1.Introduction

In [vdH03, vdH99, vdH01a, vdH02], we have started to develop a fully effective complex analysis. The aim of this theory is to evaluate constructible analytic functions to any desired precision and to continue such functions analytically whenever possible. In order to guarantee that the desired precision is indeed obtained, bound computations are an important part of this program. A key tool for doing this is Cauchy-Kovalevskaya's classical technique of majorant equations [vK75, Pet50, Car61].

In this paper, we will study this technique in a quite detailed way. Although none of the results is fundamentally new or involved, we nevertheless felt the necessity to write this paper for several reasons:

• The need for a more abstract treatment in terms of “majorant relations” .

• The need for explicit majorants which can be used in effective complex analysis.

• Our wish to extend the technique to singular differential equations.

• Our wish to obtain a better control over the precision of the majorants.

• The need for majorants in the contexts of integral transformations and convolution equations.

The first two points are dealt with in sections 2, 3 and 4. In section 2.2, we isolate a few abstract properties of “majorant relations” . It might be interesting to pursue this abstract study in more general contexts like the one from [vdH01b]. In sections 2.3 and 2.4, we also mention some simple, but useful explicit majorants. In sections 3 and 4 we give a detailed account of the Cauchy-Kovalevskaya theorem, with a strong emphasis on “majorant-theoretic properties”. The obtained majorants are quite precise, so that they can be applied to effective complex analysis.

We present some new results in section 5. In sections 5.1 and 5.2, we show how to use the majorant technique in the case of regular singular equations. This improves the treatment in [vdH01a]. In section 5.3 we consider the problem of finding “good” majorants for solutions to algebraic differential equations, in the sense that the radius of convergence of the majorant should be close to the radius of convergence of the actual solution. This problem admits a fully adequate solution in the linear case (see sections 3.4 and 3.5), but becomes much harder in the non-linear setting:

Theorem 1.1. [DL89] Given a power series with rational coefficients, which is the unique solution of an algebraic differential equation

with rational coefficients and rational initial conditions, one cannot in general decide whether the radius of convergence of is or .

In section 5.3 we will nevertheless show how to compute majorants whose radii of convergence approximate the radius of convergence of the actual solution up to any precision (see section 5.3), but without controlling this precision. This result is analogous to theorem 11 in [vdH03].

Our final motivation for this paper was to publish some of the majorants we found while developing a multivariate theory of resurgent functions. In this context, a central problem is the resolution of convolution equations and the analytic continuation of the solutions. At the moment, this study is still at a very embryonary stage, because there do not exist natural isotropic equivalents for majors and minors, and we could not yet prove all necessary bounds in order to construct a general multivariate resummation theory.

Nevertheless, in a fixed Cartesian system of coordinates, multivariate convolution products are naturally defined and in section 6 we prove several explicit majorants. If one does not merely want to study convolution equations at the origin, but also wants to consider the analytic continuation of the solutions, then it is useful to have uniform majorants on the paths where the convolution integrals are computed. Such uniform majorants are studied in section 7 and an application is given. We have also tried to consider convolution integrals in other coordinate systems. In that setting, we rather recommend to study integral operators of the form . Some majorants for such (and more general) operators are proved in section 8.

2.Majorants

2.1.Notations

Throughout this paper, vectors and matrices will be written in bold. We will consider vectors as column matrices or -tuples and systematically use the following notations:

We will denote by the set of power series in with coefficients in . We will sometimes consider other sets of coefficients, like or . Given , the coefficient of in will be denoted by . We define to be the partial derivation with respect to and

its distinguished right inverse .

2.2.Basic algebraic properties of majorants

Given , we say that is majored by , and we write , if and

for all . More generally, if and , then we write if for all .

Proposition 2.1. We have

			(2.1)
			(2.2)

For all :

			(2.3)
			(2.4)
			(2.5)

Proposition 2.2. Let . Then

			(2.6)
			(2.7)
			(2.8)
			(2.9)

If and are such that and are defined (this is so if and ), then

(2.10)

Proof. This is a direct consequence of the fact that the coefficients of , , , and can all be expressed as polynomials in the coefficients of and with positive coefficients.

2.3.Basic explicit majorants

We will often seek for majorations of the form , where

and . For simplicity, we set and .

Proposition 2.3. For , and , we have

whence

for every power series with positive coefficients which converges at .

Proof. This is a trivial consequence from the fact that the coefficients of are increasing.

