Annihilator methods for spectral synthesis on locally compact Abelian groups

Abstract

Spectral analysis and spectral synthesis study translation invariant linear function spaces on Abelian groups. Basic function classes for this study are the exponential monomials. These function classes have been investigated on discrete Abelian groups successfully using the annihilator method. In this paper we extend this technique to non-discrete locally compact Abelian groups.

1 Introduction

Spectral analysis and spectral synthesis deal with the description of different varieties. One of the fundamental theorems on this field is due to Laurent Schwartz [1]. Recently several new results on spectral analysis and spectral synthesis have been found on discrete Abelian groups (see [2, 3]). In [4] the author formulated problems and proved results concerning spectral synthesis on locally compact Abelian groups. In [5] we made an attempt to formulate and study the basic problems of spectral analysis and spectral synthesis in the non-commutative non-discrete setting. In a former paper [6] we introduced a method of studying spectral synthesis problems using annihilators of varieties on discrete Abelian groups (see also [7]). Here we extend this method to non-discrete locally compact Abelian groups.

In this paper $\mathbb {C}$ denotes the set of complex numbers. For a locally compact Abelian group G we denote by $\mathcal C(G)$ the locally convex topological vector space of all continuous complex valued functions defined on G, equipped with the pointwise operations and with the topology of uniform convergence on compact sets. For each function f in $\mathcal C(G)$ we define by , whenever x is in G. For a subset H in $\mathcal C(G)$ we denote by the set of all functions with f in H. By a ring we always mean a commutative ring with unit.

It is known that the dual of $\mathcal C(G)$ can be identified with the space $\mathcal M_c(G)$ of all compactly supported complex Borel measures on G which is equipped with the pointwise operations and with the weak*-topology. The pairing between $\mathcal C(G)$ and $\mathcal M_c(G)$ is given by the formula

$$\begin{aligned} \langle \mu ,f\rangle =\int f\,d\mu . \end{aligned}$$

The following theorem, describing the dual of $\mathcal M_c(G)$ is fundamental. The proof can be found in [8], 17.6, p. 155 (see also [7], Theorem 3.43, p. 48).

Theorem 1

Let G be a locally compact Abelian group. For every weak*-continuous linear functional $F:\mathcal M_c(G)\rightarrow \mathbb {C}$ there exists a continuous function $f:G\rightarrow \mathbb {C}$ such that $F(\mu )=\mu (f)$ for each $\mu$ in $\mathcal M_c(G)$.

In fact, the function f in this theorem is uniquely determined by F, as it is clear from the following result.

Theorem 2

Let G be a locally compact Abelian group. The finitely supported complex measures form a weak*-closed subspace in $\mathcal M_c(G)$.

Proof

Let X be the weak*-closure of the linear space of all finitely supported complex measures in $\mathcal M_c(G)$. Assuming that X is a proper subspace, by the Hahn–Banach Theorem, there exists a nonzero weak*-continuous linear functional $F:\mathcal M_c\rightarrow \mathbb {C}$ vanishing on X. In particular, $F(\delta _y)=0$ for each y in G. However, by the previous theorem, there exists a continuous function $f:G\rightarrow \mathbb {C}$ such that $f(y)=\delta _y(f)=F(\delta _y)=0$ for each y in G, which is a contradiction, and our theorem is proved. $\square$

For each $\mu$ in $\mathcal M_c(G)$ we define by the equation whenever f is in $\mathcal C(G)$. For every subset K in $\mathcal M_c(G)$ the symbol denotes the set of all measures of the form with $\mu$ in K. The orthogonal complement of the subset H in $\mathcal C(G)$ is the set of all measures $\mu$ in $\mathcal M_c(G)$ satisfying $\mu (f)=0$ for each f in H, and it is denoted by $H^{\perp }$.The dual concept is the orthogonal complement of a set K in $\mathcal M_c(G)$ of all functions f in $\mathcal C(G)$ satisfying $\mu (f)=0$ for every $\mu$ in K, and it is denoted by $K^{\perp }$. Obviously, $H^{\perp }$, resp. $K^{\perp }$ is a closed subspace in $\mathcal M_c(G)$, resp. in $\mathcal C(G)$.

Convolution on $\mathcal M_c(G)$ is defined by

$$\begin{aligned} \int f\,d(\mu *\nu )=\int f(x+y)\,d\mu (x)\,d\nu (y) \end{aligned}$$

for each $\mu ,\nu$ in $\mathcal M_c(G)$ and x in G. Convolution converts the linear space $\mathcal M_c(G)$ into a commutative topological algebra with unit $\delta _0$, 0 being the zero in G.

We also define convolution of measures in $\mathcal M_c(G)$ with arbitrary functions in $\mathcal C(G)$ by the similar formula

$$\begin{aligned} f*\mu (x)=\int f(x-y)\,d\mu (y) \end{aligned}$$

for each $\mu$ in $\mathcal M_c(G)$, f in $\mathcal C(G)$ and x in G. The linear operators $f\mapsto \mu *f$ on $\mathcal C(G)$ are called convolution operators. It is easy to see that equipped with the action $f\mapsto f*\mu$ the space $\mathcal C(G)$ is a topological module over $\mathcal M_c(G)$. For each subset K in $\mathcal M_c(G)$ and H in $\mathcal C(G)$ we use the notation

$$\begin{aligned} K H=\{f*\mu :\, \mu \in K, f\in H\}. \end{aligned}$$

For each subset H in $\mathcal C(G)$ the annihilator of H in $\mathcal M_c(G)$ is the set

$$\begin{aligned} \mathrm {Ann\,}H=\{\mu :\, f*\mu =0\quad \text {for each}\quad f\in H\}. \end{aligned}$$

We also define the dual concept: for every subset K in $\mathcal M_c(G)$ the annihilator of K in $\mathcal C(G)$ is the set

$$\begin{aligned} \mathrm {Ann\,}K=\{f:\, f*\mu =0\quad \text {for each}\quad \mu \in K\}. \end{aligned}$$

Translation with the element y in G is the operator mapping the function f in $\mathcal C(G)$ onto its translate $\tau _yf$ defined by $\tau _yf(x)=f(x+y)$ for each x in G. Clearly, $\tau _y$ is a convolution operator, namely, it is the convolution with the measure $\delta _{-y}$: we have $\tau _y f=f*\delta _{-y}$. A subset of $\mathcal C(G)$ is called translation invariant, if it contains all translates of its elements. A closed linear subspace of $\mathcal C(G)$ is called a variety on G, if it is translation invariant. Obviously, varieties are exactly the closed submodules in $\mathcal C(G)$. As it is easy to see, is a variety whenever V is a variety. For each function f the smallest variety containing f is called the variety generated by f, or simply the variety of f, and it is denoted by $\tau (f)$, which is obviously the intersection of all varieties containing f.

