Elimination, fundamental principle and duality for analytic linear systems of partial differential-difference equations with constant coefficients

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Elimination, fundamental principle and duality for analytic linear systems of partial differential-difference

equations with constant coefficients

Henri Bourlès, Ulrich Oberst

To cite this version:

Henri Bourlès, Ulrich Oberst. Elimination, fundamental principle and duality for analytic linear

systems of partial differential-difference equations with constant coefficients. Mathematics of Control,

Signals, and Systems, Springer Verlag, 2012, �10.1007/s00498-012-0089-8�. �hal-02368155�

Elimination, fundamental principle and duality for analytic linear systems of partial

differential-difference equations with constant coefficients

published in Math. of Control, Signals and Systems 2012, DOI: 10.1007/s00498-012-0089-8

Henri Bourlès

SATIE, ENS Cachan/CNAM,

61 Avenue President Wilson, F-94230 Cachan, France email: [email protected]; tel:+33 1 40958883

and Ulrich Oberst

Institut für Mathematik, Universität Innsbruck Technikerstraße 13, A-6020 Innsbruck, Austria email: [email protected], tel: +43 512 5076073

July 26, 2012

Abstract

In this paper we investigate the solvability of inhomogeneous linear systems of partial differential-difference equations with constant coefficients and also the cor- responding duality problem in how far the solutions of the corresponding homoge- neous systems determine the equations. For ordinary delay-differential (DD) equa- tions these behavioral problems were investigated in a seminal paper by Glüsing- Lürssen (1997) and in later papers by Habets, Glüsing-Lürssen, Vettori and Zampieri.

In these papers the delay-differential operators are considered as distributions with compact support which act on smooth functions or on arbitrary distributions via convolution. The entire analytic Laplace transforms of the distributions with com- pact support play an important part in the quoted papers. In our approach the partial differential-difference operators belong to various topological operator ringsAof holomorphic functions on subsets ofCⁿand are thus studied in the frequency do- main, the arguments of these (operator) functions being interpreted as generalized frequencies. We show that the topological dualsA⁰of these operator rings with the canonical action ofAonA⁰have strong elimination and duality properties for

AA⁰-behaviors and admit concrete representations as spaces of analytic functions of systems theoretic interest. In particular, we study systems with generalized fre- quencies in the vicinity of suitable compact sets. An application to elimination for systems of periodic signals is given. We also solve an open problem of the quoted authors for DD-equations with incommensurate delays and analytic signals. Mod- ule theoretic methods in context with DD-equations have also been used by other authors, for instance by Fliess, Mounier, Rocha and Willems.

1

CONTENTS 2

AMS-classification: 93B25, 93C05, 93C23, 93C35

Keywords: partial differential-difference equation, fundamental principle, elimi- nation, duality, multidimensional system, Stein algebra

Contents

1 Introduction 2

2 Duality and elimination in a general algebraic context 6 3 Analytic rings of operators and signal modules 9

4 Analytic local algebras 14

5 Holomorphic functions on compact Stein sets 17

6 The case of Stein algebras 22

7 Partial differential-difference equations 31

8 Characteristic variety and controllability 38

9 Concluding remarks 40

1 Introduction

In this paper we investigate the solvability of inhomogeneous linear systems of partial differential-difference equations with constant coefficients and also the cor- responding duality problem in how far the solutions of the corresponding homo- geneous systems determine the equations. For ordinary delay-differential (DD) equations these behavioral problems were investigated in the seminal paper [11]

and then in [18], [13], [12], [30]. In these papers the delay-differential operatorsT are considered as distributions with compact support which act onC^∞-functions or arbitrary distributions via convolution. The Laplace transformsTb(s)are entire and play an important part in the quoted papers. In our approach the partial differential- difference operators belong to various topological operator ringsAof holomorphic functions on subsets ofCⁿand are thus studied in the frequency domain, the ar- guments of these functions being interpreted as generalized frequencies. We show that the topological dualsA⁰of these operator rings with the canonical action of AonA⁰furnish strong elimination and duality properties forAA⁰-behaviors and admit concrete representations as spaces of analytic functions of systems theoretic interest. In particular, we study systems with generalized frequencies in the vicin- ity of suitable compact sets. An application to elimination for systems of periodic signals is given. We also solve an open problem of the quoted authors for DD- equations with incommensurate delays and analytic signals. The present paper considerably extends, improves and simplifies work which was started three years ago and announced in [7]. Module theoretic methods in context with DD-equations have also been used by other authors, for instance by Fliess, Mounier [22], Rocha and Willems.

In a more general algebraic setting the properties of the title refer to generalized behaviors of signals whose components belong to a signal moduleAW whereA is a not necessarily noetherian commutative ring of operators which acts onWvia a◦w. Any submoduleU ⊆A^1×q(rows)gives rise to its orthogonalgeneralized

1 INTRODUCTION 3

AW-behavior B:=U^⊥:=n

w= (w1,· · ·, wq)^>∈W^q(columns); U◦w= 0o :=

{w∈W^q; ∀x= (x1,· · ·, xq)∈U : x◦w=x1◦w1+· · ·+xq◦wq= 0}. (1) IfU =A^1×pR, R∈A^p×q,isfinitely generated(f.g.) by the rows of a matrixR

B:=U^⊥={w∈W^q; R◦w= 0} (2) is just called anAW-behavior. IfAis noetherian behaviors and generalized be- haviors coincide. Likewise any submoduleB ⊆ W^q gives rise to its orthogonal module of equationsB^⊥ :=

x∈A^1×q; x◦ B= 0 . The mapsU 7→ U^⊥, B 7→ B^⊥form aGalois correspondenceand induce a one-one correspondence be- tween modulesUwhich satisfyU =U^⊥⊥and generalized behaviorsB=U^⊥. A theorem which characterizes the modulesUwithU =U^⊥⊥is customarily called aduality theoremin behavioral systems theory. Shankar [27, Thm. 2.3] talks about theNullstellensatz problem. IfAWis acogeneratorthen all submodulesUsatisfy U =U^⊥⊥and are therefore in one-one correspondence with generalized behav- iors.

IfP ∈A^q1×q2is a matrix and ifB=U^⊥⊆W^q2is a generalized behavior then the imageP◦ Bis not necessarily such again. If, however, imagesP◦ Bof gen- eralized behaviors are again such then we say thatAW admitseliminationwhere we generalize Willems’ terminology from standard one-dimensional continuous systems theory. In his original definition Willems considers the case

B ⊆W^q1+n, P = (idq₁,0)∈A^q1×(q1+n), P◦:W^q1+n→W^q1, (y, x)^>7→y, only and calls the components ofyresp. ofxthemanifestresp. thelatentvari- ables of the behaviorB. If especiallyP◦W^q2 = (ker(◦P))^⊥withker(◦P) = x∈A^1×q1; xP= 0 holds for allPthis signifies that an inhomogeneous lin- ear systemP◦y=uhas a solutiony∈W^q2 if and only if the given right side u∈W^q1satisfies the necessarycompatibilityorintegrability conditionsx◦u= 0 for allxwithxP = 0. Ehrenpreis introduced the terminologyfundamental prin- ciplefor this property ofAW. The moduleker(◦P)is not f.g. in general and therefore the consideration ofgeneralizedbehaviors is mandatory. These can be avoided ifAiscoherentsince thenker(◦P)and other relevant modules are f.g. An injectivemoduleAW admits elimination and satisfies the fundamental principle.

The significance of injective cogenerator signal modules for multidimensional sys- tems theory was first observed in [24]. Duality and elimination results are known and of highest importance in all approaches to linear systems theory and not only in the behavioral one. For the operator ring of complex multivariate polynomials which acts on various signal spacesW of distributions the duality between mod- ules and behaviors and the deviation ofAW from being injective or a cogenerator has been studied by Shankar in several papers, for instance in [26], [27].

Our considerations were inspired by the following simple standard example: If k[s] =k[s1,· · ·, sn]denotes then-variate polynomial algebra over any fieldkits dual space

Homk(k[s], k)∼=k[[x]] =k[[x1,· · ·, xn]]∼=k^Nn, ϕ↔ X

µ∈Nⁿ

ϕ(s^µ)x^µ↔w= (ϕ(s^µ))_µ∈

Nⁿ

(3)

with the natural action is an injective cogenerator overk[s]. The signal spacek^Nn is furnished with the natural action ofk[s]by left shifts and is one canonical sig- nal module fordiscrete multidimensionalsystems theory. In the present paper we

1 INTRODUCTION 4

prove elimination and duality results for three types oflocally convex topological algebrasAof convergent power series overk =Rork=Cwhich act on their dual spaceA⁰ofcontinuous linear functionsw:A→kwith the canonical action (a◦w)(b) =w(ab). In Section 3 we give concrete representations of the prototyp- icalA⁰which explain their systems theoretic interest. We do not endowA⁰with a topology, but consider it asA-module only. For the systems theoretic applications of this paper such topologies are not needed, although they can be defined and may sometimes be of interest. The prototypical cases are the following:

1. Stein algebras[15]: The algebraAis theC-domainO(Cⁿs)of entire holo- morphic functions innvariabless = (s1,· · ·, sn)with its Fréchet topology of compact convergence. This algebra is not noetherian, but is aStein algebra[9], [15] with various essential topological properties. The elements ofA⁰ are called analytic functionalson Cⁿ [20], [19, §4.5]. TheLaplace transforminduces an O(Cⁿs)-isomorphism

L:O(Cⁿs)⁰∼=W :=O(Cⁿx; exp), ϕ7→ L(ϕ) :=ϕ(e^s•x), with x= (x1,· · ·, xn)∈Cⁿx, s•x=s1x1+· · ·+snxn

(4) whereO(Cⁿx; exp)is the space of entire holomorphic functions inx∈Cⁿofat most exponential growthand where the action◦ofO(Cⁿs)onW is the unique extension of that by partial differentiation, i.e.,

si◦w=∂w/∂xi, especially(e^y•s◦w) (x) =w(x+y), x, y∈Cⁿ. (5) Thus the ringAcontains differentiation and translation operators with which linear systemsR◦w= 0ofpartial differential-difference equationsand their generalized solution behaviors can be defined. Note that in (4) we use the Laplace transform of analytic functionals for the construction of the interesting moduleO(Cⁿx; exp) ofsignalsand not the Laplace transform ofoperatorswhich are considered in the frequency domain from the beginning and whose argumentss∈Cⁿare therefore interpreted asgeneralized frequencies. UnlessO(Cⁿs)iscoherentwhich is not known, but unlikely forn >1(communication of O. Forster) the consideration of generalizedbehaviors is mandatory. Th. 6.2 on duality and elimination for general Stein algebras is the main result in this context: The moduleAW is an injective cogenerator for the categoryAMod^St,fof f.g.Stein modules[9], i.e., the functor HomA(−, W)is faithfully exact on this category. The submodulesU ⊆ A^1×q withU =U^⊥⊥are exactly the closed ones whose factor modulesA^1×q/U are precisely the f.g. Stein modules, up to isomorphism. F.g.Uare closed and in one- one correspondence with the correspondingAW-behaviorsU^⊥. One application of this result is Th. 6.5 on elimination for behaviors ofperiodic analytic signals defined by partial differential equations. The fundamental principle for such be- haviors was recently studied in [23]. For any f.g. subgroupG =⊕^mj=1Zy^(j)of Cⁿ we also consider the Laurent polynomial algebra of polynomial-exponential functions

C[s, σ, σ⁻¹] =⊕y∈GC[s]e^y•s⊂A=O(Cⁿs),

s= (s1,· · ·, sn), σ= (σ1,· · ·, σm), σj:=e^y(j)•s, σ⁻¹ = (σ₁⁻¹,· · ·, σ⁻¹m), (6) which is the least operator algebra containing the partial derivatives and the trans- lations byy^(j). For the casen=m= 1ofordinary delay-differential (DD) equa- tions with commensurate delaysandC^∞-signals Glüsing-Lürssen observed in the seminal paper [11] that for the validity of duality and elimination the noetherian ringC[s, σ, σ⁻¹]has to be replaced by its non-noetherian extension [3]

B:= quot C[s, σ, σ⁻¹] \

A=C(s)[σ, σ⁻¹]\

A⊂quot(A) (7)

1 INTRODUCTION 5

wherequotdenotes the quotient field and especiallyquot(A)the field ofmero- morphic functions. The book [12] is an in-depth-study of the casen =m = 1 forC^∞-signals. The casen= 1, m >1of DD-equations withincommensurate delaysandC^∞-signals was discussed in [18] and [13]. In Theorems 7.1 and 7.3 we treat duality and elimination for the casen≥1, m≥1and the signal module

BW. Th. 7.7 seems to be the first elimination result forn= 1, m >1and solves an open problem of [13], but for analytic signals only. In Th. 8.3 we characterize controllable and spectrally controllableAW-behaviors.

