# Noncoherence of a Causal Wiener Algebra  Used in Control Theory

Authors: Amol Sasane
Journal: Abstract and Applied Analysis
 (2008)

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## Abstract

Letℂ≥0:={s∈ℂ∣Re(s)≥0}, and let 𝒲+ denote the ring of all functions
f:ℂ≥0→ℂ
such that  f(s)=fa(s)+∑k=0∞fke−stk(s∈ℂ≥0), where fa∈L1(0,∞),(fk)k≥0∈ℓ1, and
0=t0<t1<t2<⋯ equipped with pointwise operations. (Here ⋅^ denotes the Laplace transform.) It is shown that the ring 𝒲+ is not coherent, answering a question of Alban Quadrat. In fact, we present two principal ideals in the domain 𝒲+ whose intersection is not finitely generated.


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## Body

## 1. Introduction

The aim of this paper is to show that the ring𝒲+ (defined below) is not coherent.We first recall the notion of a coherent
ring.Definition 1.1.
LetR be a commutative ring with identity element 1,
and let Rm=R×⋯×R (m times). Suppose that f=(f1,…,fm)∈Rm.
(1)
An element(g1,…,gm)∈Rm is called a relation onf if (1.1)g1f1+⋯+gmfm=0.(2)
Letf⊥ denote the set of all relations on f∈Rm.
(Then f⊥ is an R-submodule of the R-module Rm.)(3)
The ringR is called coherent if for all m∈ℕ and all f∈Rm, f⊥ is finitely generated, that is, there exists a d∈ℕ and there exist gj∈f⊥, j∈{1,…,d},
such that for all g∈f⊥,
there exist rj∈R, j∈{1,…,d} such  that g=r1g1+⋯+rdgd.An integral domain is coherent if and only if the
intersection of any two finitely generated ideals of the ring is again finitely
generated; see [1,
Theorem 2.3.2, page 45].The coherence of some rings of analytic functions has
been investigated in earlier works. For example, McVoy and Rubel [2] showed that the Hardy algebra H∞(𝔻) is coherent, while the disc algebra A(𝔻) is not. Mortini and von
Renteln proved that the Wiener algebra W+(𝔻) (of all absolutely convergent Taylor series in
the open unit disc) is not coherent [3]. In this article, we will show that the ring 𝒲+ (defined below, and which is useful in control
theory) is not coherent.Notation 1.
Throughout the article, we will use the
following notation:
(1.2)ℂ≥0:={s∈ℂ∣Re(s)≥0}.Definition 1.2.
Let𝒲+ denote the Banach algebra(1.3)𝒲+={f:ℂ≥0→ℂ|f(s)=fa^(s)+∑k=0∞fke−stk(s∈ℂ≥0),fa:(0,∞)→ℂ,fa∈L1(0,∞),∀k≥0,fk∈ℂ,(fk)k≥0∈ℓ1,∀k≥0,tk∈ℝ,0=t0<t1<t2<⋯}equipped with pointwise operations
and the norm(1.4)∥f∥𝒲+:=∥fa∥L1+∥(fk)k≥0∥ℓ1.Here fa^ denotes the Laplace transform of fa,
given by(1.5)fa^(s)=∫0∞e−stfa(t)dt,s∈ℂ≥0.The above algebra arises as a natural class of
transfer functions of stable distributed parameter systems in control theory;
see [4, 5].Our main result is the following.Theorem 1.3.
The ring𝒲+ is not coherent.The relevance of the coherence property in control
theory can be found in [6, 7]. We will prove Theorem 1.3 following the same method
as in the proof of the noncoherence of W+(𝔻) given by Mortini and von Renteln in [3].In Section3, we will give the proof of Theorem 1.3.
But before doing that, in Section 2, we first prove a few technical results
needed in the sequel.