Theorem 3

For each variety V in $\mathcal C(G)$ its annihilator $\mathrm {Ann\,}V$ is a closed ideal in $\mathcal M_c(G)$, and . Similarly, for each ideal I in $\mathcal M_c(G)$ its annihilator $\mathrm {Ann\,}I$ is a variety in $\mathcal C(G)$, and .

Proof

Clearly, $\mathrm {Ann\,}V$ is a closed subspace in $\mathcal M_c(G)$. For each $\mu$ in $\mathrm {Ann\,}V$, $\nu$ in $\mathcal M_c(G)$ and f in V we have

$$\begin{aligned} (\nu *\mu )*f(x)= & {} \int \,f(x-y)\,d(\nu *\mu )(y)\\= & {} \int \,\int \,f(x-u-v)\,d\mu (v) d\nu (u)=\int \,(f*\mu )(x-u)\,d\nu (u)=0, \end{aligned}$$

as $f*\mu =0$. This means $\nu *\mu$ is in $\mathrm {Ann\,}V$, and $\mathrm {Ann\,}V$ is a closed ideal in $\mathcal M_c(G)$. On the other hand, we have

hence $\mu$ is in . Conversely, if $\nu$ is in , then for each f in V we have

as is in . It follows that $\mu$ is in $\mathrm {Ann\,}V$.

For the dual statement it is clear that $\mathrm {Ann\,}I$ is a closed subspace in $\mathcal C(G)$. Moreover, if f is in $\mathrm {Ann\,}I$, y is in G and $\mu$ is in I, then $\delta _{-y}*\mu$ is in I, hence we have

$$\begin{aligned} \tau _y f*\mu =(f*\delta _{-y})*\mu =f*(\delta _{-y}*\mu )=0, \end{aligned}$$

and we infer that $\tau _y f$ is in $\mathrm {Ann\,}I$, hence $\mathrm {Ann\,}I$ is a variety. On the other hand, we have

as f annihilates $\mu$. This means that . Conversely, if g is in $\mathcal C(G)$ with the property that for each $\mu$ in I, then for each $\mu$ in I. As I is an ideal, this implies for each x in G, hence

that is g is in $\mathrm {Ann\,}I$, which proves , and the proof is complete. $\square$

Theorem 4

For each variety $V\subseteq W$ in $\mathcal C(G)$ we have $\mathrm {Ann\,}V\supseteq \mathrm {Ann\,}W$ and for each ideal $I\subseteq J$ in $\mathcal M_c(G)$ we have $\mathrm {Ann\,}I\supseteq \mathrm {Ann\,}J$. In addition, we have $\mathrm {Ann\,}(\mathrm {Ann\,}V)=V$ and $\mathrm {Ann\,}(\mathrm {Ann\,}I)\supseteq I$. In particular, $V\ne W$ implies $\mathrm {Ann\,}V\ne \mathrm {Ann\,}W$.

Proof

Let $V\subseteq W$ be varieties in $\mathcal C(G)$ and let $I\subseteq J$ be ideals in $\mathcal M_c(G)$. For every $\mu$ in $\mathrm {Ann\,}W$ and for each f in V we have that f is in W, hence $f*\mu =0$. This proves that $\mu$ is in $\mathrm {Ann\,}V$, and $\mathrm {Ann\,}V\supseteq \mathrm {Ann\,}W$. Similarly, if f is in $\mathrm {Ann\,}J$ and $\mu$ is in I, then $\mu$ is in J, hence $f*\mu =0$, which proves that f is in $\mathrm {Ann\,}I$, and $\mathrm {Ann\,}I\supseteq \mathrm {Ann\,}J$.

Assume that f is in V and $\mu$ is in $\mathrm {Ann\,}V$, then, by definition, $f*\mu =0$, hence f is in $\mathrm {Ann\,}(\mathrm {Ann\,}V)$, which proves $\mathrm {Ann\,}(\mathrm {Ann\,}V)\supseteq V$. Similarly, we have $\mathrm {Ann\,}(\mathrm {Ann\,}I)\supseteq I$.

Suppose now that $\mathrm {Ann\,}(\mathrm {Ann\,}V)\subsetneq V$. Consequently, there is a function f in $\mathrm {Ann\,}(\mathrm {Ann\,}V)$ such that f is not in V. By the Hahn–Banach Theorem, there is a $\lambda$ in $\mathcal M_c(G)$ such that , and vanishes on V. This means

whenever $\varphi$ is in V. As V is a variety, this implies, by the previous theorem, that $\lambda$ is in $\mathrm {Ann\,}V$, in particular, $f*\lambda =0$, a contradiction. This proves $\mathrm {Ann\,}(\mathrm {Ann\,}V)= V$, which also implies $\mathrm {Ann\,}V\ne \mathrm {Ann\,}W$, whenever $V\ne W$. $\square$

We note that for ideals in $\mathcal M_c(G)$ the equality $\mathrm {Ann\,}(\mathrm {Ann\,}I)=I$ does not hold in general. For this, by Theorem 3, it is enough to show that $I^{\perp \perp }=I$ does not hold, in general. The following example can be found in [3].