Proposition 2.4. Let be such that . Then

Proof. For we first notice that

Indeed, this inequality follows from the combinatorial fact that each tuple of choices of persons among (), determines a unique choice of persons among . Hence

for all .

Corollary 2.5. Let and be such that . Then

Proof. Apply the above proposition times for . Next multiply the majoration by on both sides.

2.4.Majorant spaces

For fixed and , let be the space of all analytic power series , such that there exists a majoration for some . We call a majorant space for the majorant norm given by

Proof. Part (b) follows from proposition 2.5. The other properties are easy.

2.5.Majorants of Gevrey type

Divergent power series solutions to ordinary or partial differential equations usually admit majorants of “Gevrey type”. In order to compute such majorants, it may be interesting to consider more general majorant spaces as the ones from the previous section. Given , and , we define

Then we notice that

Furthermore, for all , , and , there exist constants and with

Since divergent power series will not be studied in the sequel of this paper, we will not perform the actual computation of sharp values for and here.

3.Majorant equations and applications

3.1.Noetherian operators

Given coordinates and a subset of , we denote

Let and . An operator is said to be Noetherian if for each , there exists a finite subset of and a polynomial , such that

for every .

Remark 3.1. It can be checked that this notion of Noetherian operator coincides with the one introduced in a more general setting in [vdH01b]. In fact, some of the results of this paper can be generalized to this setting.

Given two Noetherian operators , we say that is majored by , and we write , if

for all . Notice that this implies in particular that . It also implies that is real, i.e. for all . If , then we say that is a majorating Noetherian operator. We say that is strongly majored by , and we write , if

for all . For this majoration, we interpret and as a power series in . Clearly, implies , as well as and .

Remark 3.2. In general, we do not have . A counterexample is the operator with . For strongly linear operators (see [vdH01b]), we do have .

The following proposition, which can be regarded as the operator analogue of proposition 2.2, is a again direct consequence of the classical formulas for the coefficients of , , , , and :

The composition of two Noetherian operators is again Noetherian and we have:

Proposition 3.4. Let and be Noetherian operators. Then

(3.1)

3.2.Majorant equations

A fixed point operator is a Noetherian operator , such that there exists a well-ordering on with

(3.2)

In general, the -th component of the total order will be compatible with the addition on for each .

Proposition 3.5. Let be a fixed point operator. Then the equation

(3.3)

admits a unique solution in .

Proof. We claim that the sequence admits a limit, which is a solution to the equation. Here a limit of a sequence is a series such that for all , we have for all sufficiently large .

Assume for contradiction that the sequence does not admit a limit and let be minimal for , such that is not ultimately constant. By (3.2), the sequence is ultimately constant for every . Consequently, the sequence is ultimately constant. This contradiction implies our claim.

Similarly, assume that there exists a second solution and let be minimal for such that . Then for all , by (3.2). Therefore, , and this contraction proves that is the unique solution to (3.3).

Let be two Noetherian operators and assume that . Then the equation

(3.4)

is called a majorant equation of (3.3).

Proof. We have seen in the previous proof that and . Using induction on , we observe that implies . Consequently, .

For complicated equations, it can be hard to find an explicit solution to the majorant equation (like (3.8)). In that case, one may use

Proposition 3.7. Let be two fixed point operators, such that (3.4) is a majorant equation of (3.3). Assume that is such that

Then .

Proof. Since is real, we have . The fixed point operator therefore satisfies . We conclude that .

3.3.A classical application of the majorant technique

In this section, we will prove the classical Cauchy-Kovalevskaya theorem in the case of ordinary differential equations. We will consider partial differential equations in section 4. Let be a system of convergent power series with and consider the equation

(3.5)

in , with the initial condition . This equation may be rewritten as a fixed point equation

(3.6)

which has a unique solution . Since the are convergent, there exist and with

(3.7)

By propositions (3.3) and (3.4), the equation

(3.8)

is a majorant equation for (3.6). But this equation is symmetric in the , so we have

for each . The unique solution to (3.8) is therefore given by

From proposition 3.6 we now deduce

Theorem 3.8. Let be the unique solution to (3.5) with and assume (3.7). Then

Remark 3.9. Notice that many ordinary differential equations can be reduced to (3.5). For instance, equations of the form can be solved by adding as an unknown to , together with the equation . Similarly, higher order equations can be dealt with through the introduction of new unknowns for the derivatives of unknowns. Modulo substitutions of the form with , one may also consider more general initial conditions.

