¬cients. The computation of these coe¬cients is what this problem is all

about, and we shall show in which sense this can be done.

12.1 Matrix Methods and Multisummability

We recall from Chapter 6 the following notion: Given an in¬nite matrix

A = (amn ), with amn ∈ C for m, n ∈ N0 , we call a formal power series

ˆ fn z n ∈ E [[z]] A-summable in a direction d ∈ R, if there exists a

f (z) =

sectorial region G = G(d, ±) of opening ± > 0, such that the following two

conditions hold:

∞

amn fn z n converge in discs that all contain

1) The series fm (z) = n=0

G, for every m ∈ N0 .

2) The limit f (z) = limm’∞ fm (z) exists uniformly on every closed

subsector of G.

The so de¬ned function f , which is holomorphic on G, will be called the

ˆ

A-sum of f on G. The set of all formal power series that are A-summable in

ˆ

direction d will be denoted by E {z}A,d , and we write SA,d f for the A-sum

ˆ

of f .

In this context, it is natural to call a matrix A weakly p-regular (with p

being short for power series), if every convergent series is A-summable in

every direction d to the correct sum. A necessary and su¬cient condition

for this to hold is given in the exercises. Observe, however, that weak p-

ˆ

regularity is not the same as power series regularity: Let f (z) converge,

say, for |z| < r. Then for a weakly p-regular matrix we have that to every d

ˆ

there exists G(d, ±) so that the fm converge to f = S f on G(d, ±), and the

union of these regions may be a proper subset of the disc of convergence;

see the exercises below that this can occur. The de¬nition of power series

regular matrices, however, requires this union to be the full disc.

The following problem has been solved by Beck [29, 50] for the case of

E = C , but the proofs carry over to a general Banach space: Characterize

those matrices A for which the following comparison condition (C) holds:

12.1 Matrix Methods and Multisummability 185

(C) For every Banach space E , every k > 1/2 and every d ∈ R, we

ˆ ˆ

have E {z}k,d ‚ E {z}A,d , and (Sk,d f )(z) = (SA,d f )(z) for every

ˆ

f ∈ E {z}k,d and all z where both sides are de¬ned.

Because of E {z} ‚ E {z}k,d , we see that weak p-regularity is a necessary

condition for (C), while power series regularity is not. For that reason, we

shall from now on only consider matrices A that are weakly p-regular. Also,

ˆ

note that (Sk,d f )(z) always is holomorphic on a region of opening larger

ˆ

than π/k and bisecting direction d, while (SA,d f )(z) is, in general, only

de¬ned close to the bisecting ray.

To give necessary and su¬cient conditions for (C), we introduce the

∞

following terminology: For m ≥ 0, we de¬ne km (z) = n=0 amn z n . Then

we say that A = (amn ) satis¬es the regularity condition (R), if the following

holds:

(R) The functions km (z), m ∈ N0 , are all entire, and converge compactly

to (1 ’ z)’1 , for m ’ ∞ and every z in the sector S(π, 2π).

Moreover, we say that A = (amn ) satis¬es the order condition (O), if the

km (z) are all entire functions of exponential order ¤ 1/2. Finally, we say

that A = (amn ) satis¬es the growth condition (G), if the following holds:

(G) The functions are all entire, and for every k > 1/2 and every σ with

k

1/k < σ < 2 there exist c, K > 0 such that |km (z)| ¤ c eK |z| , for

every z ∈ S(π, (2 ’ σ)π) and every m ∈ N0 .

Observe that the growth condition becomes meaningless for k ¤ 1/2, since

then the interval for σ is empty. This is why we here restrict ourselves to

k > 1/2. Also note that the constants c, K in the estimate are independent

of m.

Let a weakly p-regular matrix A be given and assume that (C) holds.

ˆ

For every k > 1/2 and d with 0 < d < 2π, the formal series fk (z) =

∞

0 “(1 + n/k) z is in C {z}k,d , and hence must be A-summable in every

n

∞

such direction d. This implies that the power series 0 anm “(1 + n/k) z n

must have a positive radius of convergence, for every such k and every

m ≥ 0. From this we conclude the existence of c, K > 0, depending on m

and k but independent of n, so that |amn | ¤ c K n /“(1 + n/k) for every

n ≥ 0. Thus, the order condition (O) follows. Moreover, by de¬nition of

A-summability we have the existence of µ, r > 0, depending on d and k, so

∞

that the functions fm,k (z) = 0 anm “(1 + n/k) z n , for m ’ ∞, converge

¯

uniformly on S(d, µ), for every d as above. A compactness argument then

shows uniform convergence, hence boundedness, of the fm,k on arbitrary

¯

closed subsectors S of S(π, 2π). Since km = Bk fm,k , we can use an estimate

as in the proof of Theorem 24 (p. 82) to show (G). Finally, interchanging

Borel transform and limit, we can conclude that the kernel functions km (z)

186 12. Other Related Questions

converge to (1’z)’1 , and convergence is locally uniform, in S(π, (2’1/k)π).

Hence, using that k can be taken arbitrarily large we see that (R) holds.

So in shorthand notation, we have shown that (C) implies (O), (R), and

(G). The converse also holds, as we now show:

Theorem 59 Let a weakly p-regular in¬nite matrix A be given. Then (C)

holds if and only if (R), (O), and (G) are satis¬ed.

Proof: One direction of the proof has already been given, so we now

assume that (R), (O), and (G) are satis¬ed. For d ∈ R and k > 1/2,

ˆ

consider a series f ∈ E {z}k,d . As shown in the proof of Theorem 41 (p. 120),

ˆ

we can decompose f into a convergent series plus ¬nitely many others

ˆ

which are moment series; so without loss of generality we can restrict f

to have coe¬cients of the form (7.1) (p. 116), with ψ ∈ A1/k,0 (G, E ), and

d + π/(2k) < arg a < d + (2 ’ 1/(2k))π; compare Remark 9 (p. 117) to see

ˆ

that then f ∈ E {z}k,d . In this case, we have

∞ a

1 du

n

fm (z) = amn fn z = ψ(u) km (z/u) ,

2πi u

0

n=0

the interchange of summation and integration being justi¬ed because of

(O). For arg u su¬ciently close to d, we can then use (G) to justify inter-

changing integration and limit as m ’ ∞ to obtain with help of (R):

a

1 ψ(u)

lim fm (z) = du,

u’z

2πi

m’∞ 0

ˆ

and the right-hand side is equal to Sk,d f . 2

Let now any multisummability type k = (k1 , . . . , kq ) be given, assuming

k1 > . . . > kq > 1/2. Then the parameters κj , given by κ1 = k1 , 1/κj =

1/kj ’ 1/kj’1 , 2 ¤ j ¤ q, automatically are larger than 1/2, so that the

Main Decomposition Theorem (p. 164) applies. This shows that the above

theorem immediately generalizes to multisummable series “ however, we

have to restrict to multidirections d = (d1 , . . . , dq ) with d1 = . . . = dq ,

because otherwise there may be no common sector on which the A-sum of

a multisummable series can be de¬ned.

Not very many matrix methods seem to satisfy the conditions (O), (R),

(G): Jurkat [144] studied the matrices J± = (jmn (±)) with jmn (±) =

exp[’δm »(±n)], where δm may be any positive sequence tending to 0 as

m ’ ∞, ± is a positive real parameter, and

»(u) = u log(u + 3) log log(u + 3).

This method had already been introduced by Hardy [113] for summation

of special power series with rapidly growing coe¬cients. It is a variant

12.2 The Method of Reduction of Rank 187

of what is called Lindel¨f ™s methods, useful for computing holomorphic

o

continuation. Jurkat showed that his method satis¬es all three conditions,

so that (C) follows. So far, this is the only method known to have this

property. On the other hand, Braun in [32] showed that no power series

method can have property (C).

Exercises: In the following exercises, consider a ¬xed matrix A = (amn )

that may not be weakly p-regular, and de¬ne km (z) as above, assuming

convergence for |z| < r, with r > 0 independent of m.

