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For more details, see Ramis™s discussion of such abstract summation pro-
cesses in [229].
In particular, the requirement that products be summed to products is
rarely satis¬ed for general summation methods, but we shall see that k-
summability does have all the required properties, making it an ideal tool
for treating formal solutions of ODE “ except that it is not powerful enough
to sum all formal series arising as solutions of ODE!
A trivial example of a summation method satisfying all the requirements
listed above is as follows: Let X = E {z} and take S = S , mapping each
convergent power series to its natural sum. Clearly, this method is too
weak in that it applies to convergent series only. Constructing a summa-
bility method suitable for formal solutions of ODE may also be viewed
as ¬nding a way of extending S to larger di¬erential algebras, since an-
other natural requirement to make is that for convergent power series a
summation method produces the natural sum.
6.1 Gevrey Asymptotics and Laplace Transform 99

6.1 Gevrey Asymptotics and Laplace Transform
In what follows we again consider a given Banach space E . We have seen
in Section 4.7 that the mapping J : As (G, E ) ’ E [[z]]s , owing to Wat-
son™s Lemma, is injective for sectorial regions G of opening more than sπ,
ˆ
meaning that given a formal series f (z), there can be at most one function
f ∈ As (G, E ) with f (z) ∼s f (z) in G. The following result will be used

in the exercises below to show that in this case the mapping cannot be
surjective, except for the trivial case of dim E = 0.
ˆ ˆˆ
Theorem 33 Let s = 1/k > 0 and f ∈ E [[z]]s , hence g (u) = (Bk f )(u)
ˆ
converges for |u| su¬ciently small. Set g = S g ; thus g(u) is holomorphic in
ˆ
some neighborhood of the origin. Then, for every ¬xed real d the following
two statements are equivalent:

(a) There exists a sectorial region G = G(d, ±) with ± > sπ, and f ∈
ˆ
As (G, E ) with f = J(f ).
(b) There exists a sector S = S(d, µ) for su¬ciently small µ > 0 so that g
admits holomorphic continuation into S and is of exponential growth
not more than k there.

Moreover, if either statement holds, then f = Lk g follows.

Proof: Follows immediately from Theorems 22“24 in the previous chapter.
2
According to the theorem, existence of f as in (a) is linked to the holo-
morphic continuation of g plus its behavior when approaching in¬nity. In
ˆ
principle, (b) can be veri¬ed for a given formal series f , and if satis¬ed,
then the function f can be computed via Laplace transform.
Existence of power series that cannot be continued beyond their disc of
convergence is well known; one such example is contained in the exercises
below.

Exercises: Let s > 0 be given, and set k = 1/s.
∞ n
1. For g (z) = 0 z 2 , show that g has radius of convergence equal to
ˆ ˆ
one. Moreover, show that g = S g cannot be holomorphically contin-
ˆ
ued beyond the unit disc.
ˆ
2. For g as above, set f = Lk g . Use Theorem 33 and the previous
ˆ ˆ
ˆ
exercise to conclude that no f ∈ As (G, C ) with f = J(f ) can exist if
the opening of G exceeds sπ. Hence the map J : As (G, E ) ’ E [[z]]s ,
for E = C , is not surjective. Generalize this to arbitrary E with
dim E > 0.
100 6. Summable Power Series

6.2 Summability in a Direction
ˆ ˆ
Let k > 0, d ∈ R and f ∈ E [[z]] be given. We say that f is k-summable
in direction d, if a sectorial region G = G(d, ±) of opening ± > π/k and
ˆ ˆ
a function f ∈ A1/k (G, E ) exist with J(f ) = f . If this is so, then f ∈
E [[z]]1/k follows from results in Section 4.5, and Watson™s Lemma (p. 75)
ˆ
guarantees uniqueness of f ∈ A1/k (G, E ). In view of Theorem 33, f is k-
ˆˆ
summable in direction d if and only if g = Bk f converges and g = S (ˆ) is
ˆ g
˜
holomorphic and of exponential growth at most k in a sector S = S(d, µ), for
some µ > 0. If this is so, we call the function f = Lk g, integrating along rays
ˆ ˆ ˆˆ
close to d, the k-sum of f in direction d and write f = Sk,d f = Lk —¦S —¦ Bk f .