3.4.First order systems of linear differential equations

It is good not to treat linear differential equations as a special case of arbitrary linear differential equations, because the radius of convergence of the computed solution may be far from optimal. So let us study the system of linear differential equations

(3.9)

in , with initial conditions , where is an by matrix with entries in . Assume that

(3.10)

where denotes the matrix whose coefficients are all , and let . Then the equation

(3.11)

is a majorant equation of

(3.12)

Now the equation (3.11) is again symmetric in the , and the fixed point of the equation

is given by

Proposition 3.6 now implies

Theorem 3.10. Let be the unique solution to (3.9) with and assume (3.10). Then

3.5.Higher order linear differential equations

The exponent in the majorant from theorem 3.10 is not always optimal. Assume for instance that we have a linear differential equation

(3.13)

with initial conditions , and where the satisfy

(3.14)

It is not easy to find a closed form solution for (3.13). For this reason, we will apply the technique from proposition 3.7.

The series is the unique solution to the fixed point equation

(3.15)

For all and , such that , the equation

(3.16)

is a majorant equation of (3.15). Let

We take for all , where

We have

Therefore, we may take

This choice ensures that (3.16) has the particularly simple solution . Proposition 3.6 therefore implies:

Theorem 3.11. Let be the unique solution to (3.13) with and assume that we have (3.14) for . Then

4.Partial differential equations

4.1.Indirect majorant equations and reduction of dimension

In order to obtain majorants for solutions of partial differential equations, it is sometimes possible to generalize Cauchy-Kovalevskaya's technique from section 3.3. However, the more the type of the original equation becomes complex, the harder the explicit resolution of the corresponding majorant equation may become. For this reason, we will introduce a technique, which allows the reduction of the majorant equation to an ordinary differential equation.

Given and , the idea is to systematically search for majorants of the form , where . The choice of depends on the region near the origin where we want a bound for . The ring monomorphism satisfies the following properties:

Proposition 4.1. Let and . Then

			(4.1)
			(4.2)
			(4.3)

The next idea is to extend the majorant technique as follows: given a majorating mapping , and two fixed point operators and , we say that the equation

(4.4)

is an indirect majorant equation of

(4.5)

if for all and we have

We will attempt to apply the following generalization of proposition 3.6 for .

Proof. Similar to the proof of proposition 3.6.

4.2.The Cauchy-Kovalevskaya theorem: equations in normal form

Consider the system of partial differential equations

(4.6)

with initial conditions

where is a tuple of convergent power series in variables. We may rewrite (4.6) as a fixed point equation

(4.7)

Let , and be such that . Then

(4.8)

with initial conditions is an indirect majorant equation of (4.11). This equation can be reinterpreted as an ordinary differential equation of the form (3.5) in unknowns . By theorem 3.8, the unique solution to (4.8) is convergent. By theorem 4.2, the unique solution to (4.6) is therefore convergent, since . We have proved the following theorem:

Remark 4.4. Like in remark 3.9, we notice that many systems of partial differential equations can be seen as variants of (4.6). For instance, modulo substitutions with , we may consider more general types of convergent initial conditions. Also, if the are allowed to satisfy differential equations

of different orders, then it suffices to differentiate these equations times and compute the corresponding initial conditions, in order to reduce this more general case to the case when all equations have the same order.

4.3.Further generalizations of the majorant technique

We did not compute an explicit majorant for , because theorem 4.3 is not the result we are really after. In fact, the right-hand side of (4.6) may also depend on all partial derivatives with , except for . However, the corresponding indirect majorant equation would no longer be a fixed point equation in the sense of section 3.2.

Consider an operator where and . We say that is Noetherian in the generalized sense, if for each , there exists a finite subset of and a convergent power series , such that

for every . All previously defined concepts naturally generalize to this setting.

For instance, given two Noetherian operators in the generalized sense, we say that is majored by , and we write , if

for all . This is in particular so, if for all , in which case we say that is strongly majored by . The concepts of majorating Noetherian operators and indirect majorant equations can be generalized in a similar way.

Proof. By induction on , we observe that for all . Consequently, .

Remark 4.6. Notice that we no longer require (4.9) to be a fixed point equation in proposition 4.5. Nevertheless, it is possible to generalize proposition 3.5 to fixed point operators in a more general sense. For instance, one may replace (3.2) by

(4.9)

and the requirement that for all , where and does not depend on . This kind of fixed point operators will be encountered naturally in the next section (modulo a change of variables). More generally, one may consider operators , whose “traces for truncated series to initial segments” are contracting.

