On the Bezout equation in the ring of periodic distributions

A corona type theorem is given for the ring R of periodic distributions in R^d in terms of the sequence of Fourier coefficients of these distributions, which have at most polynomial growth. It is also shown that the Bass stable rank and the topological stable rank of R are both equal to 1.


Introduction
The aim of this short note is to study some algebraic and topological questions associated with the "Bézout equation" where b i , a i (1 ≤ i ≤ N ) are elements of the commutative unital topological ring (D ′ A (R d ), +, * , T D ′ A (R d ) ), defined below, and e denotes the identity element (which will be the locally finite sum of Dirac distributions placed at a lattice formed by the periods in A, as explained below). The Bézout equation in rings of distributions arises in problems of robust filtering, image processing, etc, see for example [2].

The spaces D ′
A (R d ) and s ′ (Z d ). For background on periodic distributions and its Fourier series theory, we refer the reader to [4,Chapter 16] and [10, p.527-529].
Consider the space s ′ (Z d ) of all complex valued maps on Z d of at most polynomial growth, that is, where · denotes the 1-norm in R d . Then this is a unital commutative ring (s ′ (Z d ), +, ·) with pointwise operations + and ·, and the unit element being the constant function n → 1 (n ∈ Z d ).
We now equip it with a topology T s ′ (Z d ) , as follows. First, consider the locally convex topological vector space s(Z d ) of rapidly decreasing sequences: with pointwise operations, and with the topology given by the family of seminorms (That is, the topology on s(Z d ) is the weakest one making all the seminorms continuous.) Note that there is a natural duality between s ′ (Z d ) and s(Z d ), namely a bilinear form on Equip s ′ (Z d ) by its natural weak- * topology T s ′ (Z d ) as a dual of s(Z d ). This topology T s ′ (Z d ) can be described in terms of convergence of nets as follows: a net ( Then s ′ (Z d ) equipped with pointwise operations, and the above topology T s ′ (Z d ) , is a topological ring. Moreover, this is isomorphic as a topological ring to D ′ , where the latter is equipped with pointwise addition and multiplication taken as convolution, and its natural dual topology to be the set of all distributions T that satisfy S a k (T ) = T (k = 1, · · · , d).
It can be seen that for some scalars α v (T ) ∈ C, and where A is the matrix with its rows equal to the transposes of the column vectors a 1 , · · · , a d : Also, in the above, δ v denotes the usual Dirac measure with support in v: Then the Fourier coefficients α v (T ) give rise to an element in s ′ (Z d ), and vice versa, every element in s ′ (Z d ) is the set of Fourier coefficients of some periodic distribution. In this manner the topological ring In the sequel we will mostly just consider the topological ring while stating our results and demonstrating proofs, with the tacit understanding that analogous results also hold for (D ′ A (R d ), +, * , T D ′ A (R d ) ). 1.2. Main results. We will prove three results (Theorem 1.1, 1.4 and 1.8), listed below: Then the following are equivalent: (2) There exists a δ > 0 and a K ∈ N such that In light of the isomorphism between D ′ A (R d ) and s ′ (Z d ), this result can be viewed as an analogue of a result for the solvability of the Bézout equation in the ring E ′ (R d ) of compactly supported distributions recalled below (see for instance [9, Corollary 3.1]): Then the following are equivalent: (1) There exist g 1 , · · · , g N ∈ E ′ (R d ) such that (2) There are positive constants δ, K, M such that for all z ∈ C d , Our second main result is the determination of the Bass stable rank of s ′ (Z d ). This notion of stable rank was introduced by Hyman Bass in order to facilitate some computations in algebraic K-theory where it plays a role analogous to dimension; see [1], Definition 1.3 (Bass stable rank). Let R be a commutative unital ring with identity element 1. We assume that 1 = 0, that is R is not the trivial ring {0}.
Here is the reason for taking the smallest such number N : if every (N + 1)tuple is reducible, then also every (N + k)-tuple, k ≥ 1, is reducible, see [1]. With this terminology, our second main result is the following: As an immediate consequence of this result, we also have that s ′ (Z d ) is a Hermite ring, as elaborated below. Definition 1.5 (Hermite ring). A commutative unital ring R is called a Hermite ring if for every N ∈ N, and every (a 1 , · · · , a N ) ∈ U N (R), there exists a N × N matrix A ∈ R N ×N , which is invertible as an element of R N ×N , and such that its first column entries A i1 coincide with a i : Proof. This follows from the known result that commutative unital rings having stable rank at most 2 are Hermite (see for example [7,Theorem 20.13, p. 301]).
Finally, we will also determine the topological stable rank of the topological ring s ′ (Z d ). Marc Rieffel, in the seminal paper [8], introduced a notion of topological stable rank, analogous to the concept of Bass stable rank, for C * -algebras. Motivated by his definition, one can more generally consider topological rings instead of only C * -algebras, and consider the following generalization of his notion of topological stable rank. Definition 1.7 (Topological stable rank). Let R be a commutative unital topological ring with topology T . The topological stable rank, tsrR, of (R, T ) is the least integer N for which U N (R) is dense in R N . If no such N exists, then tsr R := ∞.
Our third and final main result is: The organization of the paper is as follows: In Sections 2, 3, 4, we give the proofs of Theorems 1.1, 1.4, 1.8, respectively.

Corona type theorem
Proof of Theorem 1.1.
(2)⇒(1): From the corona type condition, it follows that n → 1 |a 1 (n)| + · · · + |a N (n)| is an element of s ′ (Z d ). Next, for i = 1, · · · , N and n ∈ Z d , define where Arg denotes the principal argument of a complex number, living in (−π, π], and is taken as 0 for the complex number 0. Then we see that b 1 , · · · , b N ∈ s ′ (Z d ) and b 1 a 1 + · · · + b N a N = 1 This completes the proof.

Bass stable rank of s ′ (Z d )
Proof of Theorem 1.4. Suppose that a 1 , a 2 , b 1 , b 2 are elements of s ′ (Z d ) such that b 1 a 1 + b 2 a 2 = 1. Define u 1 ∈ s ′ (Z d ) by u 1 (n) = 1 + |a 1 (n)|, n ∈ Z d . Then by using Theorem 1.1, we can see that u 1 is invertible in s ′ (Z d ). Set A 1 := a 1 u −1 1 , that is, We note that Define B 1 by where the ǫ > 0 will be determined later. Then clearly B 1 ∈ s ′ (Z d ), and in fact is invertible in s ′ (Z d ) because it is bounded below by ǫ. Moreover, B 1 "approximates" B 1 pointwise: We also note that Since for all n ∈ Z d we have that for the choice ǫ = 1/4, it follows by Theorem 1.1, that 1 + ( . As the last expression on the right hand side is a product of invertibles from s ′ (Z d ), it follows that a 1 + ha 2 is an invertible in element in s ′ (Z d ). This shows that the Bass stable rank of s ′ (Z d ) is 1.
Note that once we have (P1), the properties (P2) and (P2 ′ ) are equivalent, thanks to the identity ze iArg(z) = |z| for any complex number z, where Arg denotes the principal argument of a complex number, living in (−π, π], and we set Arg(0) := 0. Claim: bsr(R) = 1. We carry out the same proof as above, and note the following key changes in the arguments: (1) u 1 ∈ R because |a 1 | ∈ R, using (P2) or (P2 ′ ). Moreover (P3) shows also that u 1 is invertible in R. Also, then A 1 ∈ R and B 1 ∈ R.
(2) Let's show that B 1 ∈ R. To this end, write additively B 1 = x+B 1 ·y, where As x, y are bounded, they belong to R by (P1). Thus, since R is a ring, so does B 1 = x + B 1 · y. (3) We prove that B 1 is invertible in R. To this end, factor multiplicatively B 1 = (1 + | B 1 |) · C 1 , where The first factor, 1 + | B 1 |, belongs to R, using (P2) and the fact that is increasing, and so The rightmost inequality shows that C 1 ∈ R (as it is bounded), while the leftmost inequality then gives the invertibility of C 1 in R (using (P3)). The rest of the proof is the same, mutatis mutandis, as the proof of Theorem 1.4.

4.
Topological stable rank of s ′ (Z d ) Proof of Theorem 1.8. Let a ∈ s ′ (Z d ). We will construct a net (a ǫ ) ǫ>0 with index set as the directed set (0, ∞) and the usual order of real numbers, of invertible elements a ǫ in s ′ (Z d ) such that (a ǫ ) ǫ>0 converges to a. Define for ǫ > 0 and n ∈ Z d Then |a ǫ (n)| ≥ ǫ for all n ∈ Z d , and so a ǫ is invertible in s ′ (Z d ) for all ǫ > 0. Moreover, for every b ∈ s(Z d ), we have Then each a ǫ , ǫ > 0, is not invertible in s ′ (Z d ) because it is easily seen that But the net (a ǫ ) ǫ>0 converges to the invertible element (1) n∈Z d : indeed for every b ∈ s(Z d ), we have for some constant K (and K ′ ) depending on b. In the above, we have used the inequality 0 < 1 − e −x < x for x > 0, which follows from the Mean Value Theorem.

An open problem
While we have computed the stable ranks of s ′ (Z d ) ≃ D ′ A (R d ), the stable ranks of E ′ (R d ) do not seem to be known. Via Fourier transformation, and by the Payley-Wiener-Schwartz Theorem [6, Theorem 7.3.1, p.181], (E ′ (R d ), +, * ) is isomorphic to the ring E exp of entire functions of exponential type: equipped with pointwise operations. Is bsr(E exp (C d )) < ∞? If so what is it? The Bass stable rank of E exp (C) can be shown to be at least 2 (see below).
First, we note that every unit in E exp (C) has the form u(z) = e a+bz for some constants a, b. This follows directly from the Weierstrass factorisation of such functions, but here is a direct proof: writing u(z) = e h(z) , where h denotes an entire function, we conclude that |Re(h(z))| ≤ A + B|z|, but then Schwarz's formula for the disc |ζ| = 2r and |z| = r gives h(z) = 1 2πi |ζ|=2r ζ + z ζ − z Re(h(ζ)) dζ ζ + iIm(h(0)) and so, |h(z) − iIm(h(0))| ≤ 3r r sup |ζ|=2r |Re(h(ζ))| ≤ 3(A + 2rB). Now Cauchy's estimate for the coefficients of a Taylor series shows that h must be a polynomial of degree 1, that is h(z) = a + bz for some a, b.