Consider $G=\mathbb {R}$ with the usual topology, and let I denote the ideal generated by the measures $\mu _n =\delta _0 -\delta _{1/n}$ for $n=1,2,\dots$. If f is in $I^{\perp }$, then f is periodic mod 1 / n for every n, and thus, by continuity, f must be constant. Therefore $\delta _0 -\delta _{\alpha }$ is in $I^{\perp \perp }$ for each $\alpha$ in $\mathbb {R}$. However, $\delta _0 -\delta _{\alpha }$ is not in I if $\alpha$ is irrational. Indeed, for every positive integer N there is a continuous function f such that f is periodic mod 1 / n for each $n\le N$ in $\mathbb {N}$ but f is not periodic mod $\alpha$. This implies immediately that $\delta _0 -\delta _{\alpha }$ does not belong to the ideal generated by $\mu _n$ for n in $\mathbb {N}$. However, if $\delta _0 -\delta _{\alpha }$ is I, then $\delta _0 -\delta _{\alpha }$ belongs to an ideal generated by finitely many of the measures $\mu _n$, which is not the case.

Nevertheless, the following theorem holds true (see [3]).

Theorem 5

Let G be a discrete Abelian group. Then $\mathrm {Ann\,}(\mathrm {Ann\,}I)=I$ holds for every ideal I in $\mathcal M_c(G)$.

We need the following lemma.

Lemma 1

Let G be a locally compact group, and let I be an ideal in $\mathcal M_c(G)$. Then $\mathrm {Ann\,}\bigl (\mathrm {Ann\,}(\mathrm {Ann\,}I)\bigr )=\mathrm {Ann\,}I$.

Proof

Let $V=\mathrm {Ann\,}I$, then V is a variety on G, hence, by Theorem 4, we have $\mathrm {Ann\,}(\mathrm {Ann\,}V)=V$. It follows $\mathrm {Ann\,}I=V=\mathrm {Ann\,}(\mathrm {Ann\,}V)=\mathrm {Ann\,}\bigl (\mathrm {Ann\,}(\mathrm {Ann\,}I)\bigr )$. $\square$

Now we can prove the following theorem characterizing those ideals in $\mathcal M_c(G)$ which coincide with their second annihilator.

Theorem 6

Let G be a locally compact group, and let I be an ideal in $\mathcal M_c(G)$. Then we have $\mathrm {Ann\,}(\mathrm {Ann\,}I)=I$ if and only if I is closed. Also, we have $I^{\perp \perp }=I$ if and only if I is closed.

Proof

By Theorem 3, the annihilator of each variety is closed, in particular, $J=\mathrm {Ann\,}(\mathrm {Ann\,}I)$, as the annihilator of the variety $\mathrm {Ann\,}I$, is closed, which proves the necessity of our condition.

Conversely, suppose that I is closed, and I is a proper subset of J. By Lemma 1, we have $\mathrm {Ann\,}J=\mathrm {Ann\,}I$. Let $\mu$ be in J such that $\mu$ is not in I. As the space $\mathcal M_c(G)$ with the weak*-topology is locally convex, hence, by the Hahn–Banach Theorem, there is a linear functional $\xi$ in $\mathcal M_c(G)^{*}$, such that vanishes on I and . It is known (see [8], 17.6, p. 155), that every weak*-continuous linear functional on a dual space arises from an element of the original space, that is, there is an f in $\mathcal C(G)$ with $\xi (\nu )=\nu (f)$ for each $\nu$ in $\mathcal M_c(G)$. We infer , and $\mu$ is in J, hence f is not in . On the other hand, for each $\nu$ in I, as vanishes on I, which implies that f is in , a contradiction.

The second statement is a consequence of Theorem 3. Our theorem is proved. $\square$

Corollary 1

Let G be a locally compact Abelian group. Then the mappings $V \leftrightarrow \mathrm {Ann\,}V$ and $V \leftrightarrow V^{\perp }$ set up one-to-one inclusion-reversing correspondences between the varieties in $\mathcal C(G)$ and the closed ideals in $\mathcal M_c(G)$.

By this corollary, closed ideals have special importance. In particular, the following theorem describes a class of closed ideals.

Theorem 7

Let G be a locally compact Abelian group. If V is a finite dimensional variety in $\mathcal C(G)$, then every ideal including $\mathrm {Ann\,}V$, or $V^{\perp }$ is closed in $\mathcal M_c(G)$.

Proof

By Theorem 3, it is enough to proof the statement for $I\supseteq V^{\perp }$. Obviously, $\mathcal M_c(G)/V^{\perp }$ can be identified with the dual $V^{*}$ of V, which is a finite dimensional vector space, hence every subspace of it is closed. In particular, every ideal in $\mathcal M_c(G)/V^{\perp }$ is closed. The natural homomorphism F of $\mathcal M_c(G)$ onto $\mathcal M_c(G)/V^{\perp }$ is continuous and every ideal including $V^{\perp }$ in $\mathcal M_c(G)$ is the inverse image of an ideal in $\mathcal M_c(G)/V^{\perp }$ by F, hence it is closed. $\square$

Theorem 8

Let G be a locally compact group.

• 1. For each family $(V_{\gamma })_{\gamma \in \Gamma }$ of varieties in $\mathcal C(G)$ we have

  $$\begin{aligned} \mathrm {Ann\,}\left( \sum _{\gamma \in \Gamma } V_{\gamma }\right) =\bigcap _{\gamma \in \Gamma } \mathrm {Ann\,}V_{\gamma },\quad \left( \sum _{\gamma \in \Gamma } V_{\gamma }\right) ^{\perp }=\bigcap _{\gamma \in \Gamma } V_{\gamma }^{\perp }. \end{aligned}$$

• 2. For each family $(I_{\gamma })_{\gamma \in \Gamma }$ of ideals in $\mathcal M_c(G)$ we have

  $$\begin{aligned} \mathrm {Ann\,}\left( \sum _{\gamma \in \Gamma } I_{\gamma }\right) =\bigcap _{\gamma \in \Gamma } \mathrm {Ann\,}I_{\gamma },\quad \left( \sum _{\gamma \in \Gamma } I_{\gamma }\right) ^{\perp }=\bigcap _{\gamma \in \Gamma } I_{\gamma }^{\perp }. \end{aligned}$$

We note that here $\sum _{\gamma \in \Gamma } V_{\gamma }$ denotes the topological sum of the family of varieties $(V_{\gamma })_{\gamma \in \Gamma }$, that is, the closure of the union of the sums of finite subfamilies. However, $\sum _{\gamma \in \Gamma } I_{\gamma }$ denotes the algebraic sum of the family of ideals $(I_{\gamma })_{\gamma \in \Gamma }$, that is, the ideal generated by the sums of finite subfamilies.