4.4.The Cauchy-Kovalevskaya theorem: the general case

Consider the system of partial differential equations

(4.10)

with initial condition

where is a tuple of convergent power series in variables. We may rewrite this equation as a fixed point equation

(4.11)

Let and be such that and define

Choosing sufficiently small, we may assume that

(4.12)

Then we claim that

(4.13)

is an indirect majorant equation of (4.11) for and the initial conditions , , where

Indeed, the majorant equation (4.13) is symmetric in the components of , so each satisfies the equation

(4.14)

with the initial conditions . The choice of now guarantees that maps into itself.

The equation (4.14), which was obtained mechanically from (4.11) using reduction of dimension, does not admit a simple closed form solution. Therefore, it is convenient to major it a second time by the equation

(4.15)

with the same initial conditions. This latter equation can be rewritten as an equation

of second degree in . This leads to the simple closed form majorant for :

(4.16)

We have proved:

Theorem 4.7. Let be such that , for , and which satisfy (4.12). Then the system (4.10) admits a unique solution with . Moreover, this solution satisfies

Remark 4.8. The original Cauchy-Kovalevskaya theorem, which deals with equations of arbitrary order, directly follows from the first order case. Indeed, consider a system

(4.17)

Introduce the unknowns for all and with . We have equations for all with . For each with , we also have for some with . Finally, the equations (4.17) express each in terms of the with and with and . In a similar way as in remark 4.4, one may also deal with different orders for each .

5.Complements to the majorant technique

5.1.Regular singular systems of ordinary differential equations

Consider the first order system

(5.1)

where and is a tuple of convergent power series in variables with . We search solutions to this system with . Let be the coefficient of in . Then extraction of the coefficient of in (5.1) leads to the relation

(5.2)

where is a polynomial. The matrix is invertible for all but a finite number of values of , so that (5.2) is a recurrence relation for the coefficients of whenever . For , we may see as an initial condition and the relation as an additional requirement on .

Using the notations from (5.2), the original equation (5.1) can be rewritten as a fixed point equation

(5.3)

where is a tuple of convergent power series in variables with and such that the coefficient of in vanishes for all and . Given , let us now consider the change of variables

where are computed by the recurrence relation (5.2). Then (5.3) transforms into a new equation of the form

(5.4)

such that the coefficients of and in vanish.

Now choose sufficiently large, such that . Then we may compute a with

(5.5)

for all . Choose a majorant for of the form

where . Then the equation

(5.6)

is a majorant equation of (5.3) for , where we denote

The equation (5.6) admits a rather long closed form solution. In order to simplify the majorant, we notice that (5.6) is majored by the same equation with . This leads to the majorant

We have proved:

Theorem 5.1. Let be a matrix with coefficients in , such that is invertible for all and let be such that we have (5.5). Consider the equation

where satisfies the majoration

for and . Then this equation admits a unique solution and we have

(5.7)

5.2.Regular singular linear differential equations

In the case of linear differential equations, the majorant (5.7) can be further improved. Consider a regular singular system of linear differential equations

(5.8)

where is an matrix with coefficients in , an matrix with coefficients in , and . We observe that the form of the equation is invariant under substitutions of the form . Consequently, after the computation of the first coefficients of the solution (if such a solution exists), and modulo the change of variables , we may assume without loss of generality that .

Now take with the notations from the previous section. Let and be such that and . Here denotes the matrix whose entries are all . Then, for satisfying (5.5), the equation

admits the majorant equation

This latter equation has the unique solution

We have proved:

Theorem 5.2. Let be a matrix with coefficients in , such that is invertible for all and let be such that we have (5.5). Consider the equation

where is a matrix with coefficients in and . This equation admits a unique solution and, assuming that and , we have

Remark 5.3. Since substitutions of the form do not influence the radius of convergence of , it is important to notice that the constant may be chosen arbitrarily small, when taking large enough. Consequently, we may compute majorants of the form , with as close to as we wish.

5.3.Controlling the radius of convergence

When applying the majorant technique in a straightforward way, the convergence radius of the solution of the majorant equation is usually strictly smaller than the convergence radius of the actual solution. Theorem 1.1 implies that this cannot be avoided in general in the case of non-linear differential equations. Nevertheless, if the radius of convergence of the solution is empirically known, then arbitrarily good majorants can be obtained.

Consider an algebraic differential equation

(5.9)

with and initial condition . Given , let and . We may rewrite (5.9) as an equation in .

where and its coefficients are given by

For each , let be minimal such that

for all . The function is piecewise linear. For each , the equation

(5.10)

is a majorant equation for

on . The solution to (5.10) is given by

This solution has radius of convergence

Let be such that is maximal. We claim that tends to the radius of convergence of when .

