Sets of values of equivalent almost periodic functions

Abstract

This paper presents a full generalization of Bohr’s equivalence theorem for the case of almost periodic functions, which improves a recent result that was uniquely formulated in the case of existence of an integral basis for the set of exponents of the associated Dirichlet series.

1 Introduction

In the beginnings of the 20th century, the Danish mathematician Bohr gave important steps in the understanding of general Dirichlet series, which consist of those exponential sums that take the form

$$\begin{aligned} \displaystyle \sum _{n\ge 1}a_ne^{-\lambda _n s},\ a_n\in {\mathbb {C}},\ s=\sigma +it, \end{aligned}$$

where $\{\lambda _n\}$ is a strictly increasing sequence of positive numbers tending to infinity. As a result of his investigations on these functions, he introduced an equivalence relation among them that led to the so-called Bohr’s equivalence theorem, which shows that equivalent Dirichlet series take the same values in certain vertical lines or strips in the complex plane (e.g. see [1, 4, 10, 14]).

On the other hand, Bohr also developed during the 1920s the theory of almost periodic functions, which opened a way to study a wide class of trigonometric series of the general type and even exponential series (see for example [3, 5,6,7,8]). The space of almost periodic functions in a vertical strip $U\subset {\mathbb {C}}$, which will be denoted in this paper as $AP(U,{\mathbb {C}})$, coincides with the set of the functions that can be approximated uniformly in every reduced strip of U by exponential polynomials $a_1e^{\lambda _1s}+a_2e^{\lambda _2s}+\ldots +a_ne^{\lambda _ns}$ with complex coefficients $a_j$ and real exponents $\lambda _j$ (see for example [7, Theorem 3.18]). These approximating finite exponential sums can be found by Bochner–Fejér’s summation (see, in this regard, [3, Chap. 1, Sect. 9]).

Concerning this subject we recall that exponential polynomials and general Dirichlet series constitute a particular family of exponential sums or, in other words, expressions of the type

$$\begin{aligned} P_1(p)e^{\lambda _1p}+\cdots +P_j(p)e^{\lambda _jp}+\cdots , \end{aligned}$$

where the $\lambda _j$’s are complex numbers and the $P_j(p)$’s are polynomials in the parameter p. In this respect, we established in [13, Definition 2] (see also [11, Definition 3]) a generalization of Bohr’s equivalence relation on the classes ${\mathcal {S}}_{\varLambda }$ (which we will refer to as Bohr-equivalence) consisting of exponential sums of the form

$$\begin{aligned} \sum _{j\ge 1}a_je^{\lambda _jp},\ a_j\in {\mathbb {C}},\ \lambda _j\in \varLambda , \end{aligned}$$

(1)

where $\varLambda =\{\lambda _1,\lambda _2,\ldots ,\lambda _j,\ldots \}$ is an arbitrary countable set of distinct real numbers (not necessarily unbounded) that are called exponents or frequencies. Based on this equivalence relation, and under the assumption of existence of an integral basis for the set of exponents (whose condition is defined in Sect. 2), we proved in [13, Theorem 18] that two Bohr-equivalent almost periodic functions, whose associated Dirichlet series could be assumed to have the same set of exponents, take the same values on every open vertical strip included in their strip of almost periodicity U, which constituted a first extension of Bohr’s equivalence theorem for this case of functions.

In this paper, we will consider a new approach to the question of proving that a result analogous to that of Bohr’s equivalence theorem holds in the case of almost periodic functions in $AP(U,{\mathbb {C}})$, not only for those whose set of exponents has an integral basis. The main ingredient is the equivalence relation introduced in Definition 2, denoted as ${\mathop {\sim }\limits ^{*}}$ (and which we will refer to as $^*$-equivalence), on the classes ${\mathcal {S}}_{\varLambda }$ of exponential sums of type (1) and later adapted to the case of the almost periodic functions in $AP(U,{\mathbb {C}})$. Based on this equivalence relation, which is less restrictive than Bohr-equivalence, we will show that every equivalence class in $AP(U,{\mathbb {C}})/{\mathop {\sim }\limits ^{*}}$ is connected with a certain auxiliary function that generates all the sets of values taken by any almost periodic function in the equivalence class along a given vertical line included in the strip of almost periodicity (see Proposition 4 in this paper). This improves [13, Propositions 12 and 13] whose condition of existence of an integral basis was really necessary (see [13, Remark 14]), and it leads us to formulate and prove Theorem 1 (and Corollary 2 for the Bohr-equivalence), which is the main result of this paper and constitutes a full generalization of Bohr’s equivalence theorem for the case of almost periodic functions. Hence with our new equivalence relation it is possible to overcome the restriction of the integral basis (which is not merely a technical difficulty but it is inherently a limit of Bohr’s definition) and to obtain a more general result.

2 Definitions and preliminary results

We first recall the following equivalence relation, inspired by that of [1, p.173] for the case of general Dirichlet series, which was already defined in [11, Definition 1] and [13, Definition 1].

Definition 1

Let $\varLambda$ be an arbitrary countable subset of distinct real numbers, ${\text {span}}_{{\mathbb {Q}}}(\varLambda )$ the ${\mathbb {Q}}$-vector space generated by $\varLambda$, and $\mathcal {F}$ the ${\mathbb {C}}$-vector space of arbitrary functions $\varLambda \rightarrow {\mathbb {C}}$. We define a relation $\sim$ on $\mathcal {F}$ by $a\sim b$ if there exists a ${\mathbb {Q}}$-linear map $\psi :{\text {span}}_{{\mathbb {Q}}}(\varLambda )\rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} b(\lambda )=a(\lambda )e^{i\psi (\lambda )} \quad (\lambda \in \varLambda ). \end{aligned}$$

Concerning the classes $S_\varLambda$ of exponential sums of type (1), consider the following equivalence relation, which was already introduced in [11, Definition 3$^\prime$ (mod.)] and [12, Definition 2], and is defined in terms of the equivalence relation above. From now on, we will denote as $\sharp A$ the cardinal of a set A.

Definition 2

Given $\varLambda =\{\lambda _1,\lambda _2,\ldots ,\lambda _j,\ldots \}$ a set of exponents, consider $A_1(p)$ and $A_2(p)$ two exponential sums in the class ${\mathcal {S}}_{\varLambda }$, say $A_1(p)=\sum _{j\ge 1}a_je^{\lambda _jp}$ and $A_2(p)=\sum _{j\ge 1}b_je^{\lambda _jp}.$ We will say that $A_1$ is $^*$-equivalent to $A_2$, and it will be denoted as $A_1\ {\mathop {\sim }\limits ^{*}}\ A_2$, if for each integer value $n\ge 1$, with $n\le \sharp \varLambda$, it is satisfied $a_{n}^*\sim b_{n}^*$, where $a_{n}^*,b_{n}^*:\{\lambda _1,\lambda _2,\ldots ,\lambda _{n}\}\rightarrow {\mathbb {C}}$ are the functions given by $a_{n}^*(\lambda _j):=a_j$ and $b_{n}^*(\lambda _j):=b_j$, $j=1,2,\ldots ,n$ and $\sim$ is in Definition 1.

That is, we will write $A_1\ {\mathop {\sim }\limits ^{*}} A_2$ if for each integer value $n\ge 1$, with $n\le \sharp \varLambda$, there exists a ${\mathbb {Q}}$-linear map $\psi _n:{\text {span}}_{{\mathbb {Q}}}(\{\lambda _1,\ldots ,\lambda _n\})\rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} b_j=a_je^{i\psi _n(\lambda _j)},\ j=1,\ldots ,n. \end{aligned}$$

It is clear that the relation defined in the foregoing definition is an equivalence relation.

Remark 1

It is worth noting that the equivalence relation that was used in [13, Definition 2] is different from the $^*$-equivalence (i.e. that of Definition 2 of this paper). In fact, fixed $\varLambda =\{\lambda _1,\lambda _2,\ldots ,\lambda _j,\ldots \}$ a set of exponents, and given $A_1(p)=\sum _{j\ge 1}a_je^{\lambda _jp}$ and $A_2(p)=\sum _{j\ge 1}b_je^{\lambda _jp}$ two exponential sums in the class ${\mathcal {S}}_{\varLambda }$, [13, Definition 2] consists of defining $A_1\sim A_2$ (we will say in this paper that $A_1$ is Bohr-equivalent to $A_2$) if there exists a ${\mathbb {Q}}$-linear map $\psi :{\text {span}}_{{\mathbb {Q}}}(\varLambda )\rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} b_j=a_je^{i\psi (\lambda _j)},\ j=1,2,\ldots \end{aligned}$$

As an immediate consequence of this definition, we have that if two exponential sums are Bohr-equivalent then they also are $^*$-equivalent (according to Definition 2).

Now, let $G_{\varLambda }=\{g_1, g_2,\ldots , g_k,\ldots \}$ be a basis of the ${\mathbb {Q}}$-vector space generated by a set $\varLambda =\{\lambda _1,\lambda _2,\ldots \}$ of exponents (by abuse of notation, we will say that $G_{\varLambda }$ is a basis for $\varLambda$), which yields that $G_{\varLambda }$ is linearly independent over the rational numbers and each $\lambda _j$ is expressible as a finite linear combination of terms of $G_{\varLambda }$, say

$$\begin{aligned} \lambda _j=\sum _{k=1}^{i_j}r_{j,k}g_k \quad \text{ for } \text{ some } r_{j,k}\in {\mathbb {Q}},\ i_j\in {\mathbb {N}}. \end{aligned}$$

(2)

We will say that $G_{\varLambda }$ is an integral basis for $\varLambda$ when $r_{j,k}\in {\mathbb {Z}}$ for each j, k, i.e. $\varLambda \subset {\text {span}}_{{\mathbb {Z}}}(G_{\varLambda })$. Moreover, we will say that $G_{\varLambda }$ is the natural basis for $\varLambda$, and we will denote it as $G_{\varLambda }^*$, when it is constituted by elements in $\varLambda$ as follows. Firstly, if $\lambda _1\ne 0$, then $g_1:=\lambda _1\in G_{\varLambda }^*$. Secondly, if $\{\lambda _1,\lambda _2\}$ are ${\mathbb {Q}}$-rationally independent, then $g_2:=\lambda _2\in G_{\varLambda }^*$. Otherwise, if $\{\lambda _1,\lambda _3\}$ are ${\mathbb {Q}}$-rationally independent, then $g_2:=\lambda _3\in G_{\varLambda }^*$, and so on. In this way, if $\lambda _j\in G_{\varLambda }^*$, then $r_{j,m_j}=1$ and $r_{j,k}=0$ for $k\ne m_j$, where $m_j$ is such that $g_{m_j}=\lambda _j$. In fact, each element in $G_{\varLambda }^*$ is of the form $g_{m_j}$ for j such that $\lambda _j$ is ${\mathbb {Q}}$-linear independent of the previous elements in the basis. Furthermore, if $\lambda _j\notin G_{\varLambda }^*$, then $\lambda _j=\sum _{k=1}^{i_j}r_{j,k}g_k$, where $\{g_{1},g_{2},\ldots ,g_{i_j}\}\subset \{\lambda _1,\lambda _2,\ldots ,\lambda _{j-1}\}$.

In terms of a prefixed basis for the set of exponents $\varLambda$, we next quote a first characterization of the $^*$-equivalence of two exponential sums in ${\mathcal {S}}_{\varLambda }$ (see [11, Proposition 1’ (mod.)] or [12, Proposition 1]).

Proposition 1

Given $\varLambda =\{\lambda _1,\lambda _2,\ldots ,\lambda _j,\ldots \}$ a set of exponents, consider $A_1(p)$ and $A_2(p)$ two exponential sums in the class ${\mathcal {S}}_{\varLambda }$, say $A_1(p)=\sum _{j\ge 1}a_je^{\lambda _jp}$ and $A_2(p)=\sum _{j\ge 1}b_je^{\lambda _jp}.$ Fixed a basis $G_{\varLambda }$ for $\varLambda$, for each $j=1,2,\ldots$ let $\mathbf {r}_j\in {\mathbb {R}}^{\sharp G_{\varLambda }}$ be the vector of rational components verifying (2). Then $A_1\ {\mathop {\sim }\limits ^{*}} A_2$ if and only if for each integer value $n\ge 1$, with $n\le \sharp \varLambda$, there exists a vector $\mathbf {x}_n=(x_{n,1},x_{n,2},\ldots ,x_{n,k},\ldots )\in {\mathbb {R}}^{\sharp G_{\varLambda }}$ such that $b_j=a_j e^{<\mathbf {r}_j,\mathbf {x}_n>i}$ for $j=1,2,\ldots ,n$.

Furthermore, if $G_{\varLambda }$ is an integral basis for $\varLambda$, then the following three statements are equivalent:

• (i) $A_1\ {\mathop {\sim }\limits ^{*}} A_2$;

• (ii) $A_1\sim A_2$;

• (iii) There exists $\mathbf {x}_0=(x_{0,1},x_{0,2},\ldots ,x_{0,k},\ldots )\in {\mathbb {R}}^{\sharp G_{\varLambda }}$ such that $b_j=a_j e^{\langle \mathbf {r}_j,\mathbf {x}_0\rangle i}$ for each $j\ge 1$.

From Proposition 1, it is now clear that Definition 2 and the definition of Bohr-equivalence (see Remark 1) are equivalent in the case that it is feasible to obtain an integral basis for the set of exponents $\varLambda$.

In terms of the natural basis for a set of exponents $\varLambda$, we next provide a second characterization of the $^*$-equivalence of two exponential sums in ${\mathcal {S}}_{\varLambda }$.

Proposition 2

Given $\varLambda =\{\lambda _1,\lambda _2,\ldots ,\lambda _j,\ldots \}$ a set of exponents, consider $A_1(p)$ and $A_2(p)$ two exponential sums in the class ${\mathcal {S}}_{\varLambda }$, say $A_1(p)=\sum _{j\ge 1}a_je^{\lambda _jp}$ and $A_2(p)=\sum _{j\ge 1}b_je^{\lambda _jp}.$ Fixed the natural basis $G_{\varLambda }^*=\{g_1,g_2,\ldots ,g_k,\ldots \}$ for $\varLambda$, for each $j=1,2,\ldots$ let $\mathbf {r}_j\in {\mathbb {R}}^{\sharp G_{\varLambda }^*}$ be the vector of rational components verifying (2). Then $A_1\ {\mathop {\sim }\limits ^{*}}\ A_2$ if and only if there exists $\mathbf {x}_0=(x_{0,1},x_{0,2},\ldots ,x_{0,k},\ldots )\in [0,2\pi )^{\sharp G_{\varLambda }^*}$ such that for each $j=1,2,\ldots$ it is satisfied

$$\begin{aligned} b_j=a_j e^{\langle \mathbf {r}_j,\mathbf {x}_0+\mathbf {p}_j\rangle i} \end{aligned}$$

for some $\mathbf {p}_j=(2\pi n_{j,1},2\pi n_{j,2},\ldots )\in {\mathbb {R}}^{\sharp G_{\varLambda }^*}$, with $n_{j,k}\in {\mathbb {Z}}$.

Proof

Suppose that $A_1\ {\mathop {\sim }\limits ^{*}} A_2$. Consider $I=\{1,2,\ldots ,k,\ldots : \lambda _k\in G_{\varLambda }^*\}$ and $I_n=\{1,2,\ldots ,k,\ldots ,n: \lambda _k\in G_{\varLambda }^*\}$. Let $j\in I$, then $r_{j,m_j}=1$ and $r_{j,k}=0$ for $k\ne m_j$, where $m_j$ is such that $g_{m_j}=\lambda _j$. Thus, by Proposition 1, let $\mathbf {x}_j=(x_{j,1},x_{j,2},\ldots )\in {\mathbb {R}}^{\sharp G_{\varLambda }^*}$ be a vector such that

$$\begin{aligned} b_j=a_j e^{i\langle \mathbf {r}_j,\mathbf {x}_j\rangle }=a_j e^{i\sum _{k=1}^{i_j}r_{j,k}x_{j,k}}=a_je^{ir_{j,m_j}x_{j,m_j}}=a_je^{ix_{j,m_j}}. \end{aligned}$$

(3)

Define $\mathbf {x}_0=(x_{0,1},x_{0,2},\ldots )\in {\mathbb {R}}^{\sharp G_{\varLambda }^*}={\mathbb {R}}^{\sharp I}$ as $x_{0,m_j}:=x_{j,m_j}$ for $j\in I$. Thus, by taking $\mathbf {p}_j=(0,0,\ldots )$, the result trivially holds for those j’s such that $\lambda _j\in G_{\varLambda }^*$, i.e. for $j\in I$. Now, let j be such that $\lambda _j\notin G_{\varLambda }^*$, i.e. $j\notin I$. By Proposition 1, let $\mathbf {x}_j=(x_{j,1},x_{j,2},\ldots )\in {\mathbb {R}}^{\sharp G_{\varLambda }^*}$ be a vector such that

$$\begin{aligned} b_p=a_p e^{i\langle \mathbf {r}_p,\mathbf {x}_j\rangle }=a_p e^{i\sum _{k=1}^{i_j}r_{p,k}x_{j,k}},\ p=1,2,\ldots ,j. \end{aligned}$$

Note that if $p=1,2,\ldots ,j$ is such that $\lambda _p\in G_{\varLambda }^*$, then

$$\begin{aligned} b_p=a_pe^{ir_{p,m_p}x_{j,m_p}}=a_pe^{ix_{j,m_p}}, \end{aligned}$$

which necessarily implies, by (3), that $x_{j,m_p}=x_{p,m_p}+2\pi n_{j,p}$ for some $n_{j,p}\in {\mathbb {Z}}$. Hence

$$\begin{aligned} b_j= & {} a_j e^{i\langle \mathbf {r}_j,\mathbf {x}_j\rangle }=a_j e^{i\sum _{k=1}^{i_j}r_{j,k}x_{j,k}}=a_j e^{i\sum _{p\in I_{j-1}}r_{j,m_p}x_{j,m_p}}\\= & {} a_j e^{i\sum _{p\in I_{j-1}}r_{j,m_p}(x_{p,m_p}+2\pi n_{j,p})}=a_j e^{i\langle \mathbf {r}_j,\mathbf {x}_0+\mathbf {p}_j\rangle }, \end{aligned}$$

where $\mathbf {p}_j=(2\pi n_{j,1},2\pi n_{j,2},\ldots ,0,0,\ldots )$. Moreover, by changing conveniently the vectors $\mathbf {p}_j$, we can take $\mathbf {x}_0\in [0,2\pi )^{\sharp G_{\varLambda }^*}$ without loss of generality.

Conversely, suppose the existence of $\mathbf {x}_0=(x_{0,1},x_{0,2},\ldots ,x_{0,k},\ldots )\in {\mathbb {R}}^{\sharp G_{\varLambda }^*}$ satisfying $b_j=a_j e^{\langle \mathbf {r}_j,\mathbf {x}_0+\mathbf {p}_j\rangle i}$ for some $\mathbf {p}_j=(2\pi n_{j,1},2\pi n_{j,2},\ldots )\in {\mathbb {R}}^{\sharp G_{\varLambda }^*}$, with $n_{j,k}\in {\mathbb {Z}}$. Let $r_{j,k}=\frac{p_{j,k}}{q_{j,k}}$ with $p_{j,k}$ and $q_{j,k}$ coprime integer numbers, and define $q_{n,k}:={\text {lcm}}(q_{1,k},q_{2,k},\ldots ,q_{n,k})$ for each $k=1,2,\ldots$. Thus, for each integer number $n\ge 1$, take $\mathbf {x}_n=\mathbf {x}_0+\mathbf {m}_n$, where $m_{n,k}=2\pi p_{1,k}p_{2,k}\cdots p_{n,k}q_{n,k}$, $k=1,2,\ldots$. Therefore, it is satisfied $b_j=a_j e^{\langle \mathbf {r}_j,\mathbf {x}_n\rangle i}$ for each $j=1,2,\ldots ,n$, which yields that $A_1\ {\mathop {\sim }\limits ^{*}}\ A_2$.

We next study the case where the chosen basis is not the natural one. Fixed a set $\varLambda =\{\lambda _1,\lambda _2,\ldots \}$ of exponents, let $G_{\varLambda }^*$ be the natural basis for $\varLambda$ and $G_{\varLambda }$ be an arbitrary basis for $\varLambda$. For each $j\ge 1$ let $\mathbf {r}_j$ and $\mathbf {s}_j$ be the vectors of rational components so that $\lambda _j=\langle \mathbf {r}_j,\mathbf {g}\rangle$ and $\lambda _j=\langle \mathbf {s}_j,\mathbf {h}\rangle$, with $\mathbf {g}$ and $\mathbf {h}$ the vectors associated with the basis $G_{\varLambda }^*$ and $G_{\varLambda }$, respectively. Finally, for each $k\ge 1$, let $\mathbf {t}_k$ be the vector so that $h_k=\langle \mathbf {t}_k,\mathbf {g}\rangle$, i.e.

$$\begin{aligned} T=\begin{pmatrix} t_{1,1} &{}\quad t_{1,2} &{}\quad \cdots &{}\quad t_{1,j} &{}\quad \cdots \\ t_{2,1} &{}\quad t_{2,2} &{}\quad \cdots &{}\quad t_{2,j} &{}\quad \cdots \\ \vdots &{}\quad \dots &{}\quad \ddots &{}\quad \vdots &{}\quad \cdots \\ t_{k,1} &{}\quad t_{k,2} &{}\quad \cdots &{}\quad \quad t_{k,j} &{}\quad \cdots \\ \vdots &{}\quad \dots &{}\quad \ddots &{}\quad \vdots &{}\quad \cdots \\ \end{pmatrix} \end{aligned}$$

(4)

is the change of basis matrix. Thus, for any $\mathbf {x}_0\in {\mathbb {R}}^{\sharp G_{\varLambda }}$, we have

$$\begin{aligned} \langle \mathbf {r}_j,\mathbf {x}_0\rangle =\langle \mathbf {s}_j,\mathbf {x}_1\rangle , \end{aligned}$$

where $\mathbf {x}_1$ is defined as $x_{1,k}=\langle \mathbf {t}_k,\mathbf {x}_0\rangle$ for each $k\ge 1$. Indeed,

$$\begin{aligned}&\langle \mathbf {s}_j,\mathbf {x}_1\rangle =\sum _{k}s_{j,k}x_{1,k}=\sum _k s_{j,k}\langle \mathbf {t}_k,\mathbf {x}_0\rangle = \sum _k s_{j,k}\sum _m t_{k,m}x_{0,m}\\&\quad \quad \quad \quad =\sum _mx_{0,m}\sum _k s_{j,k} t_{k,m}=\sum _m r_{j,m}x_{0,m}=\langle \mathbf {r}_j,\mathbf {x}_0\rangle \end{aligned}$$

because $r_{j,m}=\sum _k s_{j,k}t_{k,m}$. Consequently, for each j and

$$\begin{aligned} \mathbf {p}_j=(2\pi n_{j,1},2\pi n_{j,2},\ldots )\in {\mathbb {R}}^{\sharp G_{\varLambda }} \text{ with } n_{j,k}\in {\mathbb {Z}}, \end{aligned}$$

(5)

we have

$$\begin{aligned}&\langle \mathbf {r}_j,\mathbf {x}_0+\mathbf {p}_j\rangle =\langle \mathbf {r}_j,\mathbf {x}_0\rangle +\langle \mathbf {r}_j,\mathbf {p}_j\rangle =\langle \mathbf {s}_j,\mathbf {x}_1\rangle +\langle \mathbf {s}_j,\mathbf {q}_j\rangle =\langle \mathbf {s}_j, \mathbf {x}_1+\mathbf {q}_j\rangle , \end{aligned}$$

where $\mathbf {q}_j$ is defined as $q_{1,k}=\langle \mathbf {t}_k,\mathbf {p}_j\rangle$ for each $k\ge 1$, i.e. $\mathbf {q}_j$ is obtained from $T\cdot \mathbf {p}_j^t$. In this way, we have proved the following result.

Corollary 1

Given $\varLambda =\{\lambda _1,\lambda _2,\ldots ,\lambda _j,\ldots \}$ a set of exponents, consider $A_1(p)$ and $A_2(p)$ two exponential sums in the class ${\mathcal {S}}_{\varLambda }$, say $A_1(p)=\sum _{j\ge 1}a_je^{\lambda _jp}$ and $A_2(p)=\sum _{j\ge 1}b_je^{\lambda _jp}.$ Fixed a basis $G_{\varLambda }=\{g_1,g_2,\ldots ,g_k,\ldots \}$ for $\varLambda$, for each $j=1,2,\ldots$ let $\mathbf {r}_j\in {\mathbb {R}}^{\sharp G_{\varLambda }}$ be the vector of rational components verifying (2). Then $A_1\ {\mathop {\sim }\limits ^{*}}\ A_2$ if and only if there exists $\mathbf {x}_0=(x_{0,1},x_{0,2},\ldots ,x_{0,k},\ldots )\in [0,2\pi )^{\sharp G_{\varLambda }}$ such that for each $j=1,2,\ldots$ it is satisfied

$$\begin{aligned} b_j=a_j e^{\langle \mathbf {r}_j,\mathbf {x}_0+\mathbf {q}_j\rangle i} \end{aligned}$$

for some $\mathbf {q}_j\in {\mathbb {R}}^{\sharp G_{\varLambda }}$ that is of the form $\mathbf {q}_j=\mathbf {p}_j\cdot T^t$, where $T^t$ is the transpose of the change of basis matrix (4) and $\mathbf {p}_j$ is of the form (5).