Proof

If $\mu$ is in $\bigcap _{\gamma \in \Gamma } \mathrm {Ann\,}V_{\gamma }$, then $\mu$ annihilates each of the varieties $V_{\gamma }$, hence it annihilates every finite sum of these varieties, and, by continuity, $\mu$ annihilates the closure of the sums of finite subfamilies. Hence $\mu$ annihilates $\sum _{\gamma \in \Gamma } V_{\gamma }$.

Conversely, if $\mu$ annihilates $\sum _{\gamma \in \Gamma } V_{\gamma }$, then $\mu$ annihilates every subvariety of it, hence it belongs to each $\mathrm {Ann\,}V_{\gamma }$. This proves the first half of the first statement. The second half is the consequence of Theorem 3.

To prove the second statement we take f in $(\sum _{\gamma \in \Gamma } I_{\gamma })^{\perp }$. Then f is in the orthogonal complement of the sum of any finite subfamily of $(I_{\gamma })_{\gamma \in \Gamma }$, in particular, it is in the orthogonal complement of each of these ideals. Hence it belongs to $I_{\gamma }^{\perp }$ for every $\gamma$.

For the reverse inclusion we take an f which is in the orthogonal complement of each ideal $I_{\gamma }$. Then clearly, every measure in the ideal generated by finite sums of these ideals vanishes on f, hence f is in $(\sum _{\gamma \in \Gamma } I_{\gamma })^{\perp }$. This proves the second half of the second statement. The first half is the consequence of Theorem 3. $\square$

2 Exponentials

A basic function class is formed by the joint eigenfunctions of all translation operators, that is, by those nonzero continuous functions $\varphi :G\rightarrow \mathbb {C}$ satisfying

$$\begin{aligned} \tau _y\varphi =m(y)\cdot \varphi \end{aligned}$$

(1)

with some $m:G\rightarrow \mathbb {C}$, that is

$$\begin{aligned} \varphi (x*y)=m(y) \varphi (x) \end{aligned}$$

(2)

for all x, y in G. It follows $\varphi (y)=\varphi (0)\cdot m(y)$ which implies that $\varphi (0)\ne 0$ and, by (2),

$$\begin{aligned} m(x+y)=m(x) m(y) \end{aligned}$$

(3)

for each x, y in G. Nonzero continuous functions $m:G\rightarrow \mathbb {C}$ satisfying (3) for each x, y in G are called exponentials. Clearly, every exponential generates a one dimensional variety, and conversely, every one dimensional variety is generated by an exponential. Sometimes exponentials are called generalized characters.

Using translation one introduces modified differences in the following manner: for each continuous function f in $\mathcal C(G)$ and y in G we let

$$\begin{aligned} \Delta _{f;y}=\delta _{-y}-f(y)\delta _0. \end{aligned}$$

Hence $\Delta _{f;y}$ is an element of $\mathcal M_c(G)$. Products of modified differences will be denoted in the following way: for each f in $\mathcal C(G)$, for every natural number n and for arbitrary $y_1,y_2,\ldots ,y_{n+1}$ in G we let

$$\begin{aligned} \Delta _{f;y_1,y_2,\ldots ,y_{n+1}}=\Pi _{i=1}^{n+1} \bigl (\delta _{-y_i}-f(y_i)\delta _0\bigr ), \end{aligned}$$

where $\Pi$ denotes convolution. In the case $f\equiv 1$ we use the simplified notation $\Delta _y$ for $\Delta _{1;y}$ and we call it difference. Accordingly, we write $\Delta _{y_1,y_2,\ldots ,y_{n+1}}$ for $\Delta _{1;y_1,y_2,\ldots ,y_{n+1}}$ For a given f in $\mathcal C(G)$ the closed ideal generated by all modified differences of the form $\Delta _{f;y}$ with y in G is denoted by $M_f$. We have the following theorem.

Theorem 9

Let G be a locally compact Abelian group and $f:G\rightarrow \mathbb {C}$ a continuous function. The ideal $M_f$ is proper if and only if f is an exponential. In this case $M_f=\mathrm {Ann\,}\tau (f)$ is maximal, and $\mathcal M_c(G)/M_f$ is topologically isomorphic to the complex field.

Proof

As $M_f$ is closed, by Theorem 6, we have $\mathrm {Ann\,}(\mathrm {Ann\,}M_f)=M_f$ and $M_f^{\perp \perp }=M_f$.

Suppose that f is an exponential. Then $f\ne 0$, and

$$\begin{aligned} \Delta _{f;y}*f(x)=f(x+y)-f(y) f(x)=0 \end{aligned}$$

for each x, y in G, hence f is in $\mathrm {Ann\,}M_f$. As $\tau (f)$ consists of all constant multiples of f, we infer that $\tau (f)\subseteq \mathrm {Ann\,}M_f$. Moreover, if $\varphi$ is in $\mathrm {Ann\,}M_f$, then we have

$$\begin{aligned} 0=\Delta _{f;y}*\varphi (x)=\varphi (x+y)-f(y) \varphi (x) \end{aligned}$$

for each x, y in G. It follows $\varphi =\varphi (0)\cdot f$, hence $\varphi$ is in $\tau (f)$. We conclude that $\tau (f)=\mathrm {Ann\,}M_f$, and $M_f=\mathrm {Ann\,}\tau (f)$.

We define the mapping $\Phi _f:\mathcal M_c(G)\rightarrow \mathbb {C}$ by

for each $\mu$ in $\mathcal M_c(G)$. Then $\Phi _f$ is a linear mapping, $\Phi _f(\delta _0)=1$, and for each $\mu ,\nu$ in $\mathcal M_c(G)$ we have

$$\begin{aligned} \Phi _f(\mu *\nu )= & {} \int f(-x-y) d\mu (x)\,d\nu (y)\\= & {} \int f(-x)\,d\mu (x) \int f(-y)\,d\nu (y) =\Phi _f(\mu )\cdot \Phi _f(\nu ), \end{aligned}$$

hence $\Phi _f$ is an algebra homomorphism. Obviously, $\Phi _f$ maps $\mathcal M_c(G)$ onto $\mathbb {C}$, hence it is a multiplicative linear functional. We infer that $\mathrm {Ker\,}\Phi _f$ is a maximal ideal and $\mathcal M_c(G)/\mathrm {Ker\,}\Phi _f$ is isomorphic to the complex field $\mathbb {C}$. For each $\mu$ in $\mathrm {Ker\,}\Phi _f$ we have , hence for each complex number c we have

consequently $\mu$ is in $\mathrm {Ann\,}\tau (f)=M_f$. It follows $\mathrm {Ker\,}\Phi _f\subseteq M_f$, which implies that $M_f$ is a maximal ideal. We also have that $\mathrm {Ker\,}\Phi _f$ is closed, hence $\Phi _f$ is continuous. As $\Phi _f$ is also open, we have that $\mathcal M_c(G)/M_f$ is topologically isomorphic to the complex field.

Finally, if $M_f$ is proper, then $\mathrm {Ann\,}M_f$ is nonzero. Let $\varphi \ne 0$ be a function in $\mathrm {Ann\,}M_f$, then we have

$$\begin{aligned} 0=\Delta _{f;y}*\varphi (x)=\varphi (x+y)-f(y) \varphi (x), \end{aligned}$$

and in the same way like above we conclude that f is an exponential. The theorem is proved. $\square$

Given a ring R we call a maximal ideal M in R an exponential maximal ideal, if the residue ring R / M is isomorphic to the complex field. If R is a topological ring, then we require the isomorphism to be topological. From the above proof it is clear that if G is a locally compact Abelian group, then each exponential maximal ideal in $\mathcal M_c(G)$ is of the form $M_m=\mathrm {Ann\,}\tau (m)$ with some exponential m.

3 Fourier–Laplace transformation

Given the locally compact Abelian group G let $\widetilde{G}$ denote the set of all exponentials on G. Obviously, $\widetilde{G}$ is an Abelian group with respect to pointwise multiplication. We equip $\widetilde{G}$ with the compact-open topology which makes $\widetilde{G}$ a topological Abelian group. For every $\mu$ in $\mathcal M_c(G)$ we define the function $\widehat{\mu }:\widetilde{G}\rightarrow \mathbb {C}$ by

whenever m is in . Obviously, $\widehat{\mu }(m)=m*\mu (0)$. Also we have $\widehat{\mu }(m)=\Phi _m(\mu )$, where $\Phi _m$ is defined in Theorem 9 with $m=f$. The function $\widehat{\mu }$ is called the Fourier–Laplace transform of $\mu$ and the mapping $\mu \mapsto \widehat{\mu }$ is called the Fourier–Laplace transformation.

Theorem 10

Let G be a locally compact Abelian group. Then for each measure $\mu$ in $\mathcal M_c(G)$ its Fourier–Laplace transform $\widehat{\mu }$ is a continuous function on $\widetilde{G}$.

Proof

Let $(m_i)_{i\in I}$ be a generalized sequence in $\widetilde{G}$ converging to the exponential m in $\widetilde{G}$. Then uniformly on the compact set $\mathrm {supp\,}\mu$, hence we have $\widehat{\mu }_i(m)\rightarrow \widehat{\mu }$, which proves that $\widehat{\mu }$ is continuous. $\square$

Theorem 11

Let G be a locally compact Abelian group. The Fourier–Laplace transformation $\mu \rightarrow \widehat{\mu }$ is a continuous injective algebra homomorphism of $\mathcal M_c(G)$ into $\mathcal C(\widetilde{G})$, the latter equipped with the pointwise linear operations and multiplication, and with the topology of pointwise convergence.

Proof

We use the notation

$$\begin{aligned} \mathcal F(\mu )=\widehat{\mu } \end{aligned}$$

for each $\mu$ in $\mathcal M_c(G)$. Obviously, $\mathcal F:\mathcal M_c(G)\rightarrow \mathcal C(\widetilde{G})$ is a linear mapping. Suppose that $(\mu _{\alpha })_{\alpha \in A}$ is a generalized sequence in $\mathcal M_c(G)$ converging to $\mu$ in the weak*-topology. Then for each m in $\widetilde{G}$ we have , that is $\widehat{\mu }_{\alpha }(m)\rightarrow \widehat{\mu }(m)$, which gives the continuity of $\mathcal F$. Finally, for $\mu ,\nu$ in $\mathcal M_c(G)$ and m in $\widetilde{G}$ we have

hence $\mathcal F$ is an algebra homomorphism.

The injectivity of the Fourier–Laplace transformation follows from the injectivity of the Fourier transform (see e.g. [9, Section 1.5]). $\square$

The range of the Fourier–Laplace transformation in $\mathcal C(\widetilde{G})$, that is the set of all Fourier–Laplace transforms will be denoted by $\mathcal A(G)$. This is a subalgebra of $\mathcal C(\widetilde{G})$, isomorphic to $\mathcal M_c(G)$, sometimes called the Fourier algebra of G.

4 Exponential monomials

Another important function class is the one formed by the solutions of the equation

$$\begin{aligned} \Delta _{m;y_1,y_2,\ldots ,y_{n+1}}*f=0, \end{aligned}$$

(4)

where m is an exponential, n is a natural number, $f:G\rightarrow \mathbb {C}$ is a continuous function and the equation is supposed to hold for every $y_1,y_2,\ldots ,y_{n+1}$ in G. The function f is called a generalized exponential monomial. In the case $n=0$ we have that f is a constant multiple of the exponential m. We have the following result.

Theorem 12

Let G be a locally compact Abelian group and $m:G\rightarrow \mathbb {C}$ an exponential. The continuous function $f:G\rightarrow \mathbb {C}$ satisfies Eq. (4) if and only if $M_m^{n+1}\subseteq \mathrm {Ann\,}\tau (f)$.

