2πi w

˜

β±

˜

with corresponding paths β± beginning at the point w = 1/r and going

o¬ to in¬nity. The integrals remain the same when changing the paths of

integration, provided that wu, for w on the path, stays within the sector

S+ where the kernel e(wu) is de¬ned, and wt, at least for large values of t,

5.8 Convolution of Kernels 93

remains in S’ where then E(wt) is bounded. By adding a corresponding

integral from 0 to 1/r, we can with help of Cauchy™s theorem also integrate

along the ray arg w = “(π + µ)/(2k). Doing so, and using (5.15), we obtain

1/r

’1 dw 1 u

k± (t, u) + E(wt) e(wu) = . (5.19)

2πi t ’ u

2πi w

0

This shows k+ (t, u) = k’ (t, u), and k± (t, u) is analytic for u ∈ S+ and

arbitrary t, except for t = u. Hence we obtain

du

(T ’ g)(t) = f+ (t) ’ f’ (t) = f (u) k± (t, u) ,

u

γ

where the path γ is from in¬nity to the origin along the ray arg u = ’δ,

and back to in¬nity along arg u = δ. For |u| > |t|, we can deform both

rays to be along the real axis, and then these parts of the integral cancel.

Therefore, we ¬nd that γ may as well be replaced by a closed Jordan curve

encircling t and having negative orientation. This, together with (5.19) and

Cauchy™s formula, then implies T ’ g = f . 2

Exercises: In the following exercises, let e(z) be a kernel function of

order k > 0, and consider the corresponding integral operators T and T ’ .

1. Let S be a sector of in¬nite radius, and let A denote a subspace of

A(k) (S, E ) that is an asymptotic space, in the sense de¬ned on p. 75.

Show that then the image of A under the operator T is again an

asymptotic space.

2. Let G be a sectorial region of opening more than π/k, and let A de-

note a subspace of H(G, E ) which is an asymptotic space. Show that

then the image of A under the operator T ’ is again an asymptotic

space.

5.8 Convolution of Kernels

In this section, we consider two pairs of operators Tj , Tj’ of order kj with

corresponding moment functions mj (u), 1 ¤ j ¤ 2. We will try to ¬nd a

+ +

third pair T, T ’ of operators, so that T coincides with T1 —¦ T2 , resp. T =

+ ’

T2 —¦ T1 , at least when applied to the geometric series. Consequently, the

corresponding m(u) equals either the product m2 (u) m1 (u) or the quotient

m2 (u)/m1 (u). In the ¬rst case the new operators clearly will have to have

order k = (1/k1 + 1/k2 )’1 , because it follows from Stirling™s formula that

m(n) is of order “(1 + n/k). In the second case, their order will be k =

94 5. Integral Operators

(1/k2 ’ 1/k1 )’1 , hence we here have to assume k1 > k2 . Because of T ’

being the inverse of T , at least for suitable function spaces, we only have

to ¬nd T , resp. its kernel function e(z).

The following lemma shows how one can recover e(z) from its moment

sequence m(n). However, observe that we shall not be concerned with the

harder question of how to characterize such m(n) to which a kernel e(z)

exists.

Lemma 7 Let a kernel function e(z) of order k with corresponding op-

erator T be given. For f (u) = (1 ’ u)’1 , let g = T f . Then g(z) is

holomorphic in for ’π/(2k) < arg z < (2 + 1/(2k))π, is asymptotic to

∞

of Gevrey order s = 1/k there, and g(z) ’ 0 as

n

g (z) =

ˆ 0 m(n) z

z ’ ∞. Moreover,

g(z) ’ g(ze2πi ) = 2πi e(1/z), | arg z| < π/(2k). (5.20)

Proof: Holomorphy of the function g and its behavior at the origin follow

from Theorem 27 (p. 91), the behavior at in¬nity can be read o¬ from the

integral representation. To show (5.20), represent g(z) resp. g(ze2πi ) by

integrals (5.13) with „ = µ resp. „ = 2π ’ µ, for (small) µ > 0, and use the

2

Residue Theorem.

Remark 7: Observe that, according to Watson™s Lemma (p. 75), there is

only one g with g(z) ∼1/k m(n) z n in a sector of opening so large as in

=

the above lemma. So we indeed have shown that the kernel e(z) is uniquely

3

determined by (5.20).

Theorem 31 Let two kernel functions ej (z) of orders kj , with correspond-

ing moment functions mj (u) and operators Tj , 1 ¤ j ¤ 2, be given. Then

there is a unique kernel function e(z) of order k = (1/k1 +1/k2 )’1 with cor-

responding moment function m(u) = m1 (u)m2 (u). For the corresponding

integral operator T and f (u) = (1 ’ u)’1 we then have T f = (T1 —¦ T2 ) f =

(T2 —¦ T1 ) f . In particular, the function e(1/z) is given by applying T1 to the

function e2 (1/u).

Proof: Uniqueness follows from Lemma 7, resp. Remark 7. For existence,

let g(z) = (T1 —¦ T2 )f , f (u) = (1 ’ u)’1 , and de¬ne e(z) by (5.20). Inter-

changing the order of integration in (T1 —¦ T2 )f , show that e(1/z) is given by

applying T1 to e2 (1/u). Then, check that e(z) has the necessary properties

for a kernel function of this order as listed on p. 89; in particular, note that

’

the function Ep (z) can be obtained by applying T2 to the corresponding

2

function E1,p (z).

As the ¬nal result in this context, we now ¬nd the kernel corresponding

to the quotient of the moment functions:

5.8 Convolution of Kernels 95

Theorem 32 Given two kernel functions ej (z) of order kj with corre-

sponding moment functions mj (u), 1 ¤ j ¤ 2, assume k1 > k2 . Then

there exists a unique kernel function e(z) of order k = (1/k2 ’ 1/k1 )’1

corresponding to the moment function m(u) = m2 (u)/m1 (u). In particu-

’

lar, the function e(1/u) is given by an application of T1 to the function

e2 (1/z).