We next construct a generating expression for all exponential polynomials in a class ${\mathcal {G}}\in {\mathcal {S}}_{\varLambda }/{\mathop {\sim }\limits ^{*}}$. Let $2\pi {\mathbb {Z}}^{m}=\{(c_1,c_2,\ldots ,c_m)\in {\mathbb {R}}^{m}:c_k=2\pi n_k, \text{ with } n_k\in {\mathbb {Z}},\ k=1,2,\ldots ,m\}$. From Proposition 2, it is clear that the set of all exponential sums A(p) in an equivalence class ${\mathcal {G}}$ in ${\mathcal {S}}_{\varLambda }/{\mathop {\sim }\limits ^{*}}$ can be determined by a function $E_{{\mathcal {G}}}:[0,2\pi )^{\sharp G_{\varLambda }^*}\times \prod _{j\ge 1}2\pi {\mathbb {Z}}^{\sharp G_{\varLambda }^*}\rightarrow {\mathcal {S}}_{\varLambda }$ of the form

$$\begin{aligned} E_{{\mathcal {G}}}(\mathbf {x},\mathbf {p}_1,\mathbf {p}_2,\ldots ):=\sum _{j\ge 1}a_je^{\langle \mathbf {r}_j,\mathbf {x}+\mathbf {p}_j\rangle i}e^{\lambda _jp}\text {, } \mathbf {x}\in [0,2\pi )^{\sharp G_{\varLambda }^*},\ \mathbf {p}_j\in 2\pi {\mathbb {Z}}^{\sharp G_{\varLambda }^*}, \end{aligned}$$

(6)

where $a_1,a_2,\ldots ,a_j,\ldots$ are the coefficients of an exponential sum in ${\mathcal {G}}$ and the $\mathbf {r}_j$’s are the vectors of rational components associated with the natural basis $G_{\varLambda }^*$ for $\varLambda$. We recommend the reader compare the definition of the function $E_{{\mathcal {G}}}$ with that of [13, Expression (2.2)].

In particular, in this paper we are going to use Definition 2 for the case of exponential sums in ${\mathcal {S}}_{\varLambda }$ of a complex variable $s=\sigma +it$. Precisely, when the formal series in ${\mathcal {S}}_{\varLambda }$ are handled as exponential sums of a complex variable on which we fix a summation procedure, from equivalence class generating expression (6) we can consider an auxiliary function as follows (compare with [13, Definition 3]).

Definition 3

Fixed $\varLambda =\{\lambda _1,\lambda _2,\ldots ,\lambda _j,\ldots \}$ a set of frequencies or exponents, let ${\mathcal {G}}$ be an equivalence class in ${\mathcal {S}}_{\varLambda }/{\mathop {\sim }\limits ^{*}}$ and $a_1,a_2,\ldots ,a_j,\ldots$ be the coefficients of an exponential sum in ${\mathcal {G}}$. For each $j=1,2,\ldots$ let $\mathbf {r}_j$ be the vector of rational components satisfying $\lambda _j=\langle \mathbf {r}_j,\mathbf {g}\rangle =\sum _{k=1}^{q_j}r_{j,k}g_k$, where $\mathbf {g}:=(g_1,\ldots ,g_k,\ldots )$ is the vector formed by the elements of the natural basis $G_{\varLambda }^*$ for $\varLambda$. Suppose that some elements in ${\mathcal {G}}$, handled as exponential sums of a complex variable $s=\sigma +it$, are summable on at least a certain set $P\subset {\mathbb {R}}$ by some prefixed summation method. Then we define the auxiliary function $F_{{\mathcal {G}}}: P \times [0,2\pi ) ^{\sharp G_{\varLambda }^*}\times \prod _{j\ge 1}2\pi {\mathbb {Z}}^{\sharp G_{\varLambda }^*}\rightarrow {\mathbb {C}}$ associated with ${\mathcal {G}}$, relative to the basis $G_{\varLambda }^*$, as

$$\begin{aligned} F_{{\mathcal {G}}}(\sigma ,\mathbf {x},\mathbf {p}_1,\mathbf {p}_2,\ldots ):=\sum _{j\ge 1}a_je^{\langle \mathbf {r}_j,\mathbf {x}+\mathbf {p}_j\rangle i}e^{\lambda _j\sigma }\text {, } \end{aligned}$$

(7)

where $\sigma \in P,\ \mathbf {x}\in [0,2\pi )^{\sharp G_{\varLambda }^*}$, $\mathbf {p}_k\in 2\pi {\mathbb {Z}}^{\sharp G_{\varLambda }^*},$ and the series in (7) is summed by the prefixed summation method, applied at $t=0$ to the exponential sum obtained from the generating expression (6) with $p=\sigma +it$.

This auxiliary function can be immediately adapted to the case of almost periodic functions $AP(U,{\mathbb {C}})$ with the Bochner–Fejér summation method. In this case, the set P above is formed by the real projection of the strip of almost periodicity of the corresponding exponential sums (see Definition 5). In this theoretical framework, we will show later the strong link among the sets of values in the complex plane taken by a function in $AP(U,{\mathbb {C}})$, its Dirichlet series and its associated auxiliary function.

Definition 4

Consider $\varLambda =\{\lambda _1,\lambda _2,\ldots ,\lambda _j,\ldots \}$ a countable set of distinct real numbers. We will say that a function $f:U\subset {\mathbb {C}}\rightarrow {\mathbb {C}}$ is in the class $\mathcal {D}_{\varLambda }$ if it is almost periodic (in the set $AP(U,{\mathbb {C}})$) and its associated Dirichlet series is of the form

$$\begin{aligned} \sum _{j\ge 1}a_je^{\lambda _js},\ a_j\in {\mathbb {C}},\ \lambda _j\in \varLambda , \end{aligned}$$

(8)

where U is a strip of the type $\{s\in {\mathbb {C}}: \alpha<{\text {Re}}s<\beta \}$, with $-\infty \le \alpha <\beta \le \infty$.

Every almost periodic function in $AP(U,{\mathbb {C}})$ is determined by its Dirichlet series, which is of type (8). In fact it is convenient to remark that, even in the case that the sequence of the partial sums of its Dirichlet series does not converge uniformly, there exists a sequence of finite exponential sums, the Bochner–Fejér polynomials, of the type $P_k(s)=\sum _{j\ge 1}p_{j,k}a_je^{\lambda _js}$ where for each k only a finite number of the factors $p_{j,k}$ differ from zero, which converges uniformly to f in every reduced strip in U and converges formally to the Dirichlet series [3, Polynomial approximation theorem, pgs. 50,148].

Moreover, the equivalence relation of Definition 2 can be immediately adapted to the case of the functions (or classes of functions) which are identifiable by their also called Dirichlet series, in particular to the classes $\mathcal {D}_{\varLambda }$. More specifically, see [11, Sect. 4, Definition 5$^\prime$ (mod.)] referred to the Besicovitch space which contains the classes of functions which are associated with Fourier or Dirichlet series and for which the extension of our equivalence relation makes sense.

3 The auxiliary functions

If we take Definition 3 as a reference to be applied to our particular case of almost periodic functions with the Bochner–Fejér summation method, notice that to every function $f\in \mathcal {D}_{\varLambda }$, with $\varLambda$ an arbitrary set of exponents, we can associate an auxiliary function $F_f$ of countably many real variables as follows.

Definition 5

Given $\varLambda =\{\lambda _1,\lambda _2,\ldots ,\lambda _j,\ldots \}$ a set of exponents, let $f(s)\in \mathcal {D}_{\varLambda }$ be an almost periodic function in the vertical strip $\{s\in {\mathbb {C}}:\alpha<{\text {Re}}s<\beta \}$, $-\infty \le \alpha <\beta \le \infty$, whose Dirichlet series is given by $\sum _{j\ge 1}a_je^{\lambda _js}$. For each $j=1,2,\ldots$ let $\mathbf {r}_j$ be the vector of rational components satisfying the equality $\lambda _j=\langle \mathbf {r}_j,\mathbf {g}\rangle =\sum _{k=1}^{q_j}r_{j,k}g_k$, where $\mathbf {g}:=(g_1,\ldots ,g_k,\ldots )$ is the vector formed by the elements of the natural basis $G_{\varLambda }^*$ for $\varLambda$. We define the auxiliary function $F_f: (\alpha ,\beta ) \times [0,2\pi ) ^{\sharp G_{\varLambda }^*}\times \prod _{j\ge 1}2\pi {\mathbb {Z}}^{\sharp G_{\varLambda }^*}\rightarrow {\mathbb {C}}$ associated with f, relative to the basis $G_{\varLambda }^*$, as

$$\begin{aligned} F_{f}(\sigma ,\mathbf {x},\mathbf {p}_1,\mathbf {p}_2,\ldots ):=\sum _{j\ge 1}a_j e^{\lambda _j\sigma }e^{\langle \mathbf {r}_j,\mathbf {x}+\mathbf {p}_j\rangle i}\text {, } \end{aligned}$$

(9)

where $\sigma \in (\alpha ,\beta ) \text {, }\mathbf {x}\in [0,2\pi )^{\sharp G_{\varLambda }^*},\ \mathbf {p}_j\in 2\pi {\mathbb {Z}}^{\sharp G_{\varLambda }^*}$ and the series in (9) is summed by Bochner–Fejér procedure, applied at $t=0$ to the sum $\sum _{j\ge 1}a_j e^{\langle \mathbf {r}_j,\mathbf {x}+\mathbf {p}_j\rangle i}e^{\lambda _js}$.

