A characterization of generalized exponential polynomials in terms of decomposable functions

TL;DR

This paper characterizes generalized exponential polynomials on topological commutative semigroups by their decomposability properties of sums, providing a new criterion for identifying such functions.

Contribution

It establishes a necessary and sufficient condition for a continuous function to be a generalized exponential polynomial based on decomposability of its sum over multiple variables.

Findings

Characterization of generalized exponential polynomials via decomposability

Equivalent condition involving sum functions on semigroups

Provides a new perspective for identifying exponential polynomials

Abstract

Let $G$ be a topological commutative semigroup with unit. We prove that a continuous function $f\colon G\to \cc$ is a generalized exponential polynomial if and only if there is an $n\ge 2$ such that $f(x_1 +\ldots +x_n )$ is decomposable; that is, if $f(x_1 +\ldots +x_n )=\sumik u_i \cd v_i$, where the function $u_i$ only depends on the variables belonging to a set $\emp \ne E_i \subsetneq \{ x_1 \stb x_n \}$, and $v_i$ only depends on the variables belonging to $\{ x_1 \stb x_n \} \se E_i$ $(i=1\stb k)$.

Equations

$$f_i(x_1+$$

$$=\sum _{j_1,\dots ,j_i=1}^{k+1}{A}_i(x_{j_1},\dots ,x_{j_i}).$$

$$f_i(x_1+\dots +x_{k+1})=\sum _{j=1}^{k+1}{u}_{i,j}(i=1,\dots ,k),$$

$$p(x_1+\dots +x_{k+1})=f_0+\sum _{i=1}^k\sum _{j=1}^{k+1}{u}_{i,j}=\sum _{j=1}^{k+1}{u}_j\cdot 1,$$

$$F_{\omega }=\{(t_1,t_2,\dots ):t_i\in \mathbb{Z}(i=1,2,\dots ),\exists i_0,t_i=0(i>i_0)\},$$

$$q(x_1+\dots +x_n)=u_1\cdot v_1+u_2\cdot v_2,$$

$$\dim V_{p_i\cdot m_i}=\dim V_{p_i}\ge 1+\mathrm{deg}{p}_i.$$

$$\dim V_f=\sum _{i=1}^n\dim V_{p_i\cdot m_i}\ge \sum _{i=1}^n(1+\mathrm{deg}{p}_i)\ge \mathrm{deg}f.$$

$${\Delta }_{m_1,h_1}^{n_1}\dots {\Delta }_{m_k,h_k}^{n_k}f=0$$

$${\Delta }_{m,h}{p}_i\cdot m_i$$

$$=[p_i(x+h)\cdot m_i(h)-p_i(x)\cdot m(h)]\cdot m_i(x)=$$

$$=q(x)\cdot m_i(x),$$

$${\Delta }_{m,h}^{n}f=m(h{)}^n\cdot m\cdot {\Delta }_h^n(f/m)(h\in G).$$

$$f_1={\Delta }_{m_2,h_2}^{n_2}\dots {\Delta }_{m_k,h_k}^{n_k}f.$$

$${\Delta }_{m_1,h_1}^{n_1-1}{\Delta }_{m_2,h_2}^{n_2}\dots {\Delta }_{m_k,h_k}^{n_k}{\Delta }_{m_1,h}f=0$$

$${\Delta }_{m_1,h}f=\sum _{i=1}^k{p}_i\cdot m_i,$$

$${\Delta }_{m_2,h}f=\sum _{i=1}^k{q}_i\cdot m_i,$$

$${\Delta }_{m_1,h}f(x)-{\Delta }_{m_2,h}f(x)=(m_2(h)-m_1(h))\cdot f(x),$$

$$(m_2(h)-m_1(h))\cdot f=\sum _{i=1}^k(p_i-q_i)\cdot m_i.$$

$$\bigcup _{i=1}^n\mathrm{sp}{f}_i=\{m_1,\dots ,m_s\},$$

$$\sum _{i=1}^n{c}_i\cdot f_i=\sum _{j=1}^s\left(\sum _{i=1}^n{c}_i\cdot p_{i,j}\right)\cdot m_j$$

$$\sum _{j=1}^{n+1}{c}_j{e}_j=\sum _{j=1}^{n+1}{c}_j{v}_j\in V.$$

$${\Delta }_{(h)}={\Delta }_{m_1,h_1}^{n_1}\dots {\Delta }_{m_k,h_k}^{n_k}.$$

$${\Delta }_{m_1,h_1}^{n+1}\dots {\Delta }_{m_{k+s},h_{k+s}}^{n+1}{\Delta }_{(h)}\varphi =0$$

$${\Delta }_{m_1,h_1}^{n_1+n+1}\dots {\Delta }_{m_k,h_k}^{n_k+n+1}{\Delta }_{m_{k+1},h_{k+1}}^{n+1}\dots {\Delta }_{m_{k+s},h_{k+s}}^{n+1}f=0$$

$$\sum _{i\in J}(d_i-n_i)\le n.$$

$$\sum _{i\in I}(d_i-n_i)\ge \sum _{i\in J}(d_i-n_i)>n.$$

$$f=\sum _{\nu =1}^t{c}_{\nu }\cdot f_{\nu }=\sum _{i\in I}\sum _{j=n_i}^{d_i-1}{c}_{i,j}\cdot p_{i,j}\cdot m_i,$$

$$\mathrm{deg}{f}_0=\sum _{i\in J}{d}_i\le n+\sum _{i\in J}{n}_i\le n+\sum _{i=1}^k{n}_i\le n+N.$$

$$\mathrm{deg}{f}_0=-1+\sum _{i\in J}{d}_i\le n-1+\sum _{i\in J}{n}_i\le n-1+\sum _{i=1}^k{n}_i\le n+N.$$

$$\sum _{i\in J}{n}_i<\sum _{i=1}^k{n}_i\le N+1$$

$$\mathrm{deg}{f}_0=\sum _{i\in J}{d}_i\le n+\sum _{i\in J}{n}_i\le n+N.$$

Full text

1 Introduction and main results

Let $X$ be a nonempty set. A function of $n$ variables $F\colon X^{n}\to{\mathbb{C}}$ is called decomposable of order $k$ if there are functions $u_{i},v_{i}\colon X^{n}\to{\mathbb{C}}$ $(i=1,\ldots,k)$ such that $F=\sum_{i=1}^{k}u_{i}\cdot v_{i}$ and, for every $i=1,\ldots,k$, $u_{i}$ only depends on the variables belonging to a set $\emptyset\neq E_{i}\subsetneq\{x_{1},\ldots,x_{n}\}$, and $v_{i}$ only depends on the variables belonging to $\{x_{1},\ldots,x_{n}\}\setminus E_{i}$. For example, if $G$ is a semigroup and $f\colon G\to{\mathbb{C}}$ satisfies a Levi-Civita equation $f(xy)=\sum_{i=1}^{k}a_{i}(x)\cdot b_{i}(y)$ $(x,y\in G)$, then $f(xy)$ is decomposable of order $k$.