Proof

Obvious, as the modified differences $\Delta _{m;y_1,y_2,\ldots ,y_{n+1}}$ generate an ideal which is dense in $M_m^{n+1}$. $\square$

Lemma 2

Let G be a locally compact Abelian group and $f:G\rightarrow \mathbb {C}$ a nonzero continuous function. Then there exists at most one exponential m such that $M_m^{n+1}\subseteq \mathrm {Ann\,}\tau (f)$ holds for some natural number n.

Proof

As f is nonzero, $\mathrm {Ann\,}\tau (f)$ is a proper ideal. There is a maximal ideal M in $\mathcal M_c(G)$ such that $\mathrm {Ann\,}\tau (f)\subseteq M$. Suppose that $M_m^{n+1}\subseteq \mathrm {Ann\,}\tau (f)$ holds for some exponential m and natural number n. Then $M_m^{n+1}\subseteq M$. As M is maximal, it is also prime, hence we have $M_m\subseteq M$. By Theorem 9, $M_m$ is also maximal, which implies $M_m=M$. It follows that $M_m$ and M are unique with the given properties. $\square$

By the previous lemma, for a given nonzero generalized exponential monomial f there is a unique exponential m such that Eq. (4) holds. We say that f is associated to the exponential m, and the smallest natural number n in (4) is called the degree of f.

Theorem 13

Let G be a locally compact Abelian group. The continuous function $f:G\rightarrow \mathbb {C}$ is a generalized exponential monomial if and only if $\mathcal M_c(G)/\mathrm {Ann\,}\tau (f)$ is a local ring with a nilpotent exponential maximal ideal.

Proof

Let $F:\mathcal M_c(G)\rightarrow \mathcal M_c(G)/\mathrm {Ann\,}\tau (f)$ denote the natural homomorphism. If $f\ne 0$ is a generalized exponential monomial, then, by Theorem 12, we have

$$\begin{aligned} F(M_m)^{n+1}=F(M_m^{n+1})\subseteq F(\mathrm {Ann\,}\tau (f))=0, \end{aligned}$$

hence the ideal $F(M_m)$ is nilpotent, and it is obviously maximal. Clearly, it is the unique maximal ideal in the residue ring $R=\mathcal M_c(G)/\mathrm {Ann\,}\tau (f)$, by Lemma 2. Finally, we have

$$\begin{aligned} R/F(M_m)\cong \mathcal M_c(G)/M_m, \end{aligned}$$

hence $F(M_m)$ is an exponential maximal ideal. The converse statement follows in the same way. $\square$

We shall call a generalized exponential monomial simply an exponential monomial, if its variety is finite dimensional. We shall need the following lemma.

Lemma 3

Let G be a locally compact Abelian group, $f:G\rightarrow \mathbb {C}$ a continuous function, m an exponential, and k a natural number. Then for each $\varphi$ in $M_m^k\tau (f)$ the $\mathcal M_c(G)$-module generated by $\varphi +M_m^{k+1}$ in $M_m^k\tau (f)/M_m^{k+1}\tau (f)$ is one dimensional.

Proof

Let $\Phi :\mathcal M_c(G)\rightarrow \mathbb {C}$ be the multiplicative functional with the property $\mathrm {Ker\,}\Phi =\mathrm {Ann\,}\tau (m)$. For each y in G we have

$$\begin{aligned} \delta _{-y}*\bigl (\varphi +M_m^{k+1}\tau (f)\bigr )= & {} \delta _{-y}*\varphi +M_m^{k+1}\tau (f)\\= & {} (\delta _{-y}-m(y)\delta _0)\varphi +m(y) \varphi +M_m^{k+1}\tau (f)\\= & {} \Phi (\delta _{-y}) \varphi +M_m^{k+1}\tau (f), \end{aligned}$$

as $\delta _{-y}-m(y)\delta _0$ is in $M_m$, hence $(\delta _{-y}-m(y)\delta _0)\varphi$ is in $M_m^{k+1}\tau (f)$. As each $\mu$ in $\mathcal M_c(G)$ is a weak*-limit of linear combinations of measures $\delta _{-y}$, by continuity and linearity, we have

$$\begin{aligned} \mu *(\varphi +M_m^{k+1}\tau (f))=\Phi (\mu )\cdot \varphi +M_m^{k+1}\tau (f)=\Phi (\mu )\bigl (\varphi +M_m^{k+1}\tau (f)\bigr ). \end{aligned}$$

which proves our statement. $\square$

Theorem 14

Let G be a locally compact Abelian group. The continuous function $f:G\rightarrow \mathbb {C}$ is an exponential monomial if and only if $\mathcal M_c(G)/\mathrm {Ann\,}\tau (f)$ is a local Artin ring with exponential maximal ideal.

Proof

Suppose that $f\ne 0$ is an exponential monomial. Then there exists a unique exponential m such that the only maximal ideal containing $\mathrm {Ann\,}\tau (f)$ is $M_m$. It follows that $\mathcal M_c(G)/\mathrm {Ann\,}\tau (f)$ is a local ring with the exponential maximal ideal $F(M_m)$, where $F:\mathcal M_c(G)\rightarrow \mathcal M_c(G)/\mathrm {Ann\,}\tau (f)$ is the natural homomorphism.

By Theorem 7, every ideal in $\mathcal M_c(G)/\mathrm {Ann\,}\tau (f)$ is closed. It follows that every strictly descending chain of ideals in $\mathcal M_c(G)/\mathrm {Ann\,}\tau (f)$ arises from a strictly descending chain of closed ideals including $\mathrm {Ann\,}\tau (f)$ in $\mathcal M_c(G)$, and the annihilators of the ideals in this chain form a strictly ascending chain of subvarieties in $\tau (f)$. By finite dimensionality such a chain must terminate, which implies that $\mathcal M_c(G)/\mathrm {Ann\,}\tau (f)$ is an Artin ring.