Given $f\in AP(U,{\mathbb {C}})$, it was proved in [11, Lemma 3] that every function in its equivalence class is also included in $AP(U,{\mathbb {C}})$. What is more, if $\sum _{j\ge 1}a_j e^{\lambda _js}$ is the Dirichlet series of $f(s)\in AP(U,{\mathbb {C}})$, we first note that, for every choice of $\mathbf {x}\in [0,2\pi )^{\sharp G_{\varLambda }}$ and $\mathbf {p}_j\in 2\pi {\mathbb {Z}}^{\sharp G_{\varLambda }}$ with $j=1,2,\ldots$, the sum $\sum _{j\ge 1}a_j e^{\langle \mathbf {r}_j,\mathbf {x}+\mathbf {p}_j\rangle i}e^{\lambda _js}$ represents the Dirichlet series of an almost periodic function. We second note that if the Dirichlet series of $f(s)\in AP(U,{\mathbb {C}})$ converges uniformly on $U=\{s\in {\mathbb {C}}:\alpha<{\text {Re}}s<\beta \}$, then f(s) coincides with its Dirichlet series and (9) can be viewed as summation by partial sums or ordinary summation.

In addition to this, we third note that the Dirichlet series $\sum _{j\ge 1}a_je^{\lambda _js}$, associated with a certain function $f\in \mathcal {D}_{\varLambda }$, arises from its auxiliary function $F_f$ by a special choice of its variables, that is $F_f(\sigma ,t\mathbf {g},\mathbf {0},\mathbf {0},\ldots )=\sum _{j\ge 1}a_je^{\lambda _j(\sigma +it)}$.

In this respect, under the assumption that the natural basis for the set of the exponents is also an integral basis, it is clear that the vectors $\mathbf {p}_j$ do not play any role and hence the auxiliary function $F_f$, associated with f, can be taken as $F_{f}(\sigma ,\mathbf {x}):=\sum _{j\ge 1}a_j e^{\lambda _j\sigma }e^{\langle \mathbf {r}_j,\mathbf {x}\rangle i}$, $\sigma \in (\alpha ,\beta ) \text {, }\mathbf {x}\in [0,2\pi )^{\sharp G_{\varLambda }^*}$.

In general, by taking into account Corollary 1, Definition 5 can be adapted to the case of an arbitrary basis for the set of exponents. For this purpose, given a basis $G_{\varLambda }$ for $\varLambda$, let T be the change of basis matrix (4), with respect to the natural basis, and let

$$\begin{aligned} S_T=\{\mathbf {q}\in {\mathbb {R}}^{\sharp G_{\varLambda }}: \mathbf {q}=\mathbf {p}\cdot T^t,\ \text{ with } \mathbf {p} \text{ of } \text{ the } \text{ form } (5)\}. \end{aligned}$$

Definition 6

Given $\varLambda =\{\lambda _1,\lambda _2,\ldots ,\lambda _j,\ldots \}$ a set of exponents, let $f(s)\in \mathcal {D}_{\varLambda }$ be an almost periodic function in $\{s\in {\mathbb {C}}:\alpha<{\text {Re}}s<\beta \}$, $-\infty \le \alpha <\beta \le \infty$, whose Dirichlet series is given by $\sum _{j\ge 1}a_je^{\lambda _js}$. For each $j\ge 1$ let $\mathbf {s}_j$ be the vector of rational components satisfying the equality $\lambda _j=\langle \mathbf {s}_j,\mathbf {g}\rangle =\sum _{k=1}^{q_j}s_{j,k}g_k$, where $\mathbf {g}:=(g_1,\ldots ,g_k,\ldots )$ is the vector of the elements of an arbitrary basis $G_{\varLambda }$ for $\varLambda$. Then we define the auxiliary function $F_f^{G_{\varLambda }}: (\alpha ,\beta ) \times [0,2\pi ) ^{\sharp G_{\varLambda }}\times \prod _{j\ge 1}S_T\rightarrow {\mathbb {C}}$ associated with f, relative to the basis $G_{\varLambda }$, as

$$\begin{aligned} F_f^{G_{\varLambda }}(\sigma ,\mathbf {x},\mathbf {q}_1,\mathbf {q}_2,\ldots ):=\sum _{j\ge 1}a_j e^{\lambda _j\sigma }e^{\langle \mathbf {s}_j,\mathbf {x}+\mathbf {q}_j\rangle i}\text {, } \end{aligned}$$

(10)

where $\sigma \in (\alpha ,\beta ) \text {, }\mathbf {x}\in [0,2\pi )^{\sharp G_{\varLambda }}$, $\mathbf {q}_j\in S_T$, and the series in (10) is summed by Bochner–Fejér procedure, applied at $t=0$ to the sum $\sum _{j\ge 1}a_j e^{\langle \mathbf {r}_j,\mathbf {x}\rangle i}e^{\lambda _js}$.

If we take the natural basis, it is obvious that the auxiliary functions $F_f$ and $F_f^{G_{\varLambda }^*}$ of the respective Definitions 5 and 6 coincide.

We next show a characterization of the property of $^*$-equivalence of functions in the classes $\mathcal {D}_{\varLambda }$ in terms of the auxiliary function relative to the natural basis.

Proposition 3

Given $\varLambda =\{\lambda _1,\lambda _2,\ldots ,\lambda _j,\ldots \}$ a set of exponents, let $f_1$ and $f_2$ be two almost periodic functions in the class $\mathcal {D}_{\varLambda }$. Let $\mathbf {g}:=(g_1,g_2,\ldots ,g_k,\ldots )$ be the vector of the elements of the natural basis $G_{\varLambda }^*$ for $\varLambda$. Then $f_1$ is $^*$-equivalent to $f_2$ if and only if there exist some $\mathbf {y}\in {\mathbb {R}}^{\sharp G_\varLambda ^*}$ and $\mathbf {p}_j\in 2\pi {\mathbb {Z}}^{\sharp G_{\varLambda }^*}$, $j=1,2,\ldots$, such that

$$\begin{aligned} f_2(\sigma +it)=F_{f_1}(\sigma ,\mathbf {y}+t\mathbf {g},\mathbf {p}_1,\mathbf {p}_2,\ldots ) \quad \ \text{ for } \text{ all } \quad \sigma +it\in U. \end{aligned}$$

Proof

Let $\sum _{j\ge 1}a_je^{\lambda _js}$ and $\sum _{j\ge 1}b_je^{\lambda _js}$ be the Dirichlet series associated with $f_1$ and $f_2$, respectively. Let U be an open vertical strip such that $f_2\in AP(U,{\mathbb {C}})$. If $f_1\ {\mathop {\sim }\limits ^{*}}\ f_2$, then Proposition 2 assures the existence of $\mathbf {x}_0\in [0,2\pi )^{\sharp G_\varLambda ^*}$ such that for each $j=1,2,\ldots$ it is satisfied $b_j=a_j e^{\langle \mathbf {r}_j,\mathbf {x}_0+\mathbf {p}_j'\rangle i}$ for some $\mathbf {p}_j'\in 2\pi {\mathbb {Z}}^{\sharp G_\varLambda ^*}$. Now, let $P_k(s)=\sum _{j\ge 1}p_{j,k}b_je^{\lambda _js}$, $k=1,2,\ldots$, be the Bochner–Fejér polynomials associated with $f_2$ (recall that they converge uniformly to $f_2$ in every reduced strip in U, which yields $p_{j,k}\rightarrow 1$ as k goes to $\infty$). Thus, given $s=\sigma +it\in U$, we have

$$\begin{aligned} f_2(\sigma +it)= & {} \lim _{k\rightarrow \infty }P_k(\sigma +it)=\lim _{k\rightarrow \infty }\sum _{j\ge 1}p_{j,k}b_je^{\lambda _j(\sigma +it)}\\= & {} \lim _{k\rightarrow \infty }\sum _{j\ge 1}p_{j,k}a_je^{i\langle \mathbf {r}_j,\mathbf {x}_0+\mathbf {p}_j'\rangle }e^{\lambda _j\sigma }e^{i\lambda _j t}\\= & {} \lim _{k\rightarrow \infty }\sum _{j\ge 1}p_{j,k}a_je^{\lambda _j\sigma }e^{i\langle \mathbf {r}_j,\mathbf {x}_0+\mathbf {p}_j'\rangle }e^{it\langle \mathbf {r}_j,\mathbf {g}\rangle }\\= & {} \lim _{k\rightarrow \infty }\sum _{j\ge 1}p_{j,k}a_je^{\lambda _j\sigma }e^{i\langle \mathbf {r}_j,\mathbf {x}_0+\mathbf {p}_j'+t\mathbf {g}\rangle }\\= & {} F_{f_1}(\sigma ,\mathbf {y}_0+t\mathbf {g},\mathbf {p}_1,\mathbf {p}_2,\ldots ), \end{aligned}$$

where $\mathbf {y}_0\in {\mathbb {R}}^{\sharp G_\varLambda ^*}$ and $\mathbf {p}_j\in 2\pi {\mathbb {Z}}^{\sharp G_{\varLambda }^*}$ are chosen so that $\mathbf {x}_0+t\mathbf {g}+\mathbf {p}_j'=\mathbf {y}_0+t\mathbf {g}+\mathbf {p}_j$, with $\mathbf {y}_0+t\mathbf {g}\in [0,2\pi )^{\sharp G_\varLambda ^*}$.