The following remarkable result was proved by Ekaterina Shulman in [5, Theorem 6]. Let $G$ be a topological semigroup with unit. If a continuous function $f\colon G\to{\mathbb{C}}$ is such that $f(x_{1}\cdots x_{n})$ is decomposable for some $n>1$, then $f$ is an almost matrix function.

A function is a matrix function if $f(xy)$ satisfies a Levi-Civita equation. The function $f$ is an almost matrix function if, for every finite set $E\subset G$, there is a finite dimensional subspace of $C(G)$ containing $f$ and invariant under the subsemigroup generated by $E$. It was also shown by Shulman that if $G$ is the union of an increasing net of topologically $p$-generated subsemigroups, then, whenever $f\in C(G)$ is such that $f(x_{1}\cdots x_{n})$ is decomposable for some $n>1$, then $f$ is a matrix function. In particular, if $f\in C({\mathbb{R}}^{p})$ is such that $f(x_{1}+\ldots+x_{n})$ is decomposable for some $n>1$, then $f$ is an exponential polynomial ([5, Corollary 8]).

In this note our aim is to prove that in the commutative case, if $f(x_{1}+\ldots+x_{n})$ is decomposable, then $f$ is necessarily a generalized exponential polynomial. The relevant definitions are as follows.

Let $G$ be an Abelian topological semigroup with unit, written additively. Let $C(G)$ denote the set of complex valued continuous functions defined on $G$. The difference operator $\Delta_{h}$ is defined by $\Delta_{h}f(x)=f(x+h)-f(x)$ $(x\in G)$ for every $h\in G$ and $f\colon G\to{\mathbb{C}}$. A function $f\in C(G)$ is a generalized polynomial, if there is an $n\geq 0$ such that $\Delta_{h_{1}}\ldots\Delta_{h_{n+1}}f=0$ for every $h_{1},\ldots,h_{n+1}\in G$. The smallest $n$ with this property is the degree of $f$, denoted by ${\rm deg}\,f$. The degree of the identically zero function is $-1$. The set of generalized polynomials will be denoted by GP.

A function $m\in C(G)$ is an exponential, if $m\neq 0$ and $m(x+y)=m(x)\cdot m(y)$ for every $x,y\in G$. The functions of the form $\sum_{i=1}^{s}p_{i}\cdot m_{i}$, where $p_{i}\in\hbox{GP}$ and $m_{i}$ is an exponential for every $i=1,\ldots,s$ are called generalized exponential polynomials. The set of generalized exponential polynomials will be denoted by GEP.

In the rest of the paper we assume that $G$ is a commutative unital semigroup written additively. Our main result is the following.

Theorem 1**.**

For every continuous function $f\in C(G)$ the following are equivalent.

1. (i)

There is an $n\geq 2$ such that $f(x_{1}+\ldots+x_{n})$ is decomposable. 2. (ii)

$f$* is a generalized exponential polynomial.*

It is easy to see that if $f(x_{1}+\ldots+x_{n})$ is decomposable, then so is $f(x_{1}+\ldots+x_{n^{\prime}})$ for every $n^{\prime}>n$. Thus (i) is equivalent to the statement that $f(x_{1}+\ldots+x_{n})$ is decomposable for every $n$ large enough.

Note that the class GEP is contained in the class of almost matrix functions, and the containment, in general, is strict (see [5, example 1, p. 18]). Therefore, Theorem 1 is more precise than [5, Theorem 6] (in the commutative case).

Let $f=\sum_{i=1}^{s}p_{i}\cdot m_{i}$, where $m_{1},\ldots,m_{s}$ are distinct exponentials and $p_{1},\ldots,p_{s}$ are nonzero generalized polynomials. Then the degree of $f$ is defined by ${\rm deg}\,f=\sum_{i=1}^{s}(1+{\rm deg}\,p_{i})$ if $m_{i}\neq 1$ for every $i=1,\ldots,s$, and by ${\rm deg}\,f=-1+\sum_{i=1}^{s}(1+{\rm deg}\,p_{i})$ if $m_{i}=1$ for one of the $i$’s. If $f=0$, then we put ${\rm deg}\,f=-1$.

It is well-known that every $f\in\hbox{\rm GEP}$, $f\neq 0$ has a unique representation of the form $\sum_{i=1}^{s}p_{i}\cdot m_{i}$, where $m_{1},\ldots,m_{s}$ are distinct exponentials and $p_{1},\ldots,p_{s}$ are nonzero generalized polynomials. (For Abelian groups this is proved in [6, Lemma 4.3, p. 41] and in [4, Lemma 6]. It is easy to check that the proof of [4, Lemma 6] works in Abelian semigroups as well.) Thus ${\rm deg}\,f$ is well-defined for every $f\in\hbox{\rm GEP}$. It is clear that ${\rm deg}\,f$ equals the usual degree of $f$ if $f\in\hbox{\rm GP}$.

The following result gives the direction (ii)$\Longrightarrow$(i) of Theorem 1.

Theorem 2**.**

Let $f=\sum_{i=1}^{s}p_{i}\cdot m_{i}$, where $p_{i}\in\hbox{\rm GP}$, $p_{i}\neq 0$ $(i=1,\ldots,s)$, and $m_{1},\ldots,m_{s}$ are distinct exponentials. Put $n_{0}=\max_{1\leq i\leq s}{\rm deg}\,p_{i}$ and $k={\rm deg}\,f$.