Proof. Given and , there exists a constant with

In particular,

Now the radius of convergence of is also for all . Consequently, there exists a constant with

for all . Since we also have

it follows that

where denotes the degree of . In other words, for a suitable constant , which does not depend on , we have

For , it follows that

In particular, for all sufficiently large , we have . Consequently, for all sufficiently large , we have . Since is true for all , we conclude that tends to .

6.Convolution products

6.1.Sharp power series

A sharp power series is a sum

where the are such that does not depend on whenever and where is the Dirac distribution on those with . We denote by the set of sharp power series. The majorant relation naturally extends to sharp power series , by setting if and only if for all . Given , we define

We will also write and .

Let and denote . Given , we will denote by the result of the substitution of for each with in . In particular, if is a sharp power series, then and . Given and such that , we will often abbreviate and for (if the are clear from the context). In particular, . One should be careful not confuse with . If and , then for all .

6.2.Convolution products

For convergent power series and , the -dimensional convolution product is defined by

(6.1)

where

In particular, for all we have

This relation allows us to extend the definition of to in a coefficient-wise way:

(6.2)

Given a function which is analytic at a point , we denote by the translate of at , i.e. . A convolution integral at a translated point can be decomposed in parts:

			(6.3)

Remark 6.1. The formula (6.3) can be reinterpreted using shuffle products by interpreting the -dimensional vector as a shuffle of and (and similarly for ). Using “shuffle notation”, the equation (6.3) becomes

More generally, given and taking , , we define the degenerated convolution product by

(6.4)

where the integral is taken over the -dimensional block . In particular, and . We have the following analogue for (6.3):

(6.5)

The definitions of the convolution product and degenerate convolution products extend to the case when and :

(6.6)

For :

(6.7)

6.3.Majorants for convolution products

Propositions 2.2 and 3.3 still hold if one replaces the componentwise product by any of the componentwise convolution products . More interesting explicit majorants follow below.

Proposition 6.2. Let be two univariate power series. Then

(6.8)

Proof. By strong bilinearity, it suffices to prove (6.8) in the case when and . Then (6.2) implies that

We also have

Since the cardinality of is bounded by , it follows that

Since , it follows that

as desired.

Proposition 6.3. For each there exists a constant such that

			(6.9)
			(6.10)

for all .

Proof. We first observe that the general case reduces to the case when using

Furthermore, setting , for and letting , we have

Since , and for all , it follows that

We conclude by (6.8).

Proposition 6.4. For all and we have

(6.11)

Proof. The final majoration (6.11) again only needs to be proved for . With the notations from the previous proposition, we have

Hence

and the majoration again follows from (6.8).

Proposition 6.5. Let be such that . Then for all and we have

Proof. We may write

Hence

using (6.11) and the fact that .

7.Uniform majorants

7.1.Uniform majorants

Consider a compact subset of such that for every . Let be the multi-radius of , so that each is minimal with the property that for all . Given , we say that is uniformly majored by on , and we write , if for all .

Remark 7.1. Uniform majorants can be seen as a special form of parametric majorants. If is a parameter space and and are analytic functions near , then we write , if for all . In the above case, we thus have

Proposition 7.2. For all analytic functions and on and , we have

			(7.1)
			(7.2)
			(7.3)
			(7.4)

Proof. The bounds apply pointwise.

Proposition 7.3. For all analytic functions and on , all and all we have

(7.5)

In particular,

(7.6)

If is a sharp analytic function on , then we also have

(7.7)

Proof. In the equation (6.5), let be such that and . Then we have

Now for fixed , the following majorations in hold:

Consequently, (9.3) implies

Since this bound holds for all , we obtain

Summing over all and with , this yields

Since for every , we thus obtain

This proves (7.5). From proposition 6.5, it also follows that

This proves (7.6). We finally have

for all and .

Proposition 7.4. Let , let and let be such that . Assume that and are analytic functions on resp. , such that

for all . Then the function

is majored by

(7.8)

In particular, if and , then

(7.9)

If is a sharp analytic function, and , then we also have

(7.10)

Proof. We have

By our hypotheses and (7.5), the following uniform majoration holds for all :

Since , the majoration (7.8) follows by integration over . The bound (7.10) follows from proposition 6.5 in a similar way as (7.6).