Conversely, suppose the existence of $\mathbf {y}_0\in {\mathbb {R}}^{\sharp G_\varLambda }$ and $\mathbf {p}_j\in 2\pi {\mathbb {Z}}^{\sharp G_{\varLambda }^*}$, for $j=1,2,\ldots$, such that $f_2(\sigma +it)=F_{f_1}(\sigma ,\mathbf {y}_0+t\mathbf {g},\mathbf {p}_1,\mathbf {p}_2,\ldots )$ for any $\sigma +it\in U$. Hence

$$\begin{aligned}&\lim _{k\rightarrow \infty }\sum _{j\ge 1}p_{j,k}b_je^{\lambda _j(\sigma +it)}=F_{f_1}(\sigma ,\mathbf {y}_0+t\mathbf {g},\mathbf {p}_1,\mathbf {p}_2,\ldots )\\&\quad \quad =\sum _{j\ge 1}a_je^{i\langle \mathbf {r}_j,\mathbf {y}_0+\mathbf {p}_j\rangle }e^{\lambda _j (\sigma +it)}\ \forall \sigma +it\in U. \end{aligned}$$

Now, by the uniqueness of the coefficients of an exponential sum in $\mathcal {D}_{\varLambda }$, it is clear that $b_j=a_j e^{<\mathbf {r}_j,\mathbf {y}_0+\mathbf {p}_j>i}$ for each $j\ge 1$, which shows that $f_1\ {\mathop {\sim }\limits ^{*}}\ f_2$.

We next define the following set which will be widely used from now on.

Definition 7

Given $\varLambda =\{\lambda _1,\lambda _2,\ldots ,\lambda _j,\ldots \}$ a set of exponents, let $f(s)\in \mathcal {D}_{\varLambda }$ be an almost periodic function in an open vertical strip U, and $\sigma _0={\text {Re}}s_0$ with $s_0\in U$. We define ${\text {Img}}\left( F_f^{G_{\varLambda }}(\sigma _0,\mathbf {x},\mathbf {q}_1,\mathbf {q}_2,\ldots )\right)$ to be the set of values in the complex plane taken on by the auxiliary function $F_f^{G_{\varLambda }}(\sigma ,\mathbf {x},\mathbf {q}_1,\mathbf {q}_2,\ldots )$, relative to a prefixed basis $G_{\varLambda }$, when $\sigma =\sigma _0$; that is ${\text {Img}}\left( F_f^{G_{\varLambda }}(\sigma _0,\mathbf {x},\mathbf {q}_1,\mathbf {q}_2,\ldots )\right) =\{s\in {\mathbb {C}}:\exists \mathbf {x}\in [0,2\pi )^{\sharp G_{\varLambda }^*}\ \text{ and } \mathbf {q}_j\in S_T \text{ such } \text{ that } s=F_f^{G_{\varLambda }}(\sigma _0,\mathbf {x},\mathbf {q}_1,\mathbf {q}_2,\ldots )\}.$

We next prove that the sets of values taken on by the auxiliary function $F_f^{G_{\varLambda }}(\sigma ,\mathbf {x},\mathbf {q}_1,\mathbf {q}_2,\ldots )$ are independent of the basis $G_{\varLambda }$. The proof is similar to that of Corollary 1.

Lemma 1

Given $\varLambda$ a set of exponents and $G_{\varLambda }$ an arbitrary basis for $\varLambda$, let $f(s)\in \mathcal {D}_{\varLambda }$ be an almost periodic function in an open vertical strip U, and $\sigma _0={\text {Re}}s_0$ with $s_0\in U$. Then

$$\begin{aligned} {\text {Img}}\left( F_f^{G_{\varLambda }}(\sigma _0,\mathbf {x}, \mathbf {q}_1,\mathbf {q}_2,\ldots )\right) ={\text {Img}} \left( F_f^{G_{\varLambda }^*}(\sigma _0,\mathbf {x},\mathbf {p}_1,\mathbf {p}_2,\ldots )\right) . \end{aligned}$$

Proof

Let $\sum _{j\ge 1}a_je^{\lambda _js}$ be the Dirichlet series associated with $f(s)\in \mathcal {D}_{\varLambda }$, and $G_{\varLambda }^*$ and $G_{\varLambda }$ be the natural and an arbitrary basis for $\varLambda$, respectively. For each $j\ge 1$ let $\mathbf {r}_j$ and $\mathbf {s}_j$ be the vector of integer components so that $\lambda _j=\langle \mathbf {r}_j,\mathbf {g}\rangle$ and $\lambda _j=\langle \mathbf {s}_j,\mathbf {h}\rangle$, with $\mathbf {g}$ and $\mathbf {h}$ the vectors associated with the basis $G_{\varLambda }^*$ and $G_{\varLambda }$, respectively. Finally, for each integer $k\ge 1$, let $\mathbf {t}_k$ be the vector given by $h_k=\langle \mathbf {t}_k,\mathbf {g}\rangle$. Take $w_1\in {\text {Img}}\left( F_f^{G_{\varLambda }^*}(\sigma _0,\mathbf {x},\mathbf {p}_1,\mathbf {p}_2,\ldots )\right)$, then there exists $\mathbf {x}_1\in [0,2\pi )^{\sharp G_{\varLambda }}$ and $\mathbf {p}_j\in 2\pi {\mathbb {Z}}^{\sharp G_{\varLambda }^*}$, $j=1,2,\ldots$, $\text{ such } \text{ that } w_1=F_f^{G_{\varLambda }^*}(\sigma _0,\mathbf {x}_1,\mathbf {p}_1,\mathbf {p}_2,\ldots )$. Hence

$$\begin{aligned} w_1= & {} F_f^{G_{\varLambda }^*}(\sigma _0,\mathbf {x}_1,\mathbf {p}_1,\mathbf {p}_2,\ldots )=\sum _{j\ge 1}a_j e^{\lambda _j\sigma _0 }e^{\langle \mathbf {r}_j,\mathbf {x}_1+\mathbf {p}_j\rangle i}\\= & {} \sum _{j\ge 1}a_j e^{\lambda _j\sigma _0 }e^{\langle \mathbf {s}_j,\mathbf {x}_2+\mathbf {q}_j\rangle i}, \end{aligned}$$

where $\mathbf {q}_j$ is defined as $q_{1,k}=<\mathbf {t}_k,\mathbf {p}_j>$ for each $k\ge 1$, and $\mathbf {x}_2$ is defined as $x_{2,k}=<\mathbf {t}_k,\mathbf {x}_1>$ for each $k\ge 1$. Therefore, $w_1=F_f^{G_{\varLambda }}(\sigma _0,\mathbf {x}_2,\mathbf {q}_1,\mathbf {q}_2,\ldots )$ and $w_1\in {\text {Img}}\left( F_f^{G_{\varLambda }}(\sigma _0,\mathbf {x},\mathbf {q}_1,\mathbf {q}_2,\ldots )\right)$, which gives

$$\begin{aligned} {\text {Img}}\left( F_f^{G_{\varLambda }^*}(\sigma _0,\mathbf {x},\mathbf {p}_1,\mathbf {p}_2,\ldots )\right) \subseteq {\text {Img}}\left( F_f^{G_{\varLambda }}(\sigma _0,\mathbf {x},\mathbf {q}_1,\mathbf {q}_2,\ldots )\right) . \end{aligned}$$

An analogous argument shows that the set ${\text {Img}}\left( F_f^{G_{\varLambda }}(\sigma _0,\mathbf {x},\mathbf {q}_1,\mathbf {q}_2,\ldots )\right)$ is included in the set ${\text {Img}}\left( F_f^{G_{\varLambda }^*}(\sigma _0,\mathbf {x},\mathbf {p}_1,\mathbf {p}_2,\ldots )\right)$, which proves the result.