1. (i)

If $m_{i}\neq 1$ for every $i=1,\ldots,s$, then $f(x_{1}+\ldots+x_{n})$ is decomposable of order $k$ for every $n>n_{0}$. 2. (ii)

If $m_{i}=1$ for one of the indices $i=1,\ldots,s$, then $f(x_{1}+\ldots+x_{n})$ is decomposable of order $k+1$ for every $n>n_{0}$.

Proof.

It is enough to prove the theorem in the special case when $s=1$.

The function of $i$ variables $A\colon G^{i}\to{\mathbb{C}}$ is called $i$-additive, if it is additive in each of its variables (the other variables being fixed). A function $f_{i}\colon G\to{\mathbb{C}}$ is a monom of degree $i$, if there exists a symmetric and $i$-additive function $A_{i}$ such that $f_{i}(x)=A_{i}(x,\ldots,x)$ for every $x\in G$.

Let $p\in\hbox{GP}$ be given, and let ${\rm deg}\,p=k$. Then $p=\sum_{i=0}^{k}f_{i}$, where $f_{i}$ is a monom of degree $i$ for every $i=1,\ldots,k$, and $f_{0}$ is constant. (See [3, Theorem 3].) Suppose $f_{i}(x)=A_{i}(x,\ldots,x)$ $(x\in G)$, where $A_{i}$ is symmetric and $i$-additive for every $i=1,\ldots,k$. If $1\leq i\leq k$, then

[TABLE]

Since $i<k+1$, each term of the sum depends on the variables belonging to a proper subset of $\{x_{1},\ldots,x_{k+1}\}$. Grouping the terms appropriately we find that

[TABLE]

where $u_{i,j}$ does not depend on $x_{j}$ $(j=1,\ldots,k+1)$. Putting $u_{j}=\sum_{i=1}^{k}u_{i,j}+(f_{0}/(k+1))$, we obtain

[TABLE]

where $u_{j}$ does not depend on $x_{j}$ $(j=1,\ldots,k+1)$. Therefore, if $p\in\hbox{GP}$ and ${\rm deg}\,f=k$, then $p(x_{1}+\ldots+x_{k+1})$ is decomposable of order $k+1$, and then so is $p(x_{1}+\ldots+x_{n})$ for every $n>k$.

Let $f=p\cdot m$, where $p\in\hbox{GP}$, ${\rm deg}\,p=k\geq 0$ and $m$ is an exponential. If $m=1$, then ${\rm deg}\,f=k$, and we obtain that (ii) holds in this special case.

Suppose $m\neq 1$. Multiplying (1) by $m(x_{1}+\ldots+x_{k+1})=m(x_{1})\cdots m(x_{k+1})$, we find that $f(x_{1}+\ldots+x_{k+1})$ is decomposable of order $k+1$, and then the same is true for $f(x_{1}+\ldots+x_{n})$ for every $n>k$. By $m\neq 1$, we have ${\rm deg}\,f=k+1$, and thus $f$ satisfies (i) (with $k+1$ in place of $k$). ∎

We note that the bound $k+1$ in (ii) cannot be replaced by $k$, as the following example shows. Let $F_{\omega}$ denote the free Abelian group generated by countable infinitely many generators. We shall represent $F_{\omega}$ as

[TABLE]

where the sum of the elements $(t_{1},t_{2},\ldots)$ and $(y_{1},y_{2},\ldots)$ is $(t_{1}+y_{1},t_{2}+y_{2},\ldots)$. We take the discrete topology on $F_{\omega}$.

Let $q(x)=\sum_{i=1}^{\infty}t_{i}^{2}$ for every $x=(t_{1},t_{2},\ldots)\in F_{\omega}$. Then $q\in\hbox{GP}$ and ${\rm deg}\,q=2$. We prove that $q(x_{1}+\ldots+x_{n})$ is not decomposable of order $2$ for any $n\geq 2$. Indeed, $q(x_{1}+x_{2})$ is not decomposable (of any order), since otherwise the translates of $q$ would span a linear space of finite dimension, which is not the case (see [7]). Let $n\geq 3$, and suppose that $q(x_{1}+\ldots+x_{n})$ is decomposable of order $2$. Then we have

[TABLE]

where the function $u_{i}$ only depends on the variables belonging to a set $\emptyset\neq E_{i}\subsetneq\{x_{1},\ldots,x_{n}\}$, and $v_{i}$ only depends on the variables belonging to $\{x_{1},\ldots,x_{n}\}\setminus E_{i}$ $(i=1,2)$. It is easy to check that there are indices $1\leq j<k\leq n$ such that $x_{j}$ and $x_{k}$ are separated by both $E_{1}$ and $E_{2}$; that is, $x_{j}\in E_{1}$ and $x_{k}\notin E_{1}$ or the other way around, and $x_{j}\in E_{2}$ and $x_{k}\notin E_{2}$ or the other way around. Fixing $x_{i}$ at zero for every $i\neq j,k$ we can see that $q(x_{j}+x_{k})$ is decomposable of order $2$, which is impossible.

The following result gives the direction (i)$\Longrightarrow$(ii) of Theorem 1.

Theorem 3**.**

If $f\in C(G)$ is such that $f(x_{1}+\ldots+x_{n})$ is decomposable for some $n\geq 2$, then $f\in\hbox{\rm GEP}$. More precisely, if $f(x_{1}+\ldots+x_{n})$ is decomposable of order $k$, then $f\in\hbox{\rm GEP}$, and ${\rm deg}\,f\leq k$.

As (i) of Theorem 2 shows, the upper bound ${\rm deg}\,f\leq k$ is sharp in Theorem 3. We prove Theorem 3 in Section 3, using the results of the next section.

2 Lemmas

In this section we assume that $G$ is a commutative unital semigroup written additively. A function $f\colon G\to{\mathbb{C}}$ is a polynomial, if $f=P(a_{1},\ldots,a_{n})$, where $P\in{\mathbb{C}}[x_{1},\ldots,x_{n}]$ and $a_{1},\ldots,a_{n}$ are continuous additive functions mapping $G$ into ${\mathbb{C}}$. (These functions are called normal polynomials in [6].) The functions of the form $\sum_{i=1}^{k}p_{i}\cdot m_{i}$, where $p_{i}$ is a polynomial and $m_{i}$ is an exponential for every $i=1,\ldots,k$ are called exponential polynomials. If $f\in C(G)$ then we denote by $V_{f}$ the linear span of the translates of $f$.