As to (7.10), let us fix an with . We first notice that

where and . Consequently, we may apply (7.10) to the lower dimensional case of series in . This yields

Summing over all with , we obtain

This proves (7.10).

7.2.Majorant spaces

For fixed and , let be the space of all analytic functions on , such that there exists a majoration for some . We call a majorant space for the majorant norm given by

This norm may be extended to by

The following properties directly follow from the previous propositions:

Proposition 7.6. With the notations from proposition 7.4 and allowing to be sharp, assume that the suprema

exist. Then and

(7.17)

7.3.Local resolution of convolution p.d.e.s

Let be a compact subset of the hyperplane and let be its multi-radius (so that ). We recall that denotes the set of analytic functions on (so that each such function is analytic on a small -dimensional neighbourhood of ). We denote by the subset of of functions which do not depend on . For each and , we also define to be the operator which sends to .

Consider the linear convolution p.d.e.

(7.18)

where

• for some .

• and .

• .

We will show that (7.18) admits a unique convergent solution in for any convergent initial condition on , provided that is sufficiently large and sufficiently small. More precisely:

Theorem 7.7. Let , and . Denote

and assume that . Then (29) admits a unique solution in for any initial condition .

Proof. Consider the linear operator on defined by

so that (7.18) can be rewritten as

Given with , propositions 7.5(b) and 7.5(d) imply

and proposition 7.5(f) entails

so that

Given with , we also have

Consequently, for we obtain

In other words, the operator is contracting on , since .

Given an initial condition , let us now consider the operator

By what precedes, the operator is again contracting, whence it admits a unique fixed point in , which is the solution to (7.18), and which satisfies

Since the coefficient of in equals for any , the solution also satisfies the initial condition.

In order to see that is the unique solution to (7.18) which satisfies the initial condition, let be another such solution and assume that . Let be the lexicographic valuation of (i.e. is the valuation in of , and the valuation in of the coefficient of in , and so on). Taking the coefficient of in the equation , we obtain

where

But and , since and is the lexicographical valuation of . This contradiction proves the uniqueness of the solution.

8.Majorants for integral transforms

8.1.Majorants for parametric integrals

Let be a compact subset of a real analytic variety and consider an analytic parameterization

which is linear in . We may naturally associate a functional to by

More precisely, this functional is defined coefficient-wise on by

(8.1)

where . Let us study the growth rate of the coefficients of as a function of the growth rate of the coefficients of . We will denote

and

for each and .

Lemma 8.1. Let be such that

Then

for all and .

Proof. Let be fixed and let us show that

(8.2)

This will clearly imply the lemma, because of (8.1).

Since the mapping is linear in , there exists a matrix with . Now by the first condition of the lemma, we have

(8.3)

We claim that this equation is equivalent to

(8.4)

Indeed, (8.3)(8.4) follows by taking and using the facts that and . Inversely, if (8.4) and , then

so that and . In particular, we notice that (8.4) whence (8.3) is satisfied for if and only if it is satisfied for .

Notice also that the condition (8.4) is in it's turn equivalent to the condition

(8.5)

when interpreting and as (linear) series in . Finally, when interpreting as an element of , this latter condition is equivalent to

(8.6)

since and .

Assume now that we have (8.8) and . Then

This proves (8.2) and we completed the proof of the lemma.

8.2.Majorants for integral transforms

More generally, consider two analytic parameterizations which are linear in the first variable. Given two power series , we define by

if and are convergent and coefficient-wise by

(8.7)

in the general case, where . For fixed , the mapping is a linear integral transformation. We have

Lemma 8.2. Let , and be such that and

Then

for all and .

Proof. For fixed , we claim that

(8.8)

This will clearly implies the lemma because of (8.7).

Let and be such that and . In a similar way as in the proof of lemma 8.1, we have

and, by lemma 2.4,

whenever and .

8.3.Generalized convolution products

Given a compact subset of a real affine subspace of the set of all complex matrices, we define the corresponding generalized convolution product of convergent series and in by

(8.9)

This definition extends to the whole of in a similar way as in the case of standard convolution. Notice that if is the set of diagonal matrices with entries in .

Now choose a bijective affine parameterization of and consider the parameterizations with and

Then we have

where is the determinant of the Jacobian of the mapping . This determinant is a homogeneous polynomial in of degree , since was chosen to be affine. Notice that

(8.10)

Consequently, we have

Lemma 8.3. Let and let be such that

Then

for all and .

Proof. Let and . Then , so that

and so that

The result now follows from lemma 8.2 and (8.10).

Bibliography