On the Result of Invariance of the Closure Set of the Real Projections of the Zeros of an Important Class of Exponential Polynomials

Sepulcre, J. M.

doi:https://doi.org/10.1155/2016/3605690

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Volume 2016 | Article ID 3605690 | https://doi.org/10.1155/2016/3605690

On the Result of Invariance of the Closure Set of the Real Projections of the Zeros of an Important Class of Exponential Polynomials

J. M. Sepulcre¹

Academic Editor: Sergei Silvestrov

Received08 Dec 2015

Revised23 Mar 2016

Accepted04 May 2016

Published30 May 2016

Abstract

We provide the proof of a practical pointwise characterization of the set defined by the closure set of the real projections of the zeros of an exponential polynomial with real frequencies linearly independent over the rationals. As a consequence, we give a complete description of the set and prove its invariance with respect to the moduli of the , which allows us to determine exactly the gaps of and the extremes of the critical interval of by solving inequations with positive real numbers. Finally, we analyse the converse of this result of invariance.

1. Introduction

Throughout this paper we will consider exponential polynomials of the formwhere is an ordered set of real numbers, called frequencies of , which are linearly independent over the field of the rational numbers. It is known that is an entire function of order having infinitely many zeros located on a vertical strip (see, e.g., [1, Lemma 2.5]), the critical strip of , bounded by the real numbers and , where

It is worth noting that the research devoted to studying the zeros of exponential polynomials appears in the first third of the twentieth century in relation to the development of differential equation theory [2] (see also [3, 4]). In this respect, it is related to the study of the location of the characteristic roots of linear delay differential equations (see, e.g., [5, 6]), and, more so, the zeros of exponential polynomials are the roots of the characteristic equations of the associated functional difference equations which determine the essential spectrum of the solution operators of the associated neutral functional differential equations [7].

The study of the zeros of the class of exponential polynomials of type (1) has recently become a topic of increasing interest. In fact, in connection with other areas of study, it allowed to prove that a conjecture on the set of dimensions of fractality associated with a nonlattice fractal string is true in the case of generic nonlattice self-similar strings [8, Theorem 13], which have a clear correspondence with the exponential polynomials of type (1).

Given such a function , the bounds and allow us to define an interval , called critical interval of , which contains the closure of the set of the real parts of the zeros of , denoted byIn this respect, the density properties of the zeros of several types of exponential polynomials were studied, for example, in [9–13] and more recently in [8, 14–18]. At this point, we must also mention the high sensitivity of the bounds of the critical interval to the frequency variations. Furthermore, the results on the nature of have immediate implication to the theory of stability for neutral time delay systems. In fact, from the stability point of view, it is important to study the behaviour of the smallest upper bound of the spectrum [7, 9, 19–24]. It was shown by Avellar and Hale [9] that, in general, the smallest upper bound is not continuous with respect to the delays (i.e., arbitrarily small changes in the delays destabilise the associated neutral functional differential equation). However, the continuity is reached when the delays are rationally independent [9, Theorem 2.2] and, in that case, the set is completely characterized by [9, Theorem 3.1, Corollary 3.2].

In this way, we will suppose in this paper that the frequencies of our exponential polynomials are rationally independent in order to make good use of the accurate knowledge of the nature of the set . In fact, concerning exponential polynomials of type (1), Moreno proved in 1973 the following theorem which we quote:

(Moreno [12, Main Theorem]) “Assume that , are real numbers linearly independent over the rationals. Consider the exponential polynomial where the are complex numbