphic at the origin, completing the proof in
this case. For j в‰¤ q в€’ 1, consider any nonsingular multidirection d and the
corresponding functions f0 , . . . , fq deп¬Ѓned in Exercise 2 on p. 162. Absence
of singular multidirections of this level shows existence of a sector S of
inп¬Ѓnite radius, bisecting direction dj+1 and opening more than ПЂ/Оєj+1
in which fj is holomorphic and of exponential growth at most Оєj . For
в€’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/Оєj + 1/Оєj+1 , in a small
sector with bisecting direction dj+1 . Consequently, the operator Tj в€— Tj+1
can be applied to connect fj+1 with fjв€’1 . This is equivalent to saying that

в€’1
1 By choice of the operators, Tj+1 is the Borel operator of order Оєj+1 .
10.7 Applications of Cauchy-Heine Transforms 169

Л†Лњ Лњ
f is k-summable in the multidirection d = (d1 , . . . , djв€’1 , dj+1 , . . . , dq ), and
Л†
f0 = Sk,d f . 2
ЛњЛњ

This result has the following converse:
Лњ
Proposition 23 Let k = (k1 , . . . , kq ), q в‰Ґ 2, be admissible, and set k =
ПЂj (k) = (k1 , . . . , kjв€’1 , kj+1 , . . . , kq ) for some п¬Ѓxed j, 1 в‰¤ j в‰¤ q. Then
Л† Л†
every f в€€ E {z}k is also in E {z}k , and therefore f has no singular mul-
Лњ
tidirections of level j. In other words, a multidirection d = (d1 , . . . , dq )
Л†
is nonsingular for f , regarded as an element of E {z}k , if and only if
Лњ Л†
d = (d1 , . . . , djв€’1 , dj+1 , . . . , dq ) is nonsingular for f in E {z}k .
Лњ

Proof: The proof is obvious for j = q, so assume otherwise. For operators
T1 , . . . , Tq of respective orders Оє1 , . . . , Оєq , the collection

T1 , . . . , Tjв€’1 , Tj в€— Tj+1 , Tj+2 , . . . , Tq
Лњ
can serve as operators for k-summability. Using Lemma 18 (p. 160), one can
then replace Tj в€— Tj+1 by the iteration Tj в—¦ 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 в‰Ґ 2.
Л† Л† Лњ
1. Let f в€€ E {z}k в€© E [[z]]1/Оє , for Оє > kq . Show f в€€ E {z}k , k = ПЂq (k).
Лњ

Л†
2. Let f в€€ E {z}k have no singular multidirections of whatever level.
Л†
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 deп¬Ѓne for every admissible k =
(k1 , . . . , kq ) and d = (d1 , . . . , dq ) closed intervals I1 , . . . , Iq corresponding to
d by Ij = [dj в€’ ПЂ/(2kj ), dj + ПЂ/(2kj )], 1 в‰¤ j в‰¤ q. Admissibility of d with
respect to k then is equivalent to the inclusions I1 вЉ‚ I2 вЉ‚ . . . вЉ‚ Iq . We also
set k0 = в€ћ and recall that f (z) в€ј0 f (z) in S implies that f will converge
=Л† Л†
and be the power series expansion of f about the origin. In particular, if
Л† 0,
f = Л† then f vanishes identically.
Proposition 24 Let k = (k1 , . . . , kq ) and d = (d1 , . . . , dq ) be admissible,
Л†
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