Proposition 4**.**

If $f$ is an exponential polynomial, then ${\rm deg}\,f\leq{\rm dim}\,V_{f}$.

Proof.

If $f$ is a polynomial, then even the stronger statement ${\rm deg}\,f<{\rm dim}\,V_{f}$ follows from [6, Lemma 2.8, p. 30]. We give a simple direct proof. We prove by induction on $n={\rm deg}\,f$. If $n=-1$, then $f=0$, $V_{f}=\{0\}$, and ${\rm deg}\,f=-1<0={\rm dim}\,V_{f}$. Let $n={\rm deg}\,f\geq 0$, and suppose that the statement is true for smaller degrees. There exists an $h\in G$ such that ${\rm deg}\,\Delta_{h}f=n-1$. Fixing such an $h$ we find that $f\notin V_{\Delta_{h}f}$, since the elements of $V_{\Delta_{h}f}$ are of degree $\leq n-1$. As $V_{\Delta_{h}f}+\langle f\rangle\subset V_{f}$, it follows that ${\rm dim}\,V_{f}>{\rm dim}\,V_{\Delta_{h}f}$. By the induction hypothesis we have ${\rm dim}\,V_{\Delta_{h}f}>{\rm deg}\,\Delta_{h}f=n-1$, and thus ${\rm dim}\,V_{f}>n$. This proves the statement for polynomials.

To prove the general statement, let $f=\sum_{i=1}^{n}p_{i}\cdot m_{i}$, where $m_{1},\ldots,$ $m_{n}$ are distinct exponentials and $p_{1},\ldots,p_{n}$ are nonzero polynomials. For every $i$, we have $V_{p_{i}\cdot m_{i}}=m_{i}\cdot V_{p_{i}}=\{m_{i}\cdot p\colon p\in V_{p_{i}}\}$, and thus

[TABLE]

It is well-known that $p_{i}\cdot m_{i}\in V_{f}$ for every $i$. (See, e.g., [4, Lemma 6]. Although the lemma is stated for groups, the proof works in every Abelian semigroup.) Then we have $V_{f}=V_{p_{1}\cdot m_{1}}+\ldots+V_{p_{n}\cdot m_{n}}$, where the right hand side is a direct sum, as $V_{p_{i}\cdot m_{i}}\cap\sum_{j\neq i}V_{p_{j}\cdot m_{j}}=\{0\}$ for every $i=1,\ldots,n$. Therefore, we obtain

[TABLE]

∎

We shall need the notion of modified difference operators introduced by Almira and Székelyhidi in [2]. If $f,\phi\colon G\to{\mathbb{C}}$ and $h\in G$, then we put $\Delta_{\phi,h}f(x)=f(x+h)-\phi(h)\cdot f(x)$ for every $x\in G$. The statement of the following lemma is a consequence of [2, Theorem 17] and of [6, Theorem 8.12, p. 68]. In order to make these note as self-contained as possible, we give an independent direct proof.

Lemma 5**.**

1. (i)

Let $f=\sum_{i=1}^{k}p_{i}\cdot m_{i}$, where $p_{1},\ldots,p_{k}$ are generalized polynomials and $m_{1},\ldots,m_{k}$ are exponentials. If $n_{i}>{\rm deg}\,p_{i}$ for every $i=1,\ldots,k$, then we have

[TABLE]

for every $h_{1},\ldots,h_{k}\in G$. 2. (ii)

Let $m_{1},\ldots,m_{k}$ be different exponentials, and let $n_{1},\ldots,n_{k}$ be nonnegative integers. If $f\colon G\to{\mathbb{C}}$ is such that (2) holds for every $h_{1},\ldots,h_{k}\in G$, then $f=\sum_{i=1}^{k}p_{i}\cdot m_{i}$, where $p_{1},\ldots,p_{k}$ are generalized polynomials, and ${\rm deg}\,p_{i}<n_{i}$ for every $i=1,\ldots,k$.

Proof.

(i) If $m$ is an exponential, then we have

[TABLE]

where $q\in\hbox{GP}$, ${\rm deg}\,q\leq{\rm deg}\,p_{i}$, and ${\rm deg}\,q<{\rm deg}\,p_{i}$ if $m=m_{i}$. From this observation the statement follows by induction on $\sum_{i=1}^{k}{\rm deg}\,p_{i}$.

(ii) First we prove the statement for $k=1$. We have to prove that if $m$ is an exponential and $\Delta_{m,h}^{n}f=0$ for every $h\in G$, then $f=p\cdot m$, where $p\in\hbox{GP}$ and ${\rm deg}\,p<n$. If $n=0$, then we interpret the condition as $f=0$. In this case we have $f=0\cdot m$, where ${\rm deg}\,0=-1$, and so the statement is true. Let $n>0$. It is easy to see that for every $f\colon G\to{\mathbb{C}}$ we have $\Delta_{m,h}f(x)=m(h)\cdot m(x)\cdot\Delta_{h}(f/m)(x)$. Therefore, by induction we find

[TABLE]

If $\Delta_{m,h}^{n}f=0$ for every $h$, then we obtain $\Delta_{h}^{n}(f/m)=0$ for every $h$. By Djoković’s theorem this condition implies $f/m\in\hbox{GP}$ and ${\rm deg}\,p\leq n-1$ (see [3, Corollary 1]). Then, by $f=(f/m)\cdot m$, we obtain the statement of (ii) for $k=1$.

Suppose that $k>1$ and the statement is true for $k-1$. We argue by induction on $n=n_{1}+\ldots+n_{k}$. If $n=0$, then $f=0=\sum_{i=1}^{k}0\cdot m_{i}$, and the statement is true. Suppose that $n>0$ and that the statement is true for $n-1$. By symmetry, we may assume $n_{1}>0$. Let $f$ be a function satisfying the condition of (ii). Let $h_{2},\ldots,h_{k}\in G$ be fixed, and put

[TABLE]

Then $\Delta_{m_{1},h_{1}}^{n_{1}}f_{1}=0$ for every $h_{1}\in G$. As we saw above, this implies $f_{1}=p\cdot m_{1}$, where $p\in\hbox{GP}$ and ${\rm deg}\,p<n_{1}$. Then $\Delta_{h}\Delta_{h_{1}}^{n_{1}-1}p=0$ for every $h,h_{1}\in G$, and thus $\Delta_{m_{1},h}\Delta_{m_{1},h_{1}}^{n_{1}-1}f_{1}=0$ for every $h,h_{1}\in G$ by (3).