ˆ
Remark 8: If we are given a series f ∈ E [[z]]1/k whose k-summability in
direction d is to be investigated, and if we then want to compute its sum,
we are presented with the following problems:
ˆˆ
1. The function g(z), locally given by the convergent series Bk f , has to
be holomorphically continued into a sector S(d, µ), for some µ > 0,
which typically will be small.
2. In this sector, we have to show that g is of exponential growth not
larger than k.
3. We have to compute the integral Lk g.
At ¬rst glance, one might think that item 1 might be the major problem.
However, there are explicit methods for performing holomorphic continu-
ation, once we know that g is holomorphic in S(d, µ). Moreover, we shall
˜ ˜
show in Lemma 8 that we may even replace k by k < k, with k ’ k su¬-
ˆ˜ ˆ
ciently small; then the sum of Bk f will be an entire function, so that item 1
is no problem at all. This entire function, in general, will have too large an
exponential growth in all directions but arg u = d. So in a way, the main
3
di¬culty lies in the problem of verifying item 2.
As we shall show, this summation method has all the properties listed
at the beginning of the chapter, plus several additional ones, and we begin
with proving some which are direct consequences of the de¬nition:
Lemma 8 For every ¬xed k > 0 the following holds:
ˆ ˆ
(a) Let f be convergent. Then for every d, the series f is k-summable in
ˆ ˆ
direction d, and (Sk,d f )(z) = (S f )(z) for every z where both sides
are de¬ned.
ˆ
(b) Let f be k-summable in direction d, and let µ > 0 be su¬ciently
ˆ ˜ ˜
small. Then f is k-summable in all directions d with |d ’ d| < µ, and
ˆ ˆ
(Sk,d f )(z) = (Sk,d f )(z) for every z where both sides are de¬ned.
˜
6.2 Summability in a Direction 101

ˆ
(c) Let f be k-summable in direction d, and let µ > 0 be su¬ciently
ˆ ˆ
small. Then f is (k ’µ)-summable in direction d, and (Sk’µ,d f )(z) =
ˆ
(Sk,d f )(z) for every z where both sides are de¬ned.

Proof: For the ¬rst statement, use f (z) = (S f )(z) ∼1/k f (z) in S, for
ˆ ˆ
=
every k > 0 and every sector S of su¬ciently small radius. To prove the
second, use the de¬nition and observe that for a sectorial region G(d, ±)
˜
of opening ± > π/k and µ < ± ’ π/k, there exists G(d, ±µ ) ‚ G(d, ±)
˜
with d as above and ±µ > π/k. Finally, for (c) use that ± > π/k implies
± > π/(k ’ µ) for small µ > 0, and “(1 + N/k)/“(1 + N/(k ’ µ)) ’ 0 as
N ’ ∞. 2
The following lemma shall prove useful in later chapters where we will
study multisummablility.
ˆ
Lemma 9 Let f be k1 -summable in direction d, let k > k1 , and de¬ne k2
ˆˆ
by 1/k2 = 1/k1 ’ 1/k. Then g = Bk f is k2 -summable in direction d, and
ˆ
ˆ
Sk2 ,d g = Bk (Sk1 ,d f ).
ˆ