Proof: As above, uniqueness follows from Lemma 7, resp. Remark 7. For

’

existence, let f (u) = (1 ’ u)’1 and de¬ne g2 = T2 f . Then g = T1 g2 is

holomorphic in ’π/(2k) < arg w < (2 + 1/(2k))π, and we de¬ne e(z) by

(5.20). Interchanging the order of integration, one obtains that e(1/u) is

’

given by an application of T1 to the function e2 (1/z). To complete the

proof, one can verify the necessary properties for e(z), using Theorems 27

2

and 28.

As an application of the last theorem, we mention that for k1 = ± > 1,

e1 (z) = ± z ± exp[’z ± ], and k2 = 1, e2 (z) = z ez , the function C± (u) =

u’1 e(u) equals the kernel of Ecalle™s acceleration operator, which will be

studied in detail in Chapter 11. More kernels are constructed in the follow-

ing exercises.

Exercises:

1. Using the above results, verify existence of kernel functions corre-

sponding to moment functions of the form

“(1 + s1 u) · . . . · “(1 + sν u)

m(u) = , (5.21)

“(1 + σ1 u) · . . . · “(1 + σµ u)

ν µ

’

with positive parameters sj , σj satisfying j=1 sj σj > 0,

j=1

and ¬nd the order of the kernels.

2. Using the above results, verify existence of kernel functions of order

one corresponding to moment functions of the form

“(β1 + u) · . . . · “(βµ + u) “(1 + u)

µ ≥ 1,

m(u) = , (5.22)

“(±1 + u) · . . . · “(±µ + u)

with positive parameters ±j , βj . Relate E(z) to the generalized con-

¬‚uent hypergeometric function (p. 23).

∞

ˆ

3. Let f (z) = 0 m(n) z n , with m(n) as in (5.21), resp. (5.22), and let

k be the order of the corresponding kernel function. Show existence

of a sectorial region G of opening larger than π/k and a function

f ∈ H(G, C ) with f (z) ∼1/k f (z) in G.

ˆ

=

96 5. Integral Operators

4. For

“(±1 + s1 u) · . . . · “(±ν + sν u)

m(u) = ,

“(β1 + σ1 u) · . . . · “(βµ + σµ u)

ν

with positive parameters ±j , βj , sj , σj restricted by j=1 sj =

µ

j=1 σj , determine the radius of convergence of the series

∞

m(n) z n .

k(z) =

n=0

Moreover, show that k(z) admits holomorphic continuation along ev-

ery ray other than the positive real axis.

6

Summable Power Series

In this chapter we shall present Ramis™s concept of k-summability of for-

mal power series [225, 226]. We shall also study the classical de¬nition of

moment summability methods in a form suitable for application to formal

power series and relate these methods to k-summability. For some historical

remarks, see Chapter 14.

A general summability method may be viewed as a linear functional S

on some linear space X of sequences, or equivalently, series, with complex

entries. In many cases the functional has the following representation: Let

∞

an in¬nite matrix A = [ajk ]∞

j,k=0 be given. Then we say that a series 0 xk

∞

is A-summable if the series k=0 ajk xk converge for every j ≥ 0 and

∞

xk = lim ajk xk

SA

j’∞

k=0

exists. In this case the number SA ( xk ) is called the A-sum of the series

xk . Such summability methods are called matrix methods, and we say

that the series is summable by the method A, or for short, is A-summable.

The space X consisting of all A-summable series is called the summability

domain of the method. Observe that this terminology gives good sense even

xk with xk in a Banach space E .

in the more general situation of series

One may even replace the entries ajk of the matrix A by (continuous) linear

operators on E , but we shall not consider this here.

For our purposes it is more natural to replace the index j by a continuous

parameter T . So instead of an in¬nite matrix A we have a sequence of

functions ak (T ), T ≥ 0, which we again denote by A. Then the functional

98 6. Summable Power Series

∞

has the form SA ( xk ) = limT ’∞ k=0 ak (T ) xk . If the series we want

∞

to sum is a formal power series, hence xk = fk z k , then k=0 ak (T ) fk z k ,

in case of convergence for, say, |z| < ρ, de¬nes a family of holomorphic

functions a(z; T ) on D = D(0, ρ). It then is natural to restrict ourselves to

cases where convergence for T ’ ∞ is locally uniform in z, at least for z in

some subregion G of D, so that we obtain a holomorphic function f (z) =

ˆ

limT ’∞ a(z; T ) on G. In this case we say that the formal power series f (z) is

ˆ

A-summable on the region G, and we shall refer to the function (SA f )(z) =

ˆ

f (z) as the sum of f (z), in the sense of the summation method A. Aside

from the holomorphy of the sum, however, we need more properties of our

summability method in order to make it suitable for di¬erential equations:

ˆ

Suppose we know that the formal series f (z) satis¬es some linear di¬erential

equation, and we have seen earlier that the radius of convergence of such a

ˆ

formal solution can be zero. If f (z) is indeed summable to f (z) on G, we

should like to conclude from general properties of the summation method

that f (z) also solves the di¬erential equation. Moreover, the origin should

be a boundary point of G, and we should like to conclude that f (z) is

ˆ

asymptotic to f (z), perhaps of some Gevrey order s > 0. Thus, in order

to make a summability method suitable for formal solutions of ODE, we

require

• that its summability domain X is a di¬erential algebra, containing

all convergent power series,

• that the functional S is a homomorphism, and so is not only linear,

but maps products to products and derivatives to derivatives, and

• that the map J, de¬ned on p. 67, inverts S.