Fix a $h$ such that $m_{1}(h)\neq m_{2}(h)$. Since the difference operators commute, we find that

[TABLE]

for every $h_{1},\ldots,h_{k}$. By the induction hypothesis, this implies

[TABLE]

where $p_{i}\in\hbox{GP}$ and ${\rm deg}\,p_{i}<n_{i}$ for every $i$. By the same argument we find that

[TABLE]

where $q_{i}\in\hbox{GP}$ and ${\rm deg}\,q_{i}<n_{i}$ for every $i$. We have

[TABLE]

and thus, subtracting (5) from (4) we get

[TABLE]

Dividing by $(m_{2}(h)-m_{1}(h))$ we obtain the statement of (ii). ∎

Lemma 6**.**

Let $V$ be a linear subspace of GEP such that $V\neq\{0\}$, and ${\rm deg}\,f\leq N<\infty$ for every $f\in V$. Then there are exponentials $m_{1},\ldots,m_{k}$ and positive integers $n_{1},\ldots,n_{k}$ such that

1. (i)

$\sum_{i=1}^{k}n_{i}\leq\begin{cases}N&\text{if}\ 1\notin\{m_{1},\ldots,m_{k}\},\cr N+1&\text{if}\ 1\in\{m_{1},\ldots,m_{k}\};\end{cases}$** 2. (ii)

every $f\in V$ is of the form $\sum_{i=1}^{k}q_{i}\cdot m_{i}$, where $q_{i}\in\hbox{\rm GP}$ and ${\rm deg}\,q_{i}<n_{i}$ $(i=1,\ldots,k)$; 3. (iii)

there exists a function $f_{0}\in V$ such that $f_{0}=\sum_{i=1}^{k}p_{i}\cdot m_{i}$, where $p_{i}\in\hbox{\rm GP}$ and ${\rm deg}\,p_{i}=n_{i}-1$ for every $i=1,\ldots,k$, and 4. (iv)

(2) holds for every $f\in V$ and for every $h_{1},\ldots,h_{k}\in G$.

Proof.

If $f=\sum_{i=1}^{s}p_{i}\cdot m_{i}$, where $m_{1},\ldots,m_{s}$ are distinct exponentials and $p_{1},\ldots,p_{s}$ are nonzero generalized polynomials, then we define the spectrum of $f$ as ${\rm sp}\,f=\{m_{1},\ldots,m_{s}\}$. If $f=0$, then we let ${\rm sp}\,f=\emptyset$.

Put $S=\bigcup_{f\in V}{\rm sp}\,f$. We show that $S$ has at most $N+1$ elements. Suppose this is not true. Then there are functions $f_{1},\ldots,f_{n}\in V$ such that

[TABLE]

where $m_{1},\ldots,m_{s}$ are distinct exponentials, and $s\geq N+2$. Let $f_{i}=\sum_{j=1}^{s}p_{i,j}\cdot m_{j}$ $(i=1,\ldots,n)$, where $p_{i,j}\in\hbox{GP}$ for every $i=1,\ldots,n$ and $j=1,\ldots,s$. It follows from (6) that for every $j=1,\ldots,s$ there is an $i$ such that $p_{i,j}\neq 0$. If $c_{1},\ldots,c_{n}\in{\mathbb{C}}$, then

[TABLE]

belongs to $V$. For every $j$, the set of $n$-tuples $(c_{1},\ldots,c_{n})$ with $\sum_{i=1}^{n}c_{i}\cdot p_{i,j}=0$ constitutes a linear subspace $L_{j}$ of ${\mathbb{C}}^{n}$. The linear subspace $L_{j}$ is proper, since $p_{i,j}\neq 0$ for a suitable $i$, and thus $(\delta_{1,i},\ldots,\delta_{n,i})\notin L_{j}$, where $\delta_{k,i}$ is the Kronecker delta. Since $\bigcup_{j=1}^{s}L_{j}\subsetneq{\mathbb{C}}^{n}$, there is an $n$-tuple $(c_{1},\ldots,c_{n})$ such that $\sum_{i=1}^{n}c_{i}\cdot p_{i,j}\neq 0$ for every $j=1,\ldots,s$, and then the spectrum of $f=\sum_{i=1}^{n}c_{i}\cdot f_{i}$ equals $\{m_{1},\ldots,m_{s}\}$. Then ${\rm deg}\,f\geq s-1>N$, which contradicts the condition on $V$.

We proved that $S$ has at most $N+1$ elements. Let $S=\{m_{1},\ldots,m_{k}\}$, where $m_{1},\ldots,m_{k}$ are distinct exponentials, and $k\leq N+1$. Then every $f\in V$ is of the form $\sum_{i=1}^{k}q_{i}\cdot m_{i}$, where $q_{i}=q_{i}(f)\in\hbox{GP}$ for every $i=1,\ldots,k$. Since ${\rm deg}\,f\leq N$ for every $f\in V$, it follows that ${\rm deg}\,q_{i}(f)\leq N$ for every $f\in V$ and for every $i=1,\ldots,k$. Let $n_{i}=1+\max\{{\rm deg}\,q_{i}(f)\colon f\in V\}$ for every $i=1,\ldots,k$, and let $f_{i}\in V$ be such that ${\rm deg}\,q_{i}(f_{i})=n_{i}-1$ $(i=1,\ldots,k)$.