2.Analytic local algebras[14]: The prototypical case is the noetheriank-domain k < s >=k < s1,· · ·, sn>⊂k[[s]]of (locally) convergent power series over the real or complex fields fieldk = R,C. It is endowed with itslocally convex sequence topology[14]. Any nonzero factor algebraA = k < s > /awith its coinduced locally convex topology is called ananalytic local algebra[14]. In Th. 4.3 we show that for each suchAthe dual spaceA⁰with its naturalA-action is an injective cogenerator for the categoryAModofA-modules. This signifies that the functorHomA(−, A⁰)is faithfully exact on the category ofallA-modules and implies the strongest possible elimination and duality properties for the as- sociated behaviors as was shown already in [24] for behaviors governed by pure partial differential equations. ForA=k < s >the Laplace transform induces a k < s >-isomorphism ofk < s >⁰onto the spaceO(kxⁿ; infexp)of everywhere convergent power series ofinfra-exponential, i.e. less than exponential, growth with the action from (5) [20] which contains all polynomials in particular. For the algebrak < s−z >of power series centered atz ∈kⁿwe obtain the injective cogeneratorWz =O(kxⁿ; infexp)e^z•xwith the action from (5). Since a power seriesP ∈k < s−z >is locally convergent and defines a function for a general- ized frequencysnearzonly we interpret ak<s−z>Wz-behavior as a system with generalized frequencies in the vicinity ofzonly.

3.Noetherian rings of holomorphic functions on compact Stein subsets of a com- plex manifold: In Th. 5.6 we prove the injective cogenerator property ofAA⁰ whereA=O(K)is the ring of holomorphic functions on a compact, connected, semi-analytic Stein subsetKof a complex manifoldZ. Such anAis noetherian.

IfK is a compact semi-analytic and geometrically convex subset ofZ = Cⁿ, for instance a compact polyhedron or polydisc, the Laplace transform induces an isomorphismL: O(K)⁰ ∼= WK, ϕ 7→ϕs(e^s•x),whereWK is a subspace of O(Cⁿx; exp)of functions with explicitly described growth conditions (Th. 5.7).

Again the interpretation of aO(K)WK-behavior is that of a system with general- ized frequencies in the vicinity of the compact setK.

Technique: The theorems of this paper and their proofs rely on deep results from theTheory of functions of several complex variablesorAnalytic Geometry, in par- ticular onSteinspaces, manifolds, algebras and modules and on coherent analytic sheaves on Stein spaces [9], [19], [20], [16], [17], [29]. These results were al- ready partially used in [24] and also by Shankar and Sule [28]. Other important ingredients of the proofs are fundamental results fromFunctional Analysis, in par- ticular the Hahn/Banach theorem, the bipolar theorem, the open mapping theorem for Fréchet spaces and its generalizations to DFS-spaces (dual Fréchet-Schwartz spaces). The relevant f.g.A-modulesMare endowed with locally convex topolo- gies for which one can show the exactness of the functorM 7→M⁰and the functo- rial isomorphismM⁰∼= HomA(M, A⁰)from which the elimination properties are inferred. Except in Section 5 no knowledge of analytic sheaves is required since we collect all needed algebraic and topological properties of the relevant algebras and modules in Results 4.2, 6.1 and 8.1 from the literature with exact quotations.

In Section 5 we have to use the full machinery of several complex variables; we give precise references for all used results.

Acknowledgement: We thank O. Forster for various useful hints and explana-

2 DUALITY AND ELIMINATION IN A GENERAL ALGEBRAIC CONTEXT 6

tions. We thank the referees for their efforts in reviewing this long and technically difficult paper.

2 Duality and elimination in a general algebraic context

We add some simple remarks on duality and elimination in a general algebraic context (compare [26]). LetAW be a signal module as in the Introduction and letAModdenote the category of allA-modules with theA-linear maps. For all p, q, r,∈Nthere are theA-modules andA-bilinear forms

A^p×q, W^q×rand◦:A^p×q×W^q×r→W^p×r, (f, w)7→f◦w, f= (fi,j)i,j∈A^p×q, w= (wj,k)j,k∈W^q×r, f◦w:= ((f◦w)i,k)_i,k∈W^p×r, (f◦w)i,k:=X

j

fi,j◦wj,k, especiallyA^1×q ={rows}, W^q:=W^q×1={columns},◦: A^1×q×W^q→W.

(8) For anyA-moduleM letP(M)denote the projective geometry of M, i.e., its ordered set ofA-submodules. The bilinear form (8) induces the (order reversing) Galois correspondence(compare (1),(2))

P(A^1×q)P(W^q), U →U^⊥, B^⊥← B,

withU ⊆U^⊥⊥, B ⊆ B^⊥⊥, U^⊥=U^⊥⊥⊥, B^⊥=B^⊥⊥⊥. (9) This induces the order anti-isomorphism orduality

U ∈P(A^1×q); U=U^⊥⊥ ∼=

B ∈P(W^q); B=B^⊥⊥

U =B^⊥ ↔ B=U^⊥ . (10)

The orthogonalU^⊥is also defined for allsubsetsU ⊆ A^1×q and thenU^⊥ = (A< U >)^⊥whereA < U >is theA-submodule generated byU. Likewise B^⊥is defined for all subsets ofW^q. The submoduleU^⊥is called thegeneralized

AW-behaviordefined byU. The moduleB^⊥is called themodule of equationsof B. The duality (10) obtains the form

n

U∈P(A^1×q); U =U^⊥⊥o

∼={B ⊆W^q; Wis a generalized behavior}. (11) Consider the standardA-basisδi = (0,· · ·,1,ⁱ · · ·,0) ∈ A^1×q, i= 1,· · ·, q.

ForU ∈P A^1×q

with generalized behaviorB=U^⊥there is the obviousMal- grange isomorphism

HomA(A^1×q/U, W)∼=U^⊥=B, ϕ←→w, ϕ(δi+U) =wi. (12) SinceB=B^⊥⊥the preceding isomorphism also induces the isomorphism

HomA(A^1×q/B^⊥, W)∼=B, ϕ←→w, ϕ(δi+B^⊥) =wi. (13) The moduleA^1×q/B^⊥depends on the generalized behaviorBonly and induces the Malgrange isomorphism and is therefore calledthe moduleofB.

2 DUALITY AND ELIMINATION IN A GENERAL ALGEBRAIC CONTEXT 7

Lemma 2.1. AssumeP ∈ A^q1×q2, submodulesUi ⊆A^1×qi withU1P ⊆U2, and generalized behaviorsBi ∼=

(12)HomA(A^1×qi/Ui, W). Then

◦P :A^1×q1 →A^1×q2, x7→xP,induces theA-linear map

φ:= (◦P)ind:A^1×q1/U1→A^1×q2/U2, x7→xP , (14) and the commutative diagram

HomA(A^1×q1/U1, W) ^Hom(φ,W←− ⁾ HomA(A^1×q2/U2, W)

↓∼=(12) ↓∼=(12) B1 =U₁^⊥ ←−^P◦ B2=U₂^⊥

. (15)

This applies especially to anyU2andU1:= (◦P)⁻¹(U2), themonomorphism φ= (◦P)ind:A^1×q1/(◦P)⁻¹(U2)→A^1×q2/U2and

P◦:U2^⊥→ (◦P)⁻¹(U2)⊥

.

(16) Proof. The inclusion U1P ⊆ U2 implies thatφis well-defined. Letδ^k_i, i = 1,· · ·, qk,denote the standard bases ofA^1×qkfork= 1,2. Then

(◦P)(δ¹_i) =δ¹_iP =Pi−=

q₂

X

j=1

Pijδ_j²,henceφ(δ¹_i) =

q₂

X

j=1

Pijδ_j².Let ϕ²∈HomA(A^1×q2/U2, W)andϕ¹:= Hom(φ, W)(ϕ²) =ϕ²φ.

The Malgrange isomorphisms (12) mapϕ^kontow^k∈W^qkwithw^k_i :=ϕ^k(δ_i^k), hence

w¹_i =ϕ¹(δ_i¹) =ϕ²φ(δ¹_i) =ϕ²(

q₂

X

j=1

Pijδ²_j) =

q₂

X

j=1

Pij◦ϕ² δ_j²

=

q₂

X

j=1

Pij◦w²_j, hencew¹=P◦w²and the commutativity of (15).

The following definition is a generalization of Willems’ terminology.

Corollary and Definition 2.2. (Elimination, fundamental principle)Data of Lemma 2.1.

1. IfU2=U2^⊥⊥=B^⊥2 thenP:U2^⊥→ (◦P)⁻¹(U2)⊥

and (◦P)⁻¹(U2)⊥

is the least generalized behavior containingP◦U2^⊥.

2. By definitioneliminationholds for generalizedAW-behaviors if and only if the image of a generalized behavior is again such. By 1. this signifies that

(◦P)⁻¹(U2)^⊥

=P◦U2^⊥for allPandU2=U2^⊥⊥.

3. IfP ◦W^q2 = ker(◦P)^⊥holds for allP as before andU2 := 0, hence U₂^⊥ = W^q and(◦P)⁻¹(U2) = ker(◦P)thenAW is said to satisfy the fundamental principle. If, in addition,ker(◦P) = A^1×kQis finitely gen- erated by the rows of a matrixQ∈ A^k×q1 then the inhomogeneous linear systemP◦y =uhas a solutionyif and only ifQ◦u= 0. IfAW is a faithful module, i.e.,W^⊥= 0and thus0 = 0^⊥⊥= (W^q2)^⊥for allq2and U2:= 0⊆A^1×q2, then elimination implies the fundamental principle.

4. IfAW is an injective module then the monomorphismφin(16)gives rise to the epimorphismsHom(φ, W)andP◦ : U₂^⊥ → (◦P)⁻¹(U2)⊥

in (15)for all submodulesU2 ⊆A^1×q2, and especially elimination holds for generalizedAW-behaviors.

2 DUALITY AND ELIMINATION IN A GENERAL ALGEBRAIC CONTEXT 8

The last statement of item 1. of the preceding corollary is easily checked.

Notice that its item 2. does not imply that the image of aW-behavior is again such because it may occur in (16) thatU2is finitely generated, but(◦P)⁻¹(U2)is not sinceAis not assumed noetherian.

Definition and Corollary 2.3. (Coherence[4, §I.2, Ex. 11,12 on p.44]) By defi- nition anA-module is finitely generated (f.g.) resp. finitely presented (f.p.) if it is isomorphic to a module of the formA^1×q/U with any resp. a f.g. submoduleU. LetAMod^f AMod^{f p}

denote the full subcategories ofAModof all f.g. (f.p.) modules. AnA-moduleMis called coherent if it is f.g. and if each f.g. submodule is f.p.. This definition applies especially toAconsidered asA-module. A coherent module is f.p.. In an exact sequence0→M1→M2→M3→0all threeMiare coherent if and only two of them have this property. This implies that homomor- phisms between coherent modules have coherent kernel, image and cokernel. The ringAis coherent if and only if the annihilatorannA(a) :={f∈A; f a= 0}of each elementa∈Aand the intersection of any two f.g. ideals are again f.g. IfA is an integral domain the annihilator condition is trivially satisfied. IfAis coherent then the coherent modules are precisely the finitely presented ones, i.e., form the categoryAMod^{f p}.IfAis coherent and ifU2in (16) is f.g. the kernel

(◦P)⁻¹(U2) = ker (◦P)ind:A^1×q1→A^1×q2/U2

(17) is coherent and thus f.g. This implies that (◦P)⁻¹(U2)⊥

is a behavior.

Lemma 2.4. Assume thatAis coherent and thatW satisfies the following weak Baer condition: For anyf.g. idealaofAthe mapW → HomA(a, W), w 7→

(f7→f◦w),is surjective. If in(16)

U2is f.g. then so is(◦P)⁻¹(U2)andP◦U₂^⊥= (◦P)⁻¹(U2)^⊥ . In particular, the image of a behavior is a behavior.

Proof. Equation (17) implies that the map φfrom (16) is a monomorphism be- tween f.g. modules. It suffices to show thatHom(φ, W)is surjective. Let, more generally,U be any f.g. submodule of a f.g. moduleMand letinj :U →M be the canonical injection. We show that the restriction

HomA(M, W)→HomA(U, W), ϕ7→ϕ|U=ϕinj, (18) is surjective. SinceM admits a filtrationU0 ⊆U1 ⊆ · · · ⊆Uk =Mwith f.g.

Uiand cyclic factorsUi/Ui−1 it suffices, via induction, to prove the surjectivity of (18) for the case thatM/Uis cyclic. So letM/U =AxorM =U+Axand consider the annihilator ideal

a:= ker (A→M/U, f7→f(x+U)) ={f∈A; f x∈U}. It is f.g. sinceAandM/U are coherent. Letψ :U →M beA-linear. By the weak Baer condition there is aw∈W such that

a−→^·x U−→^ψ W, f 7→ψ(f x), has the formψ(f x) =f◦w, f∈a.

Define the map

ϕ:M =U+Ax→W, u+f x7→ψ(u) +f◦w.