Proof: Follows immediately from Theorem 23 (p. 80) and the de¬nition
2
of k1 - resp. k2 -summability in direction d.
On one hand, for k < 1/2 the k-sum of formal series is analytic in sectorial
regions of opening more than 2π, which forces us to consider such regions
on the Riemann surface of the logarithm. On the other hand, statement (c)
of the next lemma shows that k-summability does not distinguish between
directions di¬ering by multiples of 2π.
Lemma 10
ˆ
(a) Let f be k-summable in direction d, for every d ∈ (±, β), ± < β, and
ˆ ˆ
¬xed k > 0. Then (Sk,d1 f )(z) = (Sk,d2 f )(z) for every d1 , d2 ∈ (±, β)
and every z where both sides are de¬ned.
ˆ
(b) Let f be kj -summable in direction d, 1 ¤ j ¤ 2, with k1 > k2 > 0
ˆ ˜
and some ¬xed d, then f is k1 -summable in all directions d with
˜ ˆ ˆ
2 |d ’ d| ¤ π (1/k2 ’ 1/k1 ), and (Sk1 ,d f )(z) = (Sk2 ,d f )(z) for every
˜
z where both sides are de¬ned.
˜ ˆ
(c) For d = d + 2π, k-summability of f in direction d is equivalent to
k-summability of f in direction d, and (Sk,d f )(z) = (Sk,d f )(ze’2πi )
ˆ ˜ ˆ ˆ
˜
for every z where both sides are de¬ned.

Proof:
ˆˆ
(a) The function g = S (Bk f ) is holomorphic and of exponential growth
not more than k in S((± + β)/2, β ’ ±). Hence f = Lk g ∼1/k f in a
ˆ
=
102 6. Summable Power Series

region which, for every d ∈ (±, β), contains a sector of opening larger
ˆ
than π/k and bisecting direction d. So Sk,d f = f , for every such d.
(b) From the de¬nition of k2 -summability in direction d we conclude ex-
istence of f with f (z) ∼1/k2 f (z) in G(d, ±) for some ± > π/k2 . Thus,
ˆ
=
g = Bk1 f is holomorphic and of exponential growth not more than k1
in S(d, ± ’ π/k1 ), and the assumption of k1 -summability in direction
d implies that g is holomorphic at the origin. Hence statement (b)
follows.
(c) Use Exercise 4 on p. 72.
2

Exercises: Always assume k > 0.

ˆ “(1+n/k) z n is k-summable in every direction
1. Show that f (z) = 0
d = 2jπ, j ∈ Z.
ˆ ˆˆ
2. Assume that f is chosen so that g = S (Bk f ) is a rational function.
Let d be so that no poles of g lie on the ray arg u = d, and show that
ˆ
then f is k-summable in direction d.

ˆ n
3. Show that f (z) = 0 “(1 + n/k) z /“(1 + n) is k-summable in
directions d with (2j + 1/2)π < d < (2j + 3/2)π, j ∈ Z.
ˆ
4. For f as in Exercise 3, let d be such that (2j’1/2)π < d < (2j+1/2)π,
ˆ
j ∈ Z. Show that f is (resp. is not) k-summable in direction d, if k ≥ 1
(resp. 0 < k < 1).
ˆ ˆˆ ˆ
5. Assume f so that g = S (Bk f ) is as in (4.2) (p. 63). Show that f is
k-summable in all directions d = 2jπ, j ∈ Z, and is not k-summable
in the remaining directions.

ˆ “(1 + 2n) z n /“(1 + n)
6. Show that f (z) = 0

(a) is 1-summable in every direction d ∈ [’π, π) but one (which?).
(b) is 1/2-summable in every direction d with (2j + 1/2)π < d <
(2j + 3/2)π, j ∈ Z.




6.3 Algebra Properties
ˆ
Let k > 0 and d be given, and let E {z}k,d denote the set of all f that are
k-summable in direction d. The following theorems are direct consequences
from the results on Gevrey asymptotics in Section 4.5, together with the
de¬nition of k-summability in direction d.
6.3 Algebra Properties 103

ˆˆ ˆ
Theorem 34 For ¬xed, but arbitrary, k > 0 and d, let f , g1 , g2 ∈ E {z}k,d
be given. 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

Finally, if p is a natural number, then
ˆ ˆ ˆ
f (z p ) ∈ E {z}pk,d/p , Spk,d/p (f (z p )) = (Sk,d f )(z p ).

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