The set of $k$-tuples $(c_{1},\ldots,c_{k})\in{\mathbb{C}}^{k}$ such that ${\rm deg}\,\left(\sum_{j=1}^{k}c_{j}\cdot q_{i}(f_{j})\right)<n_{i}-1$ is a linear subspace $M_{i}$ of ${\mathbb{C}}^{k}$ for every $i=1,\ldots,k$. We have $M_{i}\subsetneq{\mathbb{C}}^{k}$, as $(\delta_{1,i},\ldots,\delta_{n,i})\notin M_{i}$. Thus $M_{i}$ is a proper subspace of ${\mathbb{C}}^{k}$ for every $i$. Therefore, we have $\bigcup_{i=1}^{k}M_{i}\subsetneq{\mathbb{C}}^{k}$, and there is an $k$-tuple $(c_{1},\ldots,c_{k})$ such that ${\rm deg}\,\left(\sum_{j=1}^{k}c_{j}\cdot q_{i}(f_{j})\right)=n_{i}-1$ for every $i=1,\ldots,k$. Put $f_{0}=\sum_{j=1}^{k}c_{j}\cdot f_{j}$; then $f_{0}\in V$, and (iii) holds.

We have $f_{0}=\sum_{i=1}^{k}p_{i}\cdot m_{i}$, where $p_{1},\ldots,p_{k}$ are nonzero generalized polynomials with ${\rm deg}\,p_{i}=n_{i}-1$ $(i=1,\ldots,k)$. If $1\notin\{m_{1},\ldots,m_{k}\}$, then $N\geq{\rm deg}\,f_{0}=\sum_{i=1}^{k}n_{i}$. On the other hand, if $1\in\{m_{1},\ldots,m_{k}\}$, then $N\geq{\rm deg}\,f_{0}=-1+\sum_{i=1}^{k}n_{i}$, proving (i). Statement (ii) is obvious from the construction. Finally, (iv) follows from (ii) and from Lemma 5. ∎

Lemma 7**.**

Let $n,N$ be positive integers, and suppose that

1. (i)

$V$* is a linear subspace of GEP such that ${\rm deg}\,f\leq N$ for every $f\in V$, and* 2. (ii)

$W$* is a linear subspace of $C(G)$ such that ${\rm dim}\,W\leq n$.*

Then the number of exponentials contained in $W+V$ is at most $n+N+1$.

Proof.

By the previous lemma, there are exponentials $m_{1},\ldots,m_{k}$ such that $k\leq N+1$, and every $f\in V$ is of the form $\sum_{i=1}^{k}q_{i}\cdot m_{i}$, where $q_{1},\ldots,q_{k}\in\hbox{GP}$. We prove that there are at most $n$ exponentials in $W+V$ different from $m_{1},\ldots,m_{k}$.

Suppose this is not true, and let $e_{1},\ldots,e_{n+1}\in W+V$ be distinct exponentials different from $m_{1},\ldots,m_{k}$. Let $e_{i}=w_{i}+v_{i}$, where $w_{i}\in W$ and $v_{i}\in V$ for every $i=1,\ldots,n+1$. Since ${\rm dim}\,W\leq n$, the elements $w_{1},\ldots,w_{n+1}$ are linearly dependent. That is, there are complex numbers $c_{1},\ldots,c_{n+1}$, not all zero, such that $\sum_{j=1}^{n+1}c_{j}w_{j}=0$. Then

[TABLE]

Therefore, we have $\sum_{j=1}^{n+1}c_{j}e_{j}=\sum_{i=1}^{k}q_{i}\cdot m_{i}$ with suitable $q_{1},\ldots,q_{k}\in\hbox{GP}$. However, the representation of this form is unique, and thus, if $c_{j}\neq 0$ for a $j$, then $e_{j}=m_{i}$ for some $i=1,\ldots,k$, which is impossible. ∎

Lemma 8**.**

Let $n,N$ be positive integers, and suppose that

1. (i)

$V$* is a linear subspace of GEP such that ${\rm deg}\,f\leq N$ for every $f\in V$,* 2. (ii)

$W$* is a linear subspace of $C(G)$ such that ${\rm dim}\,W\leq n$, and* 3. (iii)

$F$* is a translation invariant linear subspace of $W+V$.*

Then $F\subset\hbox{\rm GEP}$, and ${\rm deg}\,f\leq n+N$ for every $f\in F$.

Proof.

Let $m_{1},\ldots,m_{k}$ and $n_{1},\ldots,n_{k}$ be as in Lemma 6. Then we have $\sum_{i=1}^{k}n_{i}\leq N+1$ or $\sum_{i=1}^{k}n_{i}\leq N$ according to whether or not $m_{i}=1$ for one of the indices $i=1,\ldots,k$. By Lemma 7, the number of exponentials contained in $W+V$ is finite. Let $m_{k+1},\ldots,m_{k+s}$ be the exponentials contained in $W+V$ and different from $m_{1},\ldots,m_{k}$.

We show that every function in $F\cap\hbox{GEP}$ is of the form $\sum_{i=1}^{k+s}q_{i}\cdot m_{i}$, where $q_{i}\in\hbox{GP}$ for every $i$. Suppose $f\in F\cap\hbox{GEP}$ and $f=\sum_{j=1}^{t}p_{j}\cdot e_{j}$, where $e_{1},\ldots,e_{t}$ are distinct exponentials and $p_{1},\ldots,p_{t}\in\hbox{GP}\setminus\{0\}$. Since $F$ is a translation invariant linear space, it follows that $e_{j}\in F\subset W+V$ for every $j$. Thus $e_{j}$ equals one of $m_{1},\ldots,m_{k+s}$ for every $j$, proving the statement.

For every $h_{1},\ldots,h_{k}\in G$ we put $h=(h_{1},\ldots,h_{k})$ and

[TABLE]

Fix $h_{1},\ldots,h_{k}$. By the choice of the numbers $n_{i}$ and by (iv) of Lemma 6, we have $\Delta_{(h)}v=0$ for every $v\in V$.

It is clear that $F_{(h)}=\{\Delta_{(h)}f\colon f\in F\}$ is a translation invariant linear subspace of $F$. If $f\in F$, then $f=w+v$ for some $w\in W$ and $v\in V$. Then $\Delta_{(h)}f=\Delta_{(h)}w+\Delta_{(h)}v=\Delta_{(h)}w$, and thus $F_{(h)}$ is a linear subspace of $W_{(h)}=\{\Delta_{(h)}w\colon w\in W\}$. Clearly, $W_{(h)}$ is of dimension $\leq n$, and then the same is true for $F_{(h)}$.