This is well-defined since

u1+f1x=u2+f2x=⇒(f2−f1)x=u1−u2∈U =⇒f2−f1∈a=⇒ ψ(u1−u2) =ψ((f2−f1)x) = (f2−f1)◦w=⇒ψ(u1) +f1◦w=ψ(u2) +f2◦w.

Obviouslyϕis anA-linear extension ofψ, i.e.,ϕ|U=ψ.

3 ANALYTIC RINGS OF OPERATORS AND SIGNAL MODULES 9

The following lemma is well-known.

Lemma and Definition 2.5. (Willems closure, cogenerator, compare [26, §2]) If U ∈ P(A^1×q),M := A^1×q/U andB := U^⊥thenU^⊥⊥/U = B^⊥/U is the kernel of the linear map

canM : M =A^1×q/U → W^B ∼= W^HomA(M,W)

x=x+U 7→ (x◦w)w∈B 7→ (ϕ(x))_ϕ∈Hom

A(M,W). The moduleU^⊥⊥is also called theWillems closureofU with respect to (w.r.t.) W. IfU^⊥⊥ =UthenUis calledWillems closedw.r.t.W. The duality(11)thus establishes a duality between Willems closed modules and generalized behaviors.

The moduleW is called acogenerator for f.g. modulesif for every f.g. A- moduleM = A^1×q/U the mapcanM is injective or, equivalently,U = U^⊥⊥

or, in still other words, ifHomA(M, W)separates the elements ofM, hence es- peciallyHomA(M, W) 6= 0ifM 6= 0. Then(11)establishes a duality between P(A^1×q)and the set of generalized behaviors inW^q. The moduleAWis called a cogeneratorifcanM :M→W^HomA(M,W)is injective forallA-modulesM.

3 Analytic rings of operators and signal mod- ules

We introduce the rings of operators and signal spaces relevant for this paper.

Letkdenote the field of real numbersRor of complex numbersC. Letn∈Nand

k[[s]] :=k[[s1,· · ·, sn]] =







f= X

µ∈Nⁿ

fµs^µ, fµ∈k







, s^µ:=s^µ₁¹s^µ₂²· · ·s^µ_nⁿ (19) be thek-algebra of formal power series in indeterminatess1,· · ·, snand likewise k[[x]] :=k[[x1,· · ·, xn]]. Following [14] and [19] we introduce the algebras of locally convergent and of entire or everywhere convergent power series. For this purpose we consider the ordered setRⁿ+of real vectorsT = (T1,· · ·, Tn)with positive entriesTi>0. With the componentwise order this set is directed upwards and downwards. We writeT ≤Sresp.T < SifTi≤Siresp.Ti< Sifor alli.

ForT ∈Rⁿ+we introduce the poly-cylinder [14]

Z(T) :={z∈kⁿ; ∀i= 1,· · ·, n:|zi|< Ti} ⊂ Z(T) :={z∈kⁿ; ∀i= 1,· · ·, n:|zi|≤Ti},

|f|T:=X

µ

|fµ|T^µ≤ ∞, BT:={f∈k[[s]];|f|T<∞},

∀T≤S, f∈k[[s]] :|f|T≤ |f|S,henceBS⊆BT.

(20)

The algebraBTis ak-Banach subalgebra ofk[[s]]with the norm| − |T[14, Satz 1 on p.16]. Consider the algebraC⁰(Z(T))of continuousk-valued functions on the compact setZ(T)with the maximum norm||f ||T:= max_z∈Z(T) |f(z)|.

Everyf∈BTdefines the function

f:Z(T)→k, z7→f(z) := X

µ∈Nⁿ

fµz^µ, with

f∈C⁰(Z(T))and ||f||T≤|f|T, hence inj :BT⊆C⁰(Z(T)) (21)

3 ANALYTIC RINGS OF OPERATORS AND SIGNAL MODULES 10

is a contraction and thus continuous. For the complex casek = CCauchy’s inequalities [19, Th. 2.2.7] imply

|fµ|=|f^(µ)(0)|

µ! ≤||f||T T^−µ. (22) The algebras of entire or everywhere convergent resp. of locally convergent power series are then defined as

O(kⁿs) := \

T∈Rⁿ₊

BT⊂k < s >:= [

T∈Rⁿ₊

BT =







f∈k[[s]]; ∃z∈kⁿwithzi6= 0, i= 1,· · ·, n,and convergent X

µ∈Nⁿ

fµz^µ.





 (23) Both carry a natural topology. The algebraO(kⁿs)carries the initial or induced topology with respect to all inclusionsO(ksⁿ) ⊂ BT, T ∈ Rⁿ+. By this it is a Fréchet algebra and a sequence inO(kⁿ_s)converges if and only if it converges in allBT.

Remark 3.1. The inequality|| − ||T≤ | − |T implies that the natural topol- ogy onO(kⁿ_s)is finer than the topology of compact convergence fork =R,C. In the complex caseO(Cⁿs)is the algebra of holomorphic functions in variables s1,· · ·, snonCⁿ. Forf∈ O(Cⁿ)andT < Sthe inequality (22) implies

|f|T=X

µ

|fµ|T^µ=X

µ

|fµ|S^µ T

S µ

≤

X

µ

T S

µ!

||f||S=C||f||S withC:=X

µ

T S

µ (24)

and thus that the natural topology coincides with the topology of compact conver- gence. The natural topology ofO(Rⁿs)is finer, but not equal to that of compact convergence. Notice that Cauchy’s inequality (11) is not valid in the real case.

The natural topology onk < s >is thesequence topologydiscussed in [14,

§I.3.4, §I.6, §I.7]. It is the final topology induced by all injectionsBT ⊆ k <

s >. Iffk, k ∈ N,andfbelong tok < s >then thefkconverge tofin the sequence topology if and only if there is aT ∈ Rⁿ+ such thatfk, f ∈ BT and f= limkfkinBT[14, Satz I.7.8 on p.67]. A map fromk < s >to a topological space is continuous if and only if it preserves convergent sequences. The injection O(k_sⁿ)⊂k < s >is obviously continuous. The ringk < s >is noetherian [14, Satz I.5.3 on p.45] and local with the unique maximal ideal [14, I.3.1 on p.27]

m:={f∈k < s >; f(0) =f0= 0}=

n

X

i=1

k < s > si (25) and the residue fieldk < s > /m = k. With the sequence topologyk < s >

is a Hausdorff topological algebra [14, Satz I.6.3 on p.58 and Satz I.7.7 on p.66], locally convex [14, Satz I.8.7 on p.74], sequence-complete [14, Satz I.8.6 on p.74], i.e., every Cauchy sequence converges, but does not have countable neighborhood bases [14, Satz I.7.9 on p.67] and is therefore not metrizable. We use the same topologies onk < x >andO(kⁿx). Below we will discuss and essentially use the category of f.g. k < s >-modules with their sequence topology [14, Kap.II, §0,

§1].