Consequently, from now on we will use the notation ${\text {Img}}\left( F_f(\sigma _0,\mathbf {x},\mathbf {p}_1,\mathbf {p}_2,\ldots )\right)$ for the set of values taken on by the auxiliary function associated with a function $f(s)\in \mathcal {D}_{\varLambda }$. In this respect, without loss of generality, we can use the natural basis for the set of exponents $\varLambda$. Moreover, under the assumption of existence of an integral basis, we will use the notation ${\text {Img}}\left( F_f(\sigma _0,\mathbf {x})\right)$ for the set above. In fact, in this case ${\text {Img}}\left( F_f(\sigma _0,\mathbf {x})\right)$ is the same as that of [13, Definition 5] and all the results of [13] concerning integral basis are also valid for our case (see also [12, Remark 2]).

4 Main results

Given a function f(s), take the notation

$$\begin{aligned} {\text {Img}}\left( f(\sigma _0+it)\right) =\{w\in {\mathbb {C}}:\exists t\in {\mathbb {R}} \text{ such } \text{ that } s=f(\sigma _0+it)\}. \end{aligned}$$

We next show the first important result in this paper concerning the connection between our equivalence relation and the set of values in the complex plane taken on by the auxiliary function (compare with [13, Lemma 9, Propositions 12 and 13]).

Proposition 4

Given $\varLambda$ a set of exponents, let $f(s)\in \mathcal {D}_{\varLambda }$ be an almost periodic function in an open vertical strip U, and $\sigma _0={\text {Re}}s_0$ with $s_0\in U$.

• (i) If $f_1\ {\mathop {\sim }\limits ^{*}}\ f$, then ${\text {Img}}\left( f_1(\sigma _0+it)\right) \subset \overline{{\text {Img}}\left( f(\sigma _0+it)\right) }$ and

  $$\begin{aligned} \overline{{\text {Img}}\left( f(\sigma _0+it)\right) }= \overline{{\text {Img}}\left( f_1(\sigma _0+it)\right) }. \end{aligned}$$

• (ii) ${\text {Img}}\left( F_f(\sigma _0,\mathbf {x},\mathbf {p}_1,\mathbf {p}_2,\ldots )\right) =\bigcup _{f_k {\mathop {\sim }\limits ^{*}} f}{\text {Img}}\left( f_k(\sigma _0+it)\right) .$

• (iii) ${\text {Img}}\left( F_f(\sigma _0,\mathbf {x},\mathbf {p}_1,\mathbf {p}_2,\ldots )\right)$ is a closed set.

• (iv) ${\text {Img}}\left( F_f(\sigma _0,\mathbf {x},\mathbf {p}_1,\mathbf {p}_2,\ldots )\right) =\overline{{\text {Img}}\left( f_1(\sigma _0+it)\right) }$ for every $f_1\ {\mathop {\sim }\limits ^{*}}\ f$.

Proof

(i) Note that [11, Theorem 4] shows that the functions in the same equivalence class are obtained as limit points of $\mathcal {T}_f=\{f_{\tau }(s):=f(s+i\tau ):\tau \in {\mathbb {R}}\}$, that is, every function $f_1\ {\mathop {\sim }\limits ^{*}}\ f$ is the limit (in the sense of the uniform convergence on every reduced strip of U) of a sequence $\{f_{\tau _n}(s)\}$ with $f_{\tau _n}(s):=f(s+i\tau _n)$. Take $w_1\in {\text {Img}}\left( f_1(\sigma _0+it)\right)$, then there exists $t_1\in {\mathbb {R}}$ such that $w_1=f_1(\sigma _0+it_1)$. Now, given $\varepsilon >0$ there exists $\tau >0$ such that $|f_1(\sigma _0+it_1)-f_{\tau }(\sigma _0+it_1)|<\varepsilon$, which means that

$$\begin{aligned} |w_1-f(\sigma _0+i(t_1+\tau ))|<\varepsilon . \end{aligned}$$

Now it is immediate that $w_1\in \overline{{\text {Img}}\left( f(\sigma _0+it)\right) }$ and consequently

$$\begin{aligned} {\text {Img}}\left( f_1(\sigma _0+it)\right) \subset \overline{{\text {Img}}\left( f(\sigma _0+it)\right) }. \end{aligned}$$

Analogously, by symmetry we have ${\text {Img}}\left( f(\sigma _0+it)\right) \subset \overline{{\text {Img}}\left( f_1(\sigma _0+it)\right) }$, which yields that

$$\begin{aligned} \overline{{\text {Img}}\left( f(\sigma _0+it)\right) }= \overline{{\text {Img}}\left( f_1(\sigma _0+it)\right) }. \end{aligned}$$

(ii) Take $w_0\in \bigcup _{f_k{\mathop {\sim }\limits ^{*}} f}{\text {Img}}\left( f_k(\sigma _0+it)\right)$, then $w_0\in {\text {Img}}\left( f_k(\sigma _0+it)\right)$ for some $f_k\ {\mathop {\sim }\limits ^{*}}\ f$, which means that there exists $t_0\in {\mathbb {R}}$ such that

$$\begin{aligned} w_0=f_k(\sigma _0+it_0). \end{aligned}$$

Note that Proposition 3 assures the existence of a vector $\mathbf {y}_0\in {\mathbb {R}}^{\sharp G_\varLambda ^*}$ and $\mathbf {p}_j\in 2\pi {\mathbb {Z}}^{\sharp G_{\varLambda }^*}$, $j=1,2,\ldots$, such that $w_0=F_{f}(\sigma ,\mathbf {y}_0+t_0\mathbf {g},\mathbf {p}_1,\mathbf {p}_2,\ldots )$. Hence $w_0=F_{f}(\sigma _0,\mathbf {x}_0,\mathbf {p}_1,\mathbf {p}_2,\ldots )$, with $\mathbf {x}_0=\mathbf {y}_0+t_0\mathbf {g}\in [0,2\pi )^{\sharp G_{\varLambda }^*}$, which means that $w_0\in {\text {Img}}\left( F_f(\sigma _0,\mathbf {x},\mathbf {p}_1,\mathbf {p}_2,\ldots )\right)$. Conversely, if $w_0\in {\text {Img}}\left( F_f(\sigma _0,\mathbf {x},\mathbf {p}_1,\mathbf {p}_2,\ldots )\right)$, then $w_0=F_{f}(\sigma _0,\mathbf {y}_0,\mathbf {p}_1,\mathbf {p}_2,\ldots )$ for some $\mathbf {y}_0\in [0,2\pi )^{\sharp G_{\varLambda }}$ and $\mathbf {p}_j\in 2\pi {\mathbb {Z}}^{\sharp G_{\varLambda }^*}$. Take $t_0\in {\mathbb {R}}$. Since $\mathbf {y}_0=\mathbf {x}_0+t_0\mathbf {g}$, with $\mathbf {x}_0:=\mathbf {y}_0-t_0\mathbf {g}$, then

$$\begin{aligned} w_0= & {} F_{f}(\sigma _0,\mathbf {x}_0+t_0\mathbf {g},\mathbf {p}_1,\mathbf {p}_2,\ldots )=\sum _{j\ge 1}a_j e^{\lambda _j\sigma _0 }e^{\langle \mathbf {r}_j,\mathbf {x}_0+t_0\mathbf {g}+\mathbf {p}_j\rangle i}\\= & {} \sum _{j\ge 1}a_j e^{\lambda _j(\sigma _0+it_0) }e^{\langle \mathbf {r}_j,\mathbf {x}_0+\mathbf {p}_j\rangle i}. \end{aligned}$$

Hence $\sum _{j\ge 1}a_je^{\langle \mathbf {r}_j,\mathbf {x}_0+\mathbf {p}_j\rangle i} e^{\lambda _js}$ is the Dirichlet series associated with an almost periodic function $h(s)\in AP(U,{\mathbb {C}})$ such that $h\ {\mathop {\sim }\limits ^{*}}\ f$ (see [11, Lemma 3]) and hence we have that $w_0=h(\sigma _0+it_0)$ (see also [13, Remark 11]), which shows that $w_0\in \bigcup _{f_k{\mathop {\sim }\limits ^{*}} f}{\text {Img}}\left( f_k(\sigma _0+it)\right) .$