Therefore, by a well-known theorem (see [6, Theorem 10.1] and the references given on [6, p. 79]), each element of $F_{(h)}$ is an exponential polynomial. By Proposition 4, ${\rm deg}\,\phi\leq n$ for every $\phi\in F_{(h)}$. Therefore, by (i) of Lemma 5, we have

[TABLE]

for every $\phi\in F_{(h)}$ and for every $h_{k+1},\ldots,h_{k+s}\in G$. According to the definition of $\Delta_{(h)}$, this implies

[TABLE]

for every $f\in F$ and for every $h_{1},\ldots,h_{k+s}\in G$. Then, by (ii) of Lemma 5, it follows that $F\subset\hbox{\rm GEP}$ and $\sup_{f\in F}{\rm deg}\,f<\infty$. We show that ${\rm deg}\,f\leq n+N$ for every $f\in F$.

Applying Lemma 6 with $F$ in place of $V$, we find a set of indices $J\subset\{1,\ldots,k+s\}$ and positive numbers $d_{i}$ $(i\in J)$ such that every $f\in F$ is of the form $\sum_{i\in J}q_{i}\cdot m_{i}$, where $q_{i}\in\hbox{GP}$ and ${\rm deg}\,q_{i}<d_{i}$ $(i\in J)$, and there is a function $f_{0}\in F$ such that $f_{0}=\sum_{i\in J}p_{i}\cdot m_{i}$, where $p_{i}\in\hbox{GP}$ and ${\rm deg}\,p_{i}=d_{i}-1$ $(i\in J)$. Therefore, we have $\sum_{i\in J}d_{i}={\rm deg}\,f_{0}+1$ or $\sum_{i=1}^{k+s}d_{i}={\rm deg}\,f_{0}$ according to whether or not $m_{i}=1$ for one of the indices $i\in J$.

We put $n_{i}=0$ for every $i\in J\setminus\{1,\ldots,k\}$. We prove that

[TABLE]

Suppose this is not true, and let $I$ denote the set of indices $i\in J$ such that $d_{i}>n_{i}$. Then

[TABLE]

Since $f_{0}=\sum_{i\in J}p_{i}\cdot m_{i}\in F$ and $f$ is a translation invariant linear space, we have $p_{i}\cdot m_{i}\in F$ for every $i\in J$. Moreover, $V_{p_{i}\cdot m_{i}}=m_{i}\cdot V_{p_{i}}\subset F$ for every $i\in J$. For every $p\in\hbox{GP}$ we have ${\rm deg}\,\Delta_{h}p=({\rm deg}\,p)-1$ for a suitable $h\in G$, and thus $V_{p}$ contains generalized polynomials of any degree between [math] and ${\rm deg}\,p$. Let $p_{i,j}\in V_{p_{i}}$ be such that ${\rm deg}\,p_{i,j}=j$ for every $i\in I$ and $n_{i}\leq j\leq d_{i}-1$.

Let $f_{1},\ldots,f_{t}$ be an enumeration of the functions $p_{i,j}\cdot m_{i}$ with $i\in I$ and $n_{i}\leq j\leq d_{i}-1$. Then the functions $f_{1},\ldots,f_{t}$ belong to $F$, and $t=\sum_{i\in I}(d_{i}-n_{i})>n$ by (8). Since $f_{1},\ldots,f_{t}\in F$, we have $f_{\nu}=w_{\nu}+v_{\nu}$, where $w_{\nu}\in W$, $v_{\nu}\in V$ $(\nu=1,\ldots,t)$. Now ${\rm dim}\,W\leq n<t$ implies that there are complex numbers $c_{1},\ldots,c_{t}$, not all zero, such that $\sum_{\nu=1}^{t}c_{\nu}\cdot w_{\nu}=0$. Let $f=\sum_{\nu=1}^{t}c_{\nu}\cdot f_{\nu}$. Then $f\in F$, and $f=\sum_{\nu=1}^{t}c_{\nu}\cdot v_{\nu}\in V$. On the other hand,

[TABLE]

where $c_{i,j}$ $(i\in I$, $n_{i}\leq j\leq d_{i}-1)$ is an enumeration of the numbers $c_{\nu}$ $(\nu=1,\ldots,t)$. Thus $f=\sum_{i\in I}q_{i}\cdot m_{i}$, where $q_{i}=\sum_{j=n_{i}}^{d_{i}-1}c_{i,j}\cdot p_{i,j}$ $(i\in I)$. If $i\in I$ is such that $c_{i,j}$ is nonzero for at least one $n_{i}\leq j\leq d_{i}-1$, then ${\rm deg}\,q_{i}\geq n_{i}$. If $1\leq i\leq k$, then ${\rm deg}\,q_{i}<n_{i}$ by the choice of $n_{i}$, which is impossible. If $i\in J\setminus\{1,\ldots,k\}$, then $n_{i}=0$, and thus ${\rm deg}\,q_{i}\geq n_{i}$ implies $q_{i}\neq 0$, contradicting the choice of $m_{1},\ldots,m_{k}$. Both cases are impossible, so we must have (7).

Now we consider three cases. First we assume that $m_{i}\neq 1$ for every $i\in J\cup\{1,\ldots,k\}$. Then we have $\sum_{i=1}^{k}n_{i}\leq N$ and thus, by (7), we obtain

[TABLE]

Next consider the case when $m_{i}=1$ for some $i\in J$. Then we have $\sum_{i=1}^{k}n_{i}\leq N+1$ and thus, by (7), we obtain

[TABLE]

Finally, if $m_{i}=1$ for some $i\in\{1,\ldots,k\}\setminus J$, Then we have

[TABLE]

and thus, by (7), we obtain

[TABLE]

Summing up: we have ${\rm deg}\,f_{0}\leq n+N$ in each of the cases. Therefore, we have ${\rm deg}\,f\leq{\rm deg}\,f_{0}\leq n+N$ for every $f\in F$. ∎

3 Proof of Theorem 3

We follow the argument of the proof of [5, Theorem 6]. We prove by induction on $n$. If $n=2$ and $f(x_{1}+x_{2})$ is decomposable of order $k$, then $f(x_{1}+x_{2})=\sum_{i=1}^{k}u_{i}(x_{1})\cdot v_{i}(x_{2})$ for every $x_{1},x_{2}\in G$. This implies that the translates of $f$ belong to the linear space generated by the functions $u_{1},\ldots,u_{k}$, and thus $V_{f}$ is of dimension $\leq k$. Therefore, $f$ is an exponential polynomial, and ${\rm deg}\,f\leq k$ by Proposition 4.