We now introduce the signal spaces of the present paper. Letk < s >⁰ and

3 ANALYTIC RINGS OF OPERATORS AND SIGNAL MODULES 11

O(ksⁿ)⁰ denote the topological dual spaces, i.e., the k-spaces of continuousk- linear functions onk < s >resp. O(ksⁿ)into the topological fieldk. With the help of [19] it was shown in [24, Th.4.20 on p.67] that thek-bilinear map

h−,−i: k < s >×O(kxⁿ)→k, hf, wi=hX

µ

fµs^µ,X

µ

wµx^µi:=X

µ

fµwµ, (26) is well-defined and non-degenerate in the strong sense that it induces isomorphisms

k < s >⁰∼=O(kⁿx), ϕ=h−, wi ↔w=X

µ

ϕ(s^µ)x^µ k < s >∼=O(kⁿx)⁰, f=X

µ

ψ(x^µ)s^µ↔ψ=hf,−i, hence also(s↔x) O(kⁿ_s)⁰∼=k < x >, ϕ=h−, wi ↔w=X

µ

ϕ(s^µ)x^µ.

(27) The dual spaces are not considered as topological here. The proof in [24, Th.4.20 on p.67] was given fork=C, but the proof fork=Ris the same. Recall that the topology onO(Rⁿs)is finer than the topology of compact convergence. Since the algebrasO(ksⁿ)andk < s >are topological their dual spaces are modules over them with the action or scalar multiplication

(f◦ϕ)(g) :=ϕ(f g)for

(f, g∈ O(k_sⁿ), ϕ∈ O(kⁿ_s)⁰

f, g∈k < s >, ϕ∈k < s >⁰ . (28) Via the isomorphisms from (27) we obtain modulesO(kⁿ_s)k < x >andk<s>O(kⁿx) with the actions

f◦w, hg, f◦wi=hf g, wifor

(f, g∈ O(kⁿs), w∈k < x >,

f, g∈k < s >, w∈ O(kⁿx), with s^ν◦X

µ

wµx^µ=X

µ

wν+µx^µ.

(29) Therefore these actions are extensions of the left shift action of the polynomial algebrak[s]on the multi-sequence spacek^Nn=k[[x]]to the larger rings of opera- tors and smaller signal spacesO(kⁿx)⊂k < x >⊂k[[x]]. Therefore these signal modules are suitable for partial difference equations and discrete systems theory.

The canonical isomorphism

can :k[[x]]∼=k[[x]], X

µ

wµx^µ7→X

µ

wµ

µ!x^µ, (30) induces isomorphisms on the spaces of convergent power series. For this purpose we introduce growth conditions for entire holomorphic functions, first in the com- plex case. Let|| − ||denote any norm onCⁿand then define

O(Cⁿx; exp) :=n

w∈ O(Cⁿx); ∃δ >0∃Cδ>0∀x∈Cⁿ:|w(x)|≤Cδe^δ||x||o

⊃ W0:=O(Cⁿx; infexp) :=

n

w∈ O(Cⁿx); ∀δ >0∃Cδ>0∀x∈Cⁿ:|w(x)|≤Cδe^δ||x||

o .

(31) The functions inO(Cⁿx; exp)resp. inO(Cⁿx; infexp)are calledof at most expo- nential resp. ofinfra-exponential growth. Since all norms onCⁿare equivalent

3 ANALYTIC RINGS OF OPERATORS AND SIGNAL MODULES 12

the definition does not depend on the choice of|| − ||onCⁿ. With the help of [19] it was shown in [24, Th. 4.27] thatcanfrom (30) induces theC-isomorphism can :C< x >∼=O(Cⁿx; exp). (32) The same proof furnishes theC-isomorphism

can :O(Cⁿx)∼=O(Cⁿx; infexp). (33) In the real case we define

O(Rⁿx; exp) :=O(Rⁿx)\

O(Cⁿx; exp)⊃ O(Rⁿx; infexp) :=O(Rⁿx)\

O(Cⁿx; infexp).

(34) Obviously the isomorphisms (32) and (33) then also hold withCreplaced byR. By transport of structure we replace the signal spaces in equations (26) to (29) by the isomorphic ones from (32) to (34) and obtain the following

Corollary and Definition 3.2. Letk=R,C. Thek-bilinear forms h−,−i0: O(kⁿs) × O(kⁿx; exp) → k

T S

k h−,−i0: C< s > × O(kⁿx; infexp) → k , hX

µ

fµs^µ,X

µ

wµx^µi0:=X

µ

fµwµµ!,

(35)

are well-defined and non-degenerate and induce theLaplace transformisomor- phisms

L: O(k_sⁿ)⁰ ∼= O(k_xⁿ; exp)

S S

L: k < s >⁰ ∼= O(k_xⁿ; infexp) ϕ=h−, wi0 ↔ w=L(ϕ) =P

µ ϕ(s^µ)

µ! x^µ=ϕ(e^s•x)

(36)

wheres•x:=s1x1+· · ·+snxn. These areO(ksⁿ)- resp.k < s >-isomorphisms if the power series spaces inxare equipped with the action◦0defined by

hg, f◦0wi0=hf g, wi0.Then

s^µ◦0w=∂^µwand (e^z•s◦0w) (x) =w(x+z), x, z∈kⁿ, (37) where∂ := (∂/∂x1,· · ·, ∂/∂xn). The isomorphisms(32)and(33)areO(kⁿs)- resp. k < s >-linear and thus establish an isomorphism between discrete and continuous analytic signal modules and, of course,O(kⁿx; infexp)is anO(kⁿs)- submodule ofO(kⁿ_x; exp).

The formula(e^z•s◦0w) (x) = w(x+z)implies that the action ofO(kⁿs) on analytic signals realizes the translation action in particular. Therefore we later consider the subring

⊕z∈kⁿk[s]e^z•s⊂ O(ksⁿ) (38) ofpolynomial-exponential functionswhich acts onO(kⁿx; exp)bypartial differential- differenceoperators, the term difference referring to the translation action.

We finally extend the preceding considerations from the center0to arbitrary points ofkⁿ. Letz∈kⁿbe such a point andk < s−z >the ring of locally convergent power series atzwith the sequence topology which is defined like that ofk < s >.