(iii) Let $w_1,w_2,\ldots ,w_j,\ldots$ be a sequence of points in ${\text {Img}}\left( F_f(\sigma _0,\mathbf {x},\mathbf {p}_1,\mathbf {p}_2,\ldots )\right)$ tending to $w_0$. We next prove that $w_0\in {\text {Img}}\left( F_f(\sigma _0,\mathbf {x},\mathbf {p}_1,\mathbf {p}_2,\ldots )\right)$. Indeed, for each $w_j$, we deduce from (ii) the existence of $f_j\ {\mathop {\sim }\limits ^{*}}\ f$ such that $w_j\in {\text {Img}}\left( f_j(\sigma _0+it)\right)$. Now, since that $\{f_j(\sigma _0+it)\}$ is a sequence in the same equivalence class, [11, Proposition 3] assures the existence of a subsequence $\{f_{j_k}\}$ which converges to a certain function $h\ {\mathop {\sim }\limits ^{*}}\ f$. Consequently, $\{w_{j_k}\}$ tends to $w_0\in {\text {Img}}\left( h(\sigma _0+it)\right)$. Finally, again by (ii), we conclude that

$$\begin{aligned} w_0\in {\text {Img}}\left( F_f(\sigma _0,\mathbf {x},\mathbf {p}_1,\mathbf {p}_2, \ldots )\right) . \end{aligned}$$

(iv) Let $\mathbf {g}$ be the vector associated with the natural basis $G_{\varLambda }^*$. Since the Fourier series of $f_{\sigma _0}(t):=f(\sigma _0+it)$ can be obtained as $F_f(\sigma _0,t\mathbf {g},\mathbf {0},\mathbf {0},\ldots )$, with $t\in {\mathbb {R}}$, it is clear that ${\text {Img}}\left( f(\sigma _0+it)\right) \subset {\text {Img}}\left( F_f(\sigma _0,\mathbf {x},\mathbf {p}_1,\mathbf {p}_2,\ldots )\right)$. On the other hand, we deduce from (i) and (ii) that

$$\begin{aligned}&{\text {Img}}\left( f(\sigma _0+it)\right) \subset {\text {Img}}\left( F_f(\sigma _0,\mathbf {x},\mathbf {p}_1,\mathbf {p}_2,\ldots )\right) \\&\quad =\bigcup _{f_k{\mathop {\sim }\limits ^{*}} f}{\text {Img}}\left( f_k(\sigma _0+it)\right) \subset \overline{{\text {Img}}\left( f(\sigma _0+it)\right) }. \end{aligned}$$

Finally, by taking the closure and property (iii), we conclude that

$$\begin{aligned} {\text {Img}}\left( F_f(\sigma _0,\mathbf {x},\mathbf {p}_1,\mathbf {p}_2,\ldots )\right) = \overline{{\text {Img}}\left( f(\sigma _0+it)\right) }. \end{aligned}$$

Now, the result follows from property (i).

Remark 2

Given $\varLambda$ a set of exponents, let $f(s)\in \mathcal {D}_{\varLambda }$ be an almost periodic function in an open vertical strip U, and $\sigma _0={\text {Re}}s_0$ with $s_0\in U$. As a consequence of Proposition 4, one gets that

$$\begin{aligned} \overline{{\text {Img}}\left( f(\sigma _0+it)\right) }=\bigcup _{f_k {\mathop {\sim }\limits ^{*}} f}{\text {Img}}\left( f_k(\sigma _0+it)\right) . \end{aligned}$$

If we changed $^*$-equivalence (that of Definition 2) by Bohr-equivalence (that of [13, Definition 2]) and $\varLambda$ did not have an integral basis, this result would be false as [13, Remark 14] shows.

At this point we will demonstrate a result like Bohr’s equivalence theorem [1, Sect. 8.11]. Given $\varLambda$ an arbitrary set of exponents, let $f_1,f_2\in \mathcal {D}_{\varLambda }$ be two $^*$-equivalent almost periodic functions. We next show that, in every open half-plane or open vertical strip included in their region of almost periodicity, the functions $f_1$ and $f_2$ take the same set of values. In this sense, this result improves that of [13, Theorem 1] which was proved uniquely for almost periodic functions associated with sets of exponents which have an integral basis.

Theorem 1

Fixed $\varLambda$ a set of exponents, let $f_1,f_2\in \mathcal {D}_{\varLambda }$ be two $^*$-equivalent almost periodic functions in a strip $\{\sigma +it\in {\mathbb {C}}:\alpha<\sigma <\beta \}$, and consider E an open set of real numbers included in $(\alpha ,\beta )$. Then

$$\begin{aligned} \bigcup _{\sigma \in E}{\text {Img}}\left( f_1(\sigma +it)\right) =\bigcup _{\sigma \in E}{\text {Img}}\left( f_2(\sigma +it)\right) . \end{aligned}$$

That is, the functions $f_1$ and $f_2$ take the same set of values on the region $\{s=\sigma +it\in {\mathbb {C}}:\sigma \in E\}$.

Proof

Without loss of generality, suppose that $f_1$ and $f_2$ are not constant functions (otherwise it is trivial). Take $w_0\in \bigcup _{\sigma \in E}{\text {Img}}\left( f_1(\sigma +it)\right)$, then $w_0\in {\text {Img}}\left( f_1(\sigma _0+it)\right)$ for some $\sigma _0\in E$ and hence $w_0=f_1(\sigma _0+it_0)$ for some $t_0\in {\mathbb {R}}$. Furthermore, by Proposition 4, we get $w_0\in \overline{{\text {Img}}\left( f_1(\sigma _0+it)\right) }=\overline{{\text {Img}}\left( f_2(\sigma _0+it)\right) }$, which yields the existence of a sequence $\{t_n\}$ of real numbers such that

$$\begin{aligned} w_0=\lim _{n\rightarrow \infty }f_2(\sigma _0+it_n). \end{aligned}$$

Take $h_n(s):=f_2(s+it_n)$, $n\in {\mathbb {N}}$. By [11, Proposition 4], there exists a subsequence $\{h_{n_k}\}_k\subset \{h_n\}_n$ which converges uniformly on compact subsets to a function h(s), with $h\ {\mathop {\sim }\limits ^{*}}\ f_2$. Observe that

$$\begin{aligned} \lim _{k\rightarrow \infty }h_{n_k}(\sigma _0)=h(\sigma _0)=w_0. \end{aligned}$$

Therefore, by Hurwitz’s theorem [2, Sect. 5.1.3], there is a positive integer $k_0$ such that for $k>k_0$ the functions $h^*_{n_k}(s):=h_{n_k}(s)-w_0$ have at least one zero in $D(\sigma _0,\varepsilon )$ for any $\varepsilon >0$ sufficiently small. This means that for $k>k_0$ the functions $h_{n_k}(s)=f_2(s+it_{n_k})$, and hence the function $f_2(s)$, take the value $w_0$ on the region $\{s=\sigma +it:\sigma _0-\varepsilon<\sigma <\sigma _0+\varepsilon \}$ for any $\varepsilon >0$ sufficiently small (recall that E is an open set). Consequently, $w_0\in \bigcup _{\sigma \in E}{\text {Img}}\left( f_2(\sigma +it)\right)$. We analogously prove that $\bigcup _{\sigma \in E}{\text {Img}}\left( f_2(\sigma +it)\right) \subset \bigcup _{\sigma \in E}{\text {Img}}\left( f_1(\sigma +it)\right)$.

It is worth noting that [13, Example 2] also shows that a converse to Theorem 1 cannot hold by fixing an open set E in $(\alpha ,\beta )$. However, we are considering the existence of a certain converse statement of this present generalization (see the arXiv paper [9]).

Finally, from Remark 1 and Proposition 1, we know that Definition 2 and the definition of Bohr-equivalence are equivalent under the condition of existence of an integral basis for the set of exponents $\varLambda$, which means that all the results of this paper which can be formulated in terms of an integral basis are also valid under Bohr-equivalence. In fact, if two exponential sums or almost periodic functions are Bohr-equivalent, it is clear that they also are $^*$-equivalent (according to Definition 2). Consequently, we can deduce immediately from our main result (Theorem 1) that two Bohr-equivalent almost periodic functions take the same values on every open vertical strip included in their strip of almost periodicity U, i.e. [13, Theorem 18] is also true if the condition of existence of an integral basis is omitted. This constitutes an extension of Bohr’s equivalence theorem.

Corollary 2

Fixed $\varLambda$ a set of exponents, let $f_1,f_2\in \mathcal {D}_{\varLambda }$ be two Bohr-equivalent almost periodic functions in a strip $\{\sigma +it\in {\mathbb {C}}:\alpha<\sigma <\beta \}$, and consider E an open set of real numbers included in $(\alpha ,\beta )$. Then

$$\begin{aligned} \bigcup _{\sigma \in E}{\text {Img}}\left( f_1(\sigma +it)\right) =\bigcup _{\sigma \in E}{\text {Img}}\left( f_2(\sigma +it)\right) . \end{aligned}$$

That is, the functions $f_1$ and $f_2$ take the same set of values on the region $\{s=\sigma +it\in {\mathbb {C}}:\sigma \in E\}$.

For the case of Bohr’s equivalence relation, we would like to point out that the existence of a converse statement of the above result will be possible only assuming the existence of an integral basis (see [9]).