Now we assume that $f\ne 0$, and $R=\mathcal M_c(G)/\mathrm {Ann\,}\tau (f)$ is a local Artin ring with exponential maximal ideal F(M), where $F:\mathcal M_c(G)\rightarrow \mathcal M_c(G)/\mathrm {Ann\,}\tau (f)$ is the natural homomorphism, and M is a maximal ideal in $\mathcal M_c(G)$. By the isomorphism

$$\begin{aligned} R/F(M)\cong \mathcal M_c(G)/M\cong \mathbb {C}\end{aligned}$$

we have that $M=M_m$ is an exponential maximal ideal with some exponential m. It is well-known that the maximal ideal in a local Artin ring is nilpotent (see e.g. [10], Theorem 7.15, p. 426). Hence, by Theorem 13, we have that f is a generalized exponential monomial associated with the exponential m. It is enough to show that $\tau (f)$ is finite dimensional. Let n be the degree of f, which implies that $M_m^n\tau (f)\ne \{0\}$. Let $\varphi \ne 0$ be in $M_m^n\tau (f)$, then we have for each x, y in G

$$\begin{aligned} 0=(\delta _{-y}-m(y)\delta _0)*\varphi (x)=\varphi (x+y)-m(y)\varphi (x). \end{aligned}$$

Putting $x=0$ we have $\varphi =\varphi (0)\cdot m$, which means that $M_m\tau (f)$ is one dimensional. We consider the chain of $\mathcal M_c(G)$-modules

$$\begin{aligned} \tau (f), \tau (f)/M_m\tau (f),\ldots , M_m^n\tau (f)/M^{n+1}=M_m^n\tau (f), \{0\}. \end{aligned}$$

Suppose that $\tau (f)$ is infinite dimensional. Then there exists a natural number k with $0\le k\le n-1$ such that $M_m^k\tau (f)$ is infinite dimensional and $M_m^{k+1}\tau (f)$ is finite dimensional. It follows that $M_m^{k}/M_m^{k+1}\tau (f)$ is infinite dimensional. Then there exists a sequence $\varphi _1,\varphi _2,\ldots ,\varphi _l,\ldots$ in $M_m^k\tau (f)$ such that the coset $\varphi _{l+1}+M_m^{k+1}\tau (f)$ is not included in the linear span of the elements $\varphi _{j}+M_m^{k+1}\tau (f)$ for $j=1,2,\ldots ,l$ and $l=1,2,\ldots$. However, by Lemma 3, the linear span of the elements $\varphi _{j}+M_m^{k+1}\tau (f)$ for $j=1,2,\ldots ,l$ coincides with the submodule generated by these elements in $M_m^k\tau (f)/M_m^{k+1}\tau (f)$. Consequently, $\varphi _{l+1}$ is not included in the subvariety generated by the functions $\varphi _j$ with $1\le j\le l$ in $\tau (f)$, which means that these subvarieties form a strictly ascending chain, and their annihilators generate a strictly descending chain of ideals in $\mathcal M_c(G)/\mathrm {Ann\,}\tau (f)$. This contradicts the Artin property. $\square$

5 Spectral analysis

Let G be a locally compact Abelian group and V a variety in $\mathcal C(G)$. We say that spectral analysis holds for the variety V if every nonzero subvariety of V contains an exponential. We say that spectral analysis holds on G, if spectral analysis holds for $\mathcal C(G)$, that is, every nonzero variety on G contains an exponential. Clearly, if spectral analysis holds for a variety, then it holds for every subvariety of it, too.

Lemma 4

Let G be a locally compact Abelian group. If a nonzero generalized exponential monomial associated to the exponential m belongs to a variety, then m belongs to the same variety, too.

Proof

Let $f\ne 0$ be a generalized exponential monomial associated to the exponential m. Then $\mathrm {Ann\,}\tau (f)$ is a proper ideal, hence there is a maximal ideal M containing $\mathrm {Ann\,}\tau (f)$. On the other hand, we have $M_m^{n+1}\subseteq \mathrm {Ann\,}\tau (f)$ for some natural number n. It follows $M_m^{n+1}\subseteq M$. As M is maximal, it is also prime, and we infer $M_m\subseteq M$, which implies, by maximality, that $M_m=M$, and $\mathrm {Ann\,}\tau (f)\subseteq M_m$. Finally, we have

$$\begin{aligned} \mathrm {Ann\,}M_m\subseteq \mathrm {Ann\,}\mathrm {Ann\,}\tau (f)=\tau (f), \end{aligned}$$

and obviously m is in $\mathrm {Ann\,}M_m$, which proves our statement. $\square$

Theorem 15

Let G be a locally compact Abelian group. Spectral analysis holds for a variety if and only if every nonzero subvariety of it contains a nonzero generalized exponential monomial. In particular, spectral analysis holds on G if and only if every nonzero variety on G contains a nonzero generalized exponential monomial.

As the variety of a generalized exponential monomial consists of generalized exponentials, hence spectral analysis holds for the variety of every generalized exponential monomial.

Theorem 16

Let G be a locally compact Abelian group. Spectral analysis holds for a nonzero variety V on G if and only if every maximal ideal in $\mathcal M_c(G)$ which contains $\mathrm {Ann\,}V$ is exponential. In other words, spectral analysis holds for $V\ne \{0\}$ if and only if every maximal ideal in the ring $\mathcal M_c(G)/\mathrm {Ann\,}V$ is exponential. In particular, spectral analysis holds on G if and only if every maximal ideal in $\mathcal M_c(G)$ is exponential.

Proof

Indeed, the given condition is clearly necessary. Conversely, if every maximal ideal in $\mathcal M_c(G)$ which contains $\mathrm {Ann\,}V$ is exponential, and $W\subseteq V$ is a nonzero subvariety, then $\mathrm {Ann\,}W\supseteq \mathrm {Ann\,}V$, hence every maximal ideal which contains $\mathrm {Ann\,}W$ also contains $\mathrm {Ann\,}V$. The other statements are obvious. $\square$

6 Spectral synthesis

Let G be a locally compact Abelian group and V a variety in $\mathcal C(G)$. We say that the variety V is synthesizable, if the exponential monomials in V span a dense subspace. We say that spectral synthesis holds for V, if every subvariety of V is synthesizable. We say that spectral synthesis holds on G, if spectral synthesis holds for every variety on G. Clearly, if spectral synthesis holds for a variety, then spectral synthesis and spectral analysis holds for every subvariety of it.

Given a variety V in $\mathcal C(G)$ let $\mathcal I(V)$ denote the set of all closed ideals I in $\mathcal M_c(G)$ such that $\mathrm {Ann\,}V\subseteq I$ and $\mathcal M_c(G)/I$ is a local Artin ring.