## 2. Preliminaries

We first recall the definition of the Hardy algebraH∞ of the open right half plane.Definition 2.1.
LetH∞ denote the Hardy space of all bounded analytic
functions in the open right half plane equipped with the norm(2.1)∥φ∥∞:=supRe(s)>0|φ(s)|,φ∈H∞.In order to prove our main result (Theorem1.3), we
will use the relation between the convergence in H∞ versus that in 𝒲+.Lemma 2.2.
Iff∈𝒲+,
then f∈H∞ and ∥f∥∞≤∥f∥𝒲+.Proof.
Let(2.2)f(s)=fa^(s)+∑k=0∞fke−stk(s∈ℂ≥0).For s∈ℂ≥0,
we have(2.3)|fa^(s)|=|∫0∞e−stfa(t)dt|≤∫0∞e−Re(s)t|fa(t)|dt≤∫0∞1⋅|fa(t)|dt=∥fa∥L1,and moreover,(2.4)|∑k=0∞fke−stk|≤∑k=0∞|fk|e−Re(s)tk≤∑k=0∞|fk|⋅1=∥(fk)k∥ℓ1.So the result follows.The maximal ideal𝔪0 (defined below) of 𝒲+ will play an important role in the remainder
of this article.Notation.
Let𝔪0 denote the kernel of the complex algebra
homomorphism f↦f(0):𝒲+→ℂ,
that is,
𝔪0:={f∈𝒲+∣f(0)=0}.Then𝔪0 is a maximal ideal of 𝒲+,
and this maximal ideal plays an important role in the proof of our main result
in the next section. We will prove a few technical results about 𝔪0 in this section, which will be used in the
sequel. The following result is analogous to [3, Lemma 1].Lemma 2.3.
LetL≠(0) be an ideal in 𝒲+ contained in the maximal ideal 𝔪0.
If L=L𝔪0,
that is, if every function f∈L can be factorized in a product f=hg of two functions h∈L and g∈𝔪0,
then L cannot be finitely generated.Proof.
Suppose that(2.5)L=(f1,…,fN)≠(0)is a finitely generated ideal in 𝒲+ contained in the maximal ideal 𝔪0.
By our assumption, there are functions hn∈L, gn∈𝔪0 with(2.6)fn=hngn(n=1,…,N).Since hn∈L,
there exist functions qk(n)∈𝒲+ with(2.7)hn=∑k=1Nqk(n)fk(n=1,…,N;k=1,…,N).From this it follows that(2.8)∑n=1N|hn|≤NC∑n=1N|fn|=NC∑n=1N|hngn|inℂ≥0,where C is a constant chosen so that(2.9)∥qk(n)∥∞≤C,∀kandn.(Here ∥⋅∥∞ denotes the supnorm over ℂ≥0.) This implies together with the Cauchy-Schwarz inequality
that(2.10)∑n=1N|hn|2≤(∑n=1N|hn|)2≤N2C2(∑n=1N|hngn|)2≤N2C2(∑n=1N|hn|2)(∑n=1N|gn|2).This inequality holds for all s∈ℂ≥0.
With δ:=1/(N2C2),
we obtain the inequality(2.11)δ≤∑n=1N|gn(s)|2for all points s∈E,
where(2.12)E:={s∈ℂ≥0|∑n=1N|hn(s)|2>0}.Since L≠(0), E is a dense subset of ℂ≥0 (for otherwise, if s0∈ℂ≥0 is such that it has a neighbourhood V in ℂ≥0 where there is no point of E,
then each hn is identically zero in V,
and by the identity theorem for holomorphic functions, each hn is zero; consequently each fn is zero, and so L=(0),
a contradiction). So by continuity, inequality (2.11) holds in ℂ≥0. But this contradicts the fact that each gn vanishes at 0.Remark 2.4.
Lemma2.3 can be proved purely algebraically using
Nakayama's lemma. Indeed, it holds in the following more general algebraic
situation: if I is a nonzero ideal of a commutative domain D contained in a maximal ideal M and I=IM,
then I cannot be finitely generated. However, we have
given an analytic proof in our special case above.Since every maximal ideal is closed,𝔪0 is a commutative Banach subalgebra of 𝒲+,
but obviously without identity element. But there is a substitute, namely the
notion of the approximate identity, which turns out to be useful.Definition 2.5.
LetR be a commutative Banach algebra (without
identity element). One says that R has an approximate identity if
there exists a bounded sequence (en)n of elements en in R such that for any f∈R,(2.13)limn→∞∥enf−f∥=0.We will now prove the following result, which shows
that the maximal ideal𝔪0 in 𝒲+ has an approximate identity.Theorem 2.6.
Let(2.14)en:=ss+1/n,n∈ℕ.Then (en)n∈ℕ is an approximate identity for 𝔪0.The existence of an approximate identity for the
maximal ideal𝔪0 in 𝒲+ is not obvious. In order to prove Theorem 2.6,
we will need the following lemma.Lemma 2.7.
Supposef^∈𝔪0.
Then, for all ϵ>0,
there exists a p^∈𝔪0 such that p has compact support in [0,∞),
and ∥f^−p^∥𝒲+<ϵ.Proof.
Letϵ>0 be given. Suppose that(2.15)f=fa+∑k=0∞fkδ(⋅−tk),where fa∈L1(0,∞), (fk)k≥0∈ℓ1,
and 0=t0<t1<t2<⋯.
Since ∫0∞|fa(t)|dt<∞,
we can choose an M>0 large enough such that(2.16)∫M∞|fa(t)|dt<ϵ4.With pa(t):=fa(t) if t∈[0,M],
and 0 otherwise, we have that pa∈L1(0,∞) is compactly supported and(2.17)∥pa−fa∥L1<ϵ4.Furthermore, select N∈ℕ such that(2.18)∑k>N|fk|<ϵ4.Now let T∈(0,∞) be any number satisfying tN<T<tN+1,
and define(2.19)fT:=−(∫0∞pa(t)dt+∑0≤k≤Nfk).Set(2.20)p:=pa+∑0≤k≤Nfkδ(⋅−tk)+fTδ(⋅−T).Then p^∈𝒲+ and(2.21)p^(0)=∫0∞p(t)dt=∫0∞pa(t)dt+∑0≤k≤Nfk+fT=0.So p^∈𝔪0.
Clearly p has compact support contained in [0,∞).
We have(2.22)|fT|=|∫0∞pa(t)dt+∑0≤k≤Nfk|=|∫0∞fa(t)dt+∑k=0∞fk+∫0∞(pa(t)−fa(t))dt−∑k>Nfk|≤|∫0∞f(t)dt|+∥pa−fa∥L1+∑k>N|fk|=|f^(0)|+∥pa−fa∥L1+∑k>N|fk|<0+ϵ4+ϵ4=ϵ2.Thus(2.23)∥f^−p^∥𝒲+=∥fa−pa∥L1+∑k>N|fk|+|fT|<ϵ4+ϵ4+ϵ2=ϵ.This completes the proof.We are now ready to prove the existence of an