Sincek[s] =k[s−z]⊂ O(kⁿs)⊂k < s−z >and sincek[s−z]is dense in k < s−z >so isO(k_sⁿ)and therefore we identify

k < s−z >⁰⊂ O(kⁿs)⁰, ϕ=ϕ| O(kⁿs)forϕ∈k < s−z >⁰,and define Wz:=L k < s−z >⁰

⊂ O(kxⁿ)

(39)

3 ANALYTIC RINGS OF OPERATORS AND SIGNAL MODULES 13

where, according to Cor. 3.2,L : O(ksⁿ)⁰ ∼= O(kⁿx; exp)is the Laplace trans- form andW0 =O(kxⁿ; infexp). Moreover there are the topological translation isomorphisms

tz : O(kⁿs) −→ O(kⁿs)

T T

tz : k < s > −→ k < s−z >

f=P

µ∈Nⁿfµs^µ 7→ tz(f) :=f(s−z) =P

µ∈Nⁿfµ(s−z)^µ with their adjoints

t⁰_z: O(kⁿ_s)⁰ −→ O(kⁿ_s)⁰

S S

t⁰z: k < s−z >⁰ −→ k < s >⁰

, ψ7→t⁰z(ψ) :=ψtz

(40) Obviouslyt−zis the inverse oftz.

Lemma 3.3. The following diagrams ofk-isomorphisms commute:

O(kⁿs)⁰ −→^L O(kxⁿ; exp) t⁰z↑↓t⁰−z e^−z•x· ↑↓e^z•x·

O(kⁿ_s)⁰ −→^L O(k_xⁿ; exp) and

k < s >⁰ −→^L W0=O(kⁿx; infexp) t⁰z↑↓t⁰−z e^−z•x· ↑↓e^z•x· k < s−z >⁰ −→^L Wz =L(k < s−z >⁰)

, henceWz=e^z•xW0. (41) Proof. Forϕ∈ O(ksⁿ)⁰we indeed have

Lt⁰_z(ϕ) =L(ϕtz) =ϕ(tz(e^s•x)) = ϕ

e^(s−z)•x

=ϕ e^−z•xe^s•x

=e^−z•xL(ϕ).

Assume that

w∈ O(kⁿ_x; exp)with |w(x)|≤Cδe^δ||x||for someδ, Cδ>0and allx∈Cⁿ where||x||is a norm onCⁿ. Since||x−z||≤||x||+||z||we infer

|tz(w)(x)|=|w(x−z)|≤Cδe^δ||z||e^δ||x||andtz(w)∈ O(kxⁿ; exp), hence tz : O(kxⁿ; exp) ∼= O(kxⁿ; exp),

S S

tz : O(kxⁿ; infexp) ∼= O(kⁿx; infexp) .

(42) Corollary 3.4. The spaceWz=L(k < s−z >⁰) =e^z•xW0is also given as

Wz = n

w∈ O(kⁿx); ∀δ >0∃Cδ>0with |w(x)|≤Cδe<(z•x)+δ||x||o . Proof. The following equivalences hold:

w∈Wz ⇐⇒ e^−z•xw(x)∈W0=O(kⁿx; infexp) ⇐⇒

∀δ >0∃Cδ>0withe^−<(z•x)|w(x)|=|e^−z•xw(x)|≤Cδe^δ||x|| ⇐⇒

∀δ >0∃Cδ>0with |w(x)|≤Cδe<(z•x)+δ||x||

.

4 ANALYTIC LOCAL ALGEBRAS 14

We finally determine the module structures of the considered spaces. The Laplace transform isomorphism

L:O(kⁿs)⁰→(O(kxⁿ; exp),◦0) isO(kⁿs)-linear andk < s−z >⁰⊂ O(kⁿs)⁰ is anO(kⁿ_s)-submodule. Therefore

Wz=L k < s−z >⁰

⊂(O(kⁿ_x; exp),◦0) is anO(k_sⁿ)-submodule too with f◦0w, f∈ O(ksⁿ), w∈Wz, especially

s^µ◦0w=∂^µw, (e^y•s◦0w) (x) =t−y(w)(x) =w(x+y), x, y∈kⁿ. (43) But the dual modulek < s−z >⁰is also ak < s−z >-module with(f◦ψ)(g) :=

ψ(f g)which forf ∈ O(ksⁿ)coincides with the structure asO(kⁿs)-module. Via Lwe transport this structure toWzand obtain a scalar multiplication

f◦zw, f∈k < s−z >, w∈Wz, withf◦zw=f◦0wforf∈ O(ksⁿ), especiallys^µ◦0w=∂^µw, (e^y•s◦0w) (x) =t−y(w)(x) =w(x+y), x, y∈kⁿ,

(44) according to (43).

Corollary 3.5. With these structures thek-isomorphisms from(41) t⁰−z :k < s >⁰→k < s−z >⁰ ande^z•x·:W0→Wz

are semilinear with respect totz :k < s >∼=k < s−z >, f 7→f(s−z),where f= X

µ∈Nⁿ

fµs^µ∈k < s >, tz(f) = X

µ∈Nⁿ

fµ(s−z)^µ∈k < s−z >,i.e., t⁰−z(f◦ϕ) =tz(f)◦t⁰−z(ϕ), f∈k < s >, ϕ∈k < s >⁰, and e^z•x(f◦0w) =tz(f)◦z(e^z•xw), f∈k < s >, w∈W0=O(kⁿx; infexp), or

f◦zw=e^z•x t−z(f)◦0(e^−z•xw)

, f∈k < s−z >, w∈Wz. These formulas generalize the standard formula(∂−z)^µ(e^z•xw) =e^z•x∂^µw.

Proof. Due to (41) and preceding structure◦zofWzit suffices to show this for t⁰−z. But forf∈k < s >andg∈k < s−z >we get

t⁰−z(f◦ϕ)(g) = (f◦ϕ)(t−z(g)) =ϕ(f t−z(g)) = ϕt−z(tz(f)g) = tz(f)◦t⁰−z(ϕ)

(g) =⇒t⁰−z(f◦ϕ) =tz(f)◦t⁰−z(ϕ).

4 Analytic local algebras

Letkbe the field of real or of complex numbers. Ananalytic local algebra[14, Kap. II] is a nonzero factor algebra

A=k < s > /a, a(k < s >,where k < s >=k < s1,· · ·, sn>, n >0, andm=

n

X

i=1

k < s > si

(45)

are the algebra of convergent power series as discussed in Section 3 and its max- imal ideal. The algebraAis local with the unique maximal idealmA := m/a and the residue fieldk=A/mA. ObviouslyAis also noetherian. The sequence topology onk < s >induces the final topology onA =k < s > /awhich is

4 ANALYTIC LOCAL ALGEBRAS 15

called thesequence topology onA. Eachk-algebra homomorphism between an- alytic localk-algebras is continuous [14, Satz II.1.4 on p.84]. Likek < s >the algebraAis a Hausdorff, locally convex, sequence-complete topological algebra [14, Satz II.1.7 on p.85]. LetA⁰be the dual space of continuousk-linear functions fromAtok. SinceAis a topological algebraA⁰is anA-module via the standard scalar multiplication◦defined by

(f◦ϕ)(g) :=ϕ(f g), f, g∈A, ϕ∈A⁰. (46) We are going to show thatAA⁰is an injective cogenerator. We prepare this result by a more general theorem on injective cogenerators.

LetAbe a commutative noetheriank-algebra with a topology. We call an A- moduleMtopological if it is a locally convex vector space and if the multiplica- tionsA→M, a7→am, m∈M,are continuous [6, Lemma 2.19]. Assume now thatAAis a topological module in this sense (We do not need or require thatAis a topological algebra). Then all f.g. free modulesA^1×qare topologicalA-modules with the product topology and all linear maps◦R:A^1×k→A^1×q, R∈A^k×q, are continuous. We study the categoryAMod^f of all f.g. or noetherian A- modules. Any linear mapf : M1 →M2 between f.g. A-modules can be em- bedded into a commutative diagram with exact rows

A^1×k1 −→^◦R1 A^1×q1 −→^ν1 M1∼= A^1×q1/A^1×k1R1 →0

↓ ◦S ↓ ◦T ↓f

A^1×k2 −→^◦R2 A^1×q2 −→^ν2 M2∼= A^1×q2/A^1×k1R2 →0 (47)

With the final quotient topology the modulesMibecome topologicalA-modules in the above sense such that the epimorphismsνiare strict or, equivalently, open.

Sincef ν1 =ν2(◦T)is continuous andν1 is open we infer thatfis continuous too. For any moduleM ∈AMod^f,M1 :=M2 :=M andf := idM we infer thatidMis a topological isomorphism. This signifies that the topology ofMdoes not depend on the choice of the representationM ∼=A^1×q/A^1×kR. We call this topology thecanonical topologyof the f.g.A-moduleMand use this in the sequel.

Then everyA-epimorphism inAMod^fis open. For everyM∈AMod^fwe also consider the dual spaceM⁰which is again anA-module as in (46). There result two contravariant left exact functors

(−)⁰, HomA(−, A⁰) : AMod^f →AMod (48) and the functorial homomorphism

M :M⁰→HomA(M, A⁰), ϕ7→φ, φ(m)(f) :=ϕ(f m), (49) which forM =Aand thenM =A^1×qis obviously an isomorphism. The exact sequenceA^1×k −→^◦R A^1×q −→^ν M →0induces the commutative diagram with exact rows

0→ M^{0 ν}

0=Hom(ν,k)

−→ A^0q= (A^1×q)⁰ −→^R◦ A^0k

↓M ↓_A1×q ↓_A1×k

0→ HomA(M, A⁰) ^Hom(ν,A

0)

−→ HomA(A^1×q, A⁰) ^Hom(◦R,A

0)

−→ HomA(A^1×k, A⁰) (50)

where the exactness of the first row follows from the openness ofνand where the two right vertical maps are isomorphisms. This implies that alsoM is bijective, i.e.,M is a functorial isomorphism on the categoryAMod^f.

Lemma 4.1. Assume that Ais a noetherian and that AA is a topological A- module as introduced above.

4 ANALYTIC LOCAL ALGEBRAS 16

1. If for each idealaofAthe canonical topology ofacoincides with the in- duced topology fromAthenAA⁰is injective.

2. If in addition to 1. Ais Hausdorff and each maximal ideal is closed then

AA⁰is also a cogenerator.

Proof. 1. Letabe an ideal ofA. It is f.g. and thus carries the canonical topology and the induced topology fromAwhich coincide by assumption. The spacea⁰is the dual space with respect to this topology. By assumptionAis a locally convex space. The Hahn-Banach theorem implies that any function ina⁰can be extended to a function inA⁰or, in other words, the restriction mapA⁰ →a⁰is surjective.

SinceM is a functorial isomorphism also

HomA(inj, A⁰) : HomA(A, A⁰)→HomA(a, A⁰), φ7→φ|a, is surjective. From Baer’s criterion for injectivity we infer thatA⁰is injective.

2. SinceA⁰is injective its cogenerator property can be inferred if each simple module can be embedded intoA⁰. So consider any simple moduleA/mwhere mis a maximal ideal ofA. By assumptionmis closed andAis Hausdorff, hence alsoA/mis a nonzero locally convex Hausdorff space. Again by the Hahn-Banach theorem we conclude06= (A/m)⁰ ∼= HomA(A/m, A⁰). Hence there is nonzero A-linear mapA/m→A⁰which is a monomorphism sinceA/mis simple.

We return to an analytic local algebraAas introduced above. Its quoted prop- erties imply thatAAis a topological module as used in Lemma 4.1. The modules inAMod^f are calledanalytic modulesin [14, Kap.II]. Their canonical topology is again called thesequence topology. The next result shows that the assumptions of Lemma 4.1 are satisfied forA.

Result 4.2. Consider an analytic local algebraAwith its sequence topology and its categoryAMod^f of analytic modules with their canonical or sequence topol- ogy. These modules have the following properties:

1. The algebraAis even a topological algebra and anyAMis a topological module in the sense that the multiplicationA×M→Mis continuous [14, Satz II.1.10].

2. Each submodule of a moduleM ∈AMod^f is closed and especially each moduleM is Hausdorff [14, Satz II.1.10 on p.87].

3. AnyM ∈ AMod^f is locally convex [14, Satz I.8.7 on p.74 and Satz and Bemerkung on p.86].

4. The sequence topology of a submoduleNofM ∈AMod^f coincides with the topology induced from the sequence topology ofM [14, Satz II.2.9 on p.97].

Theorem 4.3. For any analytic local algebraA overk = Rork = Cand its dual spaceA⁰with the canonicalA-structure the moduleAA⁰ is an injective cogenerator.

In particular this holds for the algebrak < s−z >=k < s1−z1,· · ·, sn− zn >, z ∈ kⁿ,of convergent power series around z ∈ kⁿ,and the module Wz:=O(kxⁿ; infexp)e^z•x∼=k < s−z >⁰with its canonical structure, compare Cor. 3.4 and 3.5.

Proof. Result 4.2 implies that the assumptions of Lemma 4.1 are satisfied for an- alytic local algebrasA.