∞

0 amn fn z , assume convergence for |z| < ρ, with

n

1. For fm (z) =

0 < ρ, independent of m. Derive the integral representation fm (z) =

(1/2πi) |u|=ρ f (u) km (z/u) (du/u), for ρ < ρ and |z| < ρ r.

˜ ˜

˜

2. Show that A is weakly p-regular if and only if some r with 0 < r ¤ 1

exists, for which km (z) converge locally uniformly to (1 ’ z)’1 on

D(0, r). Conclude that power series regularity is equivalent to the

same with r = 1.

∞

3. For a ∈ C with |a| = 1, let amn = e’am j=n (am)j /j!, m, n ≥ 0.

Use the previous exercise to conclude that this A is weakly p-regular

if and only if a has positive real part, and power series regular if and

only if a = 1.

12.2 The Method of Reduction of Rank

In this section we show that in some sense a system (3.1) (p. 37) of

Poincar´ rank r ≥ 2 is equivalent to one of rank r = 1, which will be

e

called the rank-reduced system. The process of rank-reduction has been

used by Poincar´ [222] and Birkho¬ [53] in representing certain solutions

e

of systems of higher rank as Laplace integrals. Also see Turrittin [270],

Lutz [174], Balser, Jurkat, and Lutz [38], and Sch¨fke and Volkmer [242].

a

Let a system (3.1), with r ≥ 2, be given, let µr = exp[2πi/r]. With 0ν ,

resp. Iν denoting the zero, resp. identity, matrix of dimension ν, de¬ne the

rν — rν matrices D = r’1 diag [0ν , Iν , 2 Iν , . . . , (r ’ 1) Iν ], and

®

0ν 0ν . . . 0ν Iν

Iν 0ν . . . 0ν 0ν

U = . . .

. .

°. .»

. .

. . . .

0ν 0ν ... Iν 0ν

For an arbitrary fundamental solution X(z) of (3.1), set

Y (z) = z ’D U ’1 diag [X(z 1/r ), X(µr z 1/r ), . . . , X(µr’1 z 1/r )]U z D .

r

188 12. Other Related Questions

With some patience, one can then verify that Y (z) is a fundamental solution

∞

of a system z y = B(z) y, B(z) = n=0 Bn z ’n , with coe¬cients of the

following form:

®

Arn Arn+1 . . . Arn+r’2 Arn+r’1

Arn’1 Arn . . . Arn+r’3 Arn+r’2

n ≥ 0,

Bn = ,

. . . .

° »

. . . .

. . . .

Arn’r+1 Arn’r+2 ... Arn’1 Arn

where A’r+1 = . . . = A’1 = 0, and the others are as in (3.1). In particular,

the new system has Poincar´ rank one. It is named the rank-reduced system

e

corresponding to (3.1). Note that the expansion of B(z) converges for |z| >

ρ1/r , with ρ > 0 as in (3.1). For many more formulas relating (3.1) and its

rank-reduced system, see [38].

12.3 The Riemann-Hilbert Problem

Let n distinct complex numbers aj be given, which for notational conve-

nience are assumed to be nonzero. Consider linear systems of ODE of the

n

form x = A(z) x, for z ∈ G = C \ {a1 , . . . , an }, and A(z) = j=1 (z ’

aj )’1 Aj , for given matrices Aj ∈ C ν—ν . Obviously, this is a system with

singular points at a1 , . . . , an and in¬nity, which all are singularities of ¬rst

kind; compare the discussion at the end of Section 1.6 on how to deter-

mine the type of singularity at in¬nity. Every system satisfying the above

requirements will be named a Fuchsian system.

Let such a Fuchsian system be given. Consider paths γj in G, originating

from the origin and going to points “near” aj in one way or another, then

encircling aj in the positive sense along a circle of small radius, and re-

tracing themselves back to the origin. According to results from Chapter 1,

there is a unique fundamental solution X(z) of our Fuchsian system sat-

isfying X(0) = I, holomorphic in some disc about the origin. We perform

its holomorphic continuation along the path γj , ending with a fundamental

solution Xγj (z), which will in general be di¬erent from X(z). In any case,

there exists an invertible constant matrix Cγj so that Xγj (z) = X(z) Cγj .