approximate identity for the maximal ideal𝔪0 in 𝒲+.Proof of Theorem2.6.
We have(2.24)en=ss+1/n=s+1/n−1/ns+1/n=1−1n1s+1/n=1+−1ne−t/n^.Thus for
an n∈ℕ,(2.25)∥en∥𝒲+=∥−1ne−t/n∥L1+|1|=1+1=2.Given f^∈𝒲+,
and ϵ>0 arbitrarily small, in view of Lemma 2.7, we
can find a p^∈𝔪0 such that p has compact support and ∥f^−p^∥𝒲+<ϵ.
Then(2.26)∥enf^−f^∥𝒲+≤∥enp^−p^∥𝒲++∥en∥𝒲+∥f^−p^∥𝒲++∥f^−p^∥𝒲+.So it is enough to prove that(2.27)limn→∞∥enp^−p^∥𝒲+=0for all p^∈𝔪0 such that p has compact support in [0,∞).
We do this below.
We have(2.28)enp^−p^=s+1/n−1/ns+1/np^−p^=−1n1s+1/np^=−1n(e−t/n∗p^).Let C denote the convolution e−t/n∗p:(2.29)C(t):=∫0te−(t−τ)/np(τ)dτ.We note that C∈L1(0,∞),
since L1(0,∞) is an ideal in 𝒲+.
Let T>0 be such that(2.30)supp(p)⊂[0,T].We have(2.31)∥enp^−p^∥𝒲+=1n∥C∥L1=1n∫0∞|C(t)|dt=1n∫0T|C(t)|dt︸(I)+1n∫T∞|C(t)|dt︸(II).We estimate (I) as follows:(2.32)(I)=1n∫0T|C(t)|dt=1n∫0T|∫0te−(t−τ)/np(τ)dτ|dt≤1n∫0T∫0te−(t−τ)/n|p(τ)|dτdt≤1n(∫0T∫0t1⋅|p(τ)|dτdt)︸(III).Since the integral (III) does not depend on n,
we obtain that(2.33)limn→∞1n∫0T|C(t)|dt=0.Furthermore,(2.34)(II)=1n∫T∞|C(t)|dt=1n∫T∞e−t/n|∫0teτ/np(τ)dτ|dt=1n∫T∞e−t/n|∫0∞eτ/np(τ)dτ|dt(sincesupp(p)⊂[0,T])=1n∫T∞e−t/n|p^(−1n)|dt.Since p has compact support in [0,T], p^ is an entire function by the Payley-Wiener
theorem (see, e.g., [8, Theorem 7.2.3, page 122]). Consequently,(2.35)(II)=1n∫T∞e−t/n|p^(−1n)|dt=e−T/n|p^(−1n)|→n→∞1⋅|p^(0)|=1⋅0=0.This completes the proof.We will also need the following lemma, which is
basically a repetition of key steps from Browder's proof of Cohen's
factorization theorem; see [9, Theorem 1.6.5, page 74]. We will need this version
since in our application in the proof of Theorem 1.3, we are not able to use
Cohen's factorization theorem directly.Lemma 2.8.
Letf1,f2∈𝔪0,
and δ>0.
Let U(𝒲+) denote the set of all invertible elements in 𝒲+.
Then there exists a sequence (gn)n∈ℕ in 𝒲+ such that
(1)
for alln∈ℕ, gn∈U(𝒲+);(2)
(gn)n∈ℕ is convergent in 𝒲+ to a limit g∈𝔪0;(3)
for alln∈ℕ, ∥gn−1fi−gn+1−1fi∥𝒲+≤δ/2n, i=1,2.Proof.
We will first prove two general results in steps (A) and (B), which
we will use in the rest of the proof.
(A) Lete∈𝔪0 and ∥e∥𝒲+≤K,
where K>1.
Then 1−c+ce∈U(𝒲+),
where c is a number chosen such that(2.36)0<c<14K<14.Indeed,(2.37)∥cc−1e∥𝒲+<1/(4K)3/4⋅K=13<1,and so(2.38)(1−c+ce)−1=11−c∑k=0∞(cc−1)kek.
(B) Furthermore, under the same assumptions and
notation as in (A) above, we now show that if∥eF−F∥𝒲+ is small for some F,
then so is ∥EF−F∥𝒲+,
where E:=(1−c+ce)−1.
Since(2.39)1=11−c∑k=0∞(cc−1)k,we have(2.40)∥EF−F∥𝒲+=∥11−c∑k=0∞(cc−1)k(ekF−F)∥𝒲+≤11−c∑k=0∞(c1−c)k∥ekF−F∥𝒲+.But(2.41)∥ekF−F∥𝒲+=∥∑j=0k−1(ej+1F−ejF)∥𝒲+≤∑j=0k−1∥ej∥𝒲+∥eF−F∥𝒲+≤∥eF−F∥𝒲+∑j=0k−1∥e∥𝒲+j<∥eF−F∥𝒲+KkK−1.Hence(2.42)∥EF−F∥𝒲+<∥eF−F∥𝒲+11−c∑k=0∞1K−1(14(1−c))k<2K−1∥eF−F∥𝒲+.This estimate will be used in
constructing the sequence of gn's.
Let(en)n∈ℕ denote the approximate identity for 𝔪0 from Theorem 2.6. Let K>1 be such that ∥en∥𝒲+≤K for all n∈ℕ.
Choose c such that(2.43)0<c<14K<14.
We will inductively define a sequence(emk)k∈ℕ with terms from the approximate identity for 𝔪0 such that if(2.44)gn:=c∑k=1n(1−c)k−1emk+(1−c)n,then we have ∥fi−g1−1fi∥𝒲+<δ/2, i=1,2,
and
(P1)
for alln∈ℕ, gn∈U(𝒲+),(P2)
for alln∈ℕ, ∥gn−1fi−gn+1−1fi∥𝒲+<δ/2n, i=1,2.
Since (en)n∈ℕ is an approximate identity for 𝔪0,
we can choose m1 such that(2.45)∥em1fi−fi∥𝒲+<δ4(K−1),i=1,2.Define g1=cem1+1−c.
So by (A), g1∈U(𝒲+) and using the calculation in (B), we see that(2.46)∥fi−g1−1fi∥𝒲+<δ2,i=1,2.Suppose that em1,…,emn have been constructed, so that gn defined by (2.44) satisfies (P1) and (P2). We
assert that if we choose emn+1 such that(2.47)∥emn+1fi−fi∥𝒲+(i=1,2),∥emn+1emk−emk∥𝒲+(1≤k≤n)are sufficiently small, then gn+1 defined by (2.44) satisfies (P1) and (P2),
completing the induction step.
Indeed, ifE:=(1−c+cemn+1)−1,
we have(2.48)gn=E−1c∑k=1n(1−c)k−1Eemk+(1−c)n,gn+1=E−1(c∑k=1n(1−c)k−1Eemk+(1−c)n).Let Gn be defined by(2.49)Gn=c∑k=1n(1−c)k−1Eemk+(1−c)n.Then we have(2.50)∥Gn−gn∥𝒲+<c∑k=1n(1−c)k−1∥Eemk−emk∥𝒲+<max1≤k≤n∥Eemk−emk∥𝒲+<2K−1max1≤k≤n∥emn+1emk−emk∥𝒲+.Hence Gn∈U(𝒲+) and moreover ∥Gn−1−gn−1∥𝒲+ is small, provided only that ∥emn+1emk−emk∥𝒲+ is small for k=1,…,n.
(Indeed, this is because U(𝒲+) is an open set in 𝒲+.)
Sincegn+1=E−1Gn,
we have then gn+1∈U(𝒲+), gn+1−1=Gn−1E,
and so for i=1,2,(2.51)∥gn+1−1fi−gn−1fi∥𝒲+=∥Gn−1Efi−gn−1fi∥𝒲+≤∥Gn−1Efi−gn−1Efi∥𝒲++∥gn−1Efi−gn−1fi∥𝒲+≤∥Gn−1−gn−1∥𝒲+∥Efi∥𝒲++∥gn−1∥𝒲+∥Efi−fi∥𝒲+.Moreover, recall that by (B), we
know that(2.52)∥Efi−fi∥𝒲+≤2K−1∥emn+1fi−fi∥𝒲+,i=1,2.Thus if ∥emn+1fi−fi∥𝒲+ (i=1,2) and ∥emn+1emk−emk∥𝒲+ (1≤k≤n) are sufficiently small, we will have ∥gn+1−1fi−gn−1fi∥𝒲+ as small as we please. This completes the
induction step.
Since∥emk∥𝒲+≤K, 0<1−c<1,
and 𝒲+ is a Banach algebra, it follows that(2.53)gn→c∑k=1∞(1−c)k−1emk=:g∈𝔪0,and the proof is completed.