Theorem 17

Let G be a locally compact Abelian group. The variety V in $\mathcal C(G)$ is synthesizable if and only if

$$\begin{aligned} \mathrm {Ann\,}V=\bigcap \mathcal I(V). \end{aligned}$$

(5)

Proof

Suppose that V is synthesizable. Then

$$\begin{aligned} V=\sum _{\varphi \in V} \tau (\varphi ), \end{aligned}$$

where the summation is extended for all exponential monomials $\varphi$ in V. By Theorem 8, it follows

$$\begin{aligned} \mathrm {Ann\,}V=\bigcap _{\varphi \in V} \mathrm {Ann\,}\tau (\varphi )=\bigcap _{\mathrm {Ann\,}V\subseteq \mathrm {Ann\,}\tau (\varphi )} \mathrm {Ann\,}\tau (\varphi ). \end{aligned}$$

By Theorem 14, the set of the annihilators $\mathrm {Ann\,}\tau (\varphi )$ where $\varphi$ is an exponential monomial in V is identical with the set $\mathcal I(V)$, which proves the theorem. $\square$

Theorem 18

Let G be a locally compact Abelian group and $f:G\rightarrow \mathbb {C}$ a generalized exponential monomial, which is not an exponential monomial. Then $\tau (f)$ is non-synthesizable.

Proof

We show that if

$$\begin{aligned} \mathrm {Ann\,}\tau (f)=\bigcap \mathcal F, \end{aligned}$$

where $\mathcal F$ is a family of closed ideals, then there is an I in $\mathcal F$ with $I=\mathrm {Ann\,}\tau (f)$.

Suppose that $f\ne 0$ is a generalized exponential monomial of degree $n\ge 1$ associated to the exponential m. Then $\mathrm {Ann\,}I$ is a subvariety of $\tau (f)$, hence it consists of generalized exponential monomials of degree at most n, which are associated to m, too. We also have that $M_m \tau (\varphi )$ consists of generalized exponential monomials of degree at most $n-1$, which are associated to m. For each y in G we have

$$\begin{aligned} \tau _y f=\delta _{-y}*f=(\delta _{-y}-m(y)\delta _0)*f+m(y) f, \end{aligned}$$

hence $\tau _y f$ is in the linear space $X=M_m\tau (f)+\mathbb {C}f$, which is closed in $\mathcal C(G)$. Indeed, if $(\varphi _{\alpha }+c_{\alpha } f)_{\alpha \in A}$ is a generalized sequence in X, with $\varphi _{\alpha }$ in $M_m\tau (f)$ and $c_{\alpha }$ in $\mathbb {C}$, which converges to $\psi$ in $\mathcal C(G)$, then for each $\mu$ in $M_m^n$ the generalized sequence $(\mu *\varphi _{\alpha }+c_{\alpha } \,\mu *f)_{\alpha \in A}$ converges to $\mu *\psi$. The function $\mu *f$ is in $M_m^n \tau (f)$, which is different from $\{0\}$, as the degree of f is n. This means that we can choose $\mu$ such that $\mu *f\ne 0$. On the other hand, as $\varphi _{\alpha }$ is in $M_m\tau (f)$, we have that $\mu *\varphi _{\alpha }$ is in $M_m^{n+1} \tau (f)=\{0\}$, that is $\mu *\varphi _{\alpha }=0$. It follows that $(c_\alpha )_{\alpha \in A}$ converges to some c in $\mathbb {C}$, which implies that $(\varphi _{\alpha })_{\alpha \in A}$ converges to a function in $M_m\tau (f)$, and $\psi$ is in X.

As X is closed, we have $\tau (f)\subseteq M_m\tau (f)+\mathbb {C}f$, in fact $\tau (f)= M_m\tau (f)+\mathbb {C}f$. On the other hand, $M_m\tau (f)\cap \mathbb {C}f=\{0\}$, hence $\tau (f)$ is the direct sum of the closed subspaces $M_m \tau (f)$ and $\mathbb {C}f$.

Now suppose that

$$\begin{aligned} \mathrm {Ann\,}\tau (f)=\bigcap \mathcal F, \end{aligned}$$

where $\mathcal F$ is a family of closed ideals, then

$$\begin{aligned} \tau (f)=\sum _{I\in \mathcal F} \mathrm {Ann\,}I. \end{aligned}$$

As f is of degree n, there must be an I in $\mathcal F$ such that $\mathrm {Ann\,}I$ includes a generalized monomial of degree n. By the direct decomposition of $\tau (f)$ we have

$$\begin{aligned} \mathrm {Ann\,}I=(\mathrm {Ann\,}I\cap M_m \tau (f))+(\mathrm {Ann\,}I\cap \mathbb {C}f). \end{aligned}$$

If $\mathrm {Ann\,}I\cap \mathbb {C}f\ne \{0\}$, then f is in $\mathrm {Ann\,}I$ and $\mathrm {Ann\,}I=\tau (f)$, and $I=\mathrm {Ann\,}\tau (f)$. However, $\mathrm {Ann\,}I\cap \mathbb {C}f=\{0\}$ is impossible, because in this case $\mathrm {Ann\,}I\subseteq M_m \tau (f)$, which consist of generalized exponential monomials of degree at most $n-1$, a contradiction. This proves that there is an I in $\mathcal F$ such that $\mathrm {Ann\,}\tau (f)=I$. Assume now that $\tau (f)$ is synthesizable. By Theorem 17, we have a representation of $\tau (f)$ as the intersection of the ideals in the family $\mathcal F=\mathcal I(\tau (f))$. We have seen above that in this case $\mathrm {Ann\,}\tau (f)$ must be in $\mathcal F$, but, by Theorem 14, this is impossible, as $\mathcal M_c(G)/\mathrm {Ann\,}\tau (f)$ is a not local Artin ring with exponential maximal ideal. $\square$

Corollary 2

Let G be a locally compact Abelian group. If a variety in $\mathcal C(G)$ contains a generalized exponential monomial, which is not an exponential monomial, then spectral synthesis fails to hold for the variety.