Let $n>2$, and suppose that the statement is true for $n-1$. Suppose that $f(x_{1}+\ldots+x_{n})=\sum_{i=1}^{k}u_{i}\cdot v_{i}$, where $u_{i}$ only depends on the variables belonging to a nonempty set $E_{i}\subsetneq\{x_{1},\ldots,x_{n}\}$, and $v_{i}$ only depends on the variables belonging to $\{x_{1},\ldots,x_{n}\}\setminus E_{i}$ $(i=1,\ldots,k)$. Switching $u_{i}$ and $v_{i}$ if necessary, we may assume that $x_{n}\notin E_{i}$ for every $i$. We may also assume that $E_{i}=\{x_{1},\ldots,x_{n-1}\}$ for every $i\leq p$, and $E_{i}\subsetneq\{x_{1},\ldots,x_{n-1}\}$ for $p<i\leq k$.

If $p=0$, then substituting $x_{n}=0$ in $f(x_{1}+\ldots+x_{n})=\sum_{i=1}^{k}u_{i}\cdot v_{i}$ we can see that $f(x_{1}+\ldots+x_{n-1})$ is decomposable of order $k$. Then, by the induction hypothesis, we have $f\in\hbox{\rm GEP}$, and ${\rm deg}\,f\leq k$, and we are done. Therefore, we may assume that $p>0$. Note that the functions $v_{1},\ldots,v_{p}$ only depend on $x_{n}$. We have

[TABLE]

We may assume that the functions $v_{1},\ldots,v_{p}$ are linearly independent. Then there are elements $g_{1},\ldots,g_{p}\in G$ such that the determinant ${\rm det}\,|v_{i}(g_{j})|$ is nonzero (see [1, Lemma 1, p. 229]). Substituting $x_{n}=g_{\nu}$ into (9) for every $\nu=1,\ldots,p$ we obtain a system of linear equations with unknowns $u_{1},\ldots,u_{p}$. Since the determinant of the system is nonzero, we find that each of $u_{1},\ldots,u_{p}$ is a linear combination of the right hand sides. None of $u_{1},\ldots,u_{k}$ depends on $x_{n}$, so we obtain

[TABLE]

where $f_{1},\ldots,f_{p}$ are functions of one variable mapping $G$ into ${\mathbb{C}}$, and $w_{i,j}$ only depends on the variables belonging to the nonempty set $\{x_{1},\ldots,x_{n-1}\}\setminus E_{j}$ for every $i=1,\ldots,p$ and $j=p+1,\ldots,k$.

Substituting the right hand sides of (10) into $f(x_{1}+\ldots+x_{n})=\sum_{i=1}^{k}u_{i}\cdot v_{i}$ and rearranging the terms we obtain

[TABLE]

where $z_{j}=\sum_{i=1}^{p}(w_{i,j}\cdot v_{i}(x_{n})+v_{j})$. Note that, for every $j=p+1,\ldots,k$, $z_{j}$ only depends on the variables belonging to $\{x_{1},\ldots,x_{n}\}\setminus E_{j}$, since each function appearing in the definition of $z_{j}$ has this property.

Let $\Sigma$ denote the set of $(k+1)$-tuples $(\phi,a_{1},\ldots,a_{p},b_{p+1},\ldots,b_{k})$ satisfying the following conditions:

1. (i)

$\phi\in C(G)$ and $a_{1},\ldots,a_{p}\colon G\to{\mathbb{C}}$, 2. (ii)

for every $j=p+1,\ldots,k$, $b_{j}\colon G^{n}\to{\mathbb{C}}$ is a function only depending on the variables belonging to $\{x_{1},\ldots,x_{n}\}\setminus E_{j}$, and 3. (iii)

[TABLE]

for every $x_{1},\ldots,x_{n}\in G$.

It is clear that $\Sigma$ is a linear space over ${\mathbb{C}}$ under addition and multiplication by complex numbers coordinate-wise.

Let $F$ denote the set of functions $\phi\in C(G)$ such that

[TABLE]

for some $a_{1},\ldots,a_{p},b_{p+1},\ldots,b_{k}$. By (11), we have $f\in F$. Since $\Sigma$ is a linear space over ${\mathbb{C}}$, so is $F$.

We show that $F$ is translation invariant. Let $\phi\in F$ and $h\in G$ be arbitrary. Let $(\phi,a_{1},\ldots,a_{p},b_{p+1},\ldots,b_{k})\in\Sigma$. Then (12) holds for every $x_{1},\ldots,x_{n}\in G$. Replacing $x_{n}$ by $x_{n}+g$ we obtain

[TABLE]

where $\overline{b}_{j}=T_{(0,\ldots,0,g)}b_{j}$. It is clear that $\overline{b}_{j}$ only depends on the variables belonging to $\{x_{1},\ldots,x_{n}\}\setminus E_{j}$. Therefore, we have

[TABLE]

and $T_{g}\phi\in F$.

Let $V$ denote the set of functions $\phi-\sum_{i=1}^{p}a_{i}(0)\cdot f_{i}$ such that (13) holds for suitable functions $b_{p+1},\ldots,b_{k}$. It is clear that $V$ is a linear space. If $v\in V$, then (12) gives

[TABLE]

where $\tilde{b}_{j}$ is obtained from $b_{j}$ by putting $x_{n}=0$. Since $E_{j}$ is a proper subset of $\{x_{1},\ldots,x_{n-1}\}$ for every $j=p+1,\ldots,k$, we find that $v(x_{1}+\ldots+x_{n-1})$ is decomposable of order $k-p$. By the induction hypothesis it follows that $V\subset\hbox{GEP}$ and ${\rm deg}\,v\leq k-p$ for every $v\in V$.

Let $W$ denote the linear span of $f_{1},\ldots,f_{p}$. It is clear that every $\phi\in F$ is of the form $w+v$, where $w\in W$ and $v\in V$. Therefore, by Lemma 8, we have $\phi\in\hbox{GEP}$ and ${\rm deg}\,\phi\leq p+(k-p)=k$. Since $f\in F$, the proof is complete. $\square$

Acknowledgement The author was supported by the Hungarian National Foundation for Scientific Research, Grant No. K124749.