## 3. Noncoherence of𝒲+

Proof of Theorem1.3.
We will use the characterization that an integral domain is
coherent if and only if the intersection of any two finitely generated ideals
of the ring is again finitely generated; see [1, Theorem 2.3.2, page 45]. In fact, we present two
finitely generated ideals I and J such that I∩J is  not finitely generated.
Letp,S be given by(3.1)p=(1−e−s)3,S=e−(1+e−s)/(1−e−s).Clearly we have p∈𝔪0.
It is known (see, e.g., [3, Remark after Theorem 1,
page 224]) that(3.2)(1−z)3e−(1+z)/(1−z)∈W+(𝔻):={f(z)=∑n=0∞anzn(z∈𝔻¯)|∑n=0∞|an|<∞}.Here 𝔻¯:={z∈ℂ∣|z|≤1}.
So if an's are defined via(3.3)(1−z)3e−(1+z)/(1−z)=a0+a1z+a2z2+a3z3+⋯,z∈𝔻,then we have(3.4)∑k=0∞|ak|<∞.If Re(s)>0,
then e−s∈𝔻,
and so from (3.3), we have(3.5)pS=a0+a1e−s+a2e−2s+a3e−3s+⋯,Re(s)>0.Since ∑k=0∞|ak|<∞,
the right-hand side in (3.5) belongs to 𝒲+.
So pS∈𝒲+.
We define the idealsI=(p) and J=(pS) of 𝒲+.
Let(3.6)K:={pSf∣f∈𝒲+andSf∈𝒲+}.We claim that K=I∩J.
Trivially K⊂I∩J.
To prove the reverse inclusion, let g∈I∩J.
Then there exist two functions f and h in 𝒲+ such that g=ph=pSf.
Since p≠0 and 𝒲+ is an integral domain, we obtain that Sf=h∈𝒲+.
So g∈K.
LetL denote the ideal(3.7)L:={f∈𝒲+∣Sf∈𝒲+}.Then K:=pSL.
Since S has a singularity at s=0,
it follows that L⊂𝔪0.
We will show that L=L𝔪0.
Let f∈L.
We would like to factor f=hg with h∈L and g∈𝔪0.
Applying Lemma 2.8 with f1:=f∈𝔪0 and f2:=Sf∈𝔪0,
for any δ>0,
there exists a sequence (gn)n∈ℕ in 𝒲+ such that
(1)
for alln∈ℕ, gn∈U(𝒲+);(2)
(gn)n∈ℕ is convergent in 𝒲+ to a limit g∈𝔪0;(3)
for alln∈ℕ,(3.8)∥gn−1f−gn+1−1f∥𝒲+≤δ2n,∥gn−1Sf−gn+1−1Sf∥𝒲+≤δ2n.
Put(3.9)hn:=gn−1f,Hn:=gn−1Sf.Then hn∈𝔪0.
Also Hn∈𝔪0,
since |S| is bounded by 1 on Re(s)>0 and f(0)=0.
The estimates above imply that (hn)n∈ℕ and (Hn)n∈ℕ are Cauchy sequences in 𝒲+.
Since 𝔪0 is closed, they converge to elements h and H,
respectively, in 𝔪0,
that is, hn=gn−1f→h and Hn=gn−1Sf=Shn→H.
Since convergence in 𝒲+ implies convergence in H∞ (Lemma 2.2), it follows that(3.10)hn→H∞h(sincehn→𝒲+h),Shn→H∞Sh(sincehn→H∞h,S∈H∞),Shn→H∞H(sinceHn→𝒲+H)and so by the uniqueness of the
limit of the sequence (Shn)n∈ℕ in H∞,
we have Sh=H.
Also, in 𝒲+-norm we have(3.11)f=limn→∞hngn=hg,since multiplication is
continuous in the Banach algebra 𝒲+.
Since h and Sh=H belong to 𝔪0⊂𝒲+,
we see that h∈L.
Moreover, as g∈𝔪0,
we have got the desired factorization and L=L𝔪0.
ButL≠(0),
since p∈L.
By Lemma 2.3, it follows that L cannot be finitely generated. Therefore, pSL=I∩J is not finitely generated.Remark 3.1.
The idealL in the above proof can be interpreted as an ideal of denominators; see [10, page 396]. Using the fact
that pS∈𝒲+,
we have S∈Q(𝒲+),
where Q(𝒲+) denotes the field of fractions of 𝒲+.
We can then consider the fractional
idealM:=𝒲++𝒲+S of 𝒲+ (see [11, page 19]) and the ideal of denominatorsL of S,
namely L=𝒲+:M={d∈𝒲+∣dS∈𝒲+}.
Based on the results in [12, Theorem 3, Example 3], it
follows that S∈Q(𝒲+) does not admit a weak coprime factorization,
since L is not a principal ideal of 𝒲+.
In particular, S does not admit a coprime factorization, that
is, there do not exist d,x,y,n∈𝒲+ such that d≠0, S=n/d, and dx−ny=1.
Moreover, S is not
internally stabilizable, since otherwise L would be generated by two elements. Finally,
the fact that L is not finitely generated implies that 𝒲+ is not a greatest common divisor domain: indeed,
were it the case that 𝒲+ is a greatest common divisor domain, then by
[12, Corollary 3],
every element in Q(𝒲+) would admit a weak coprime factorization.



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*Source: 459310-2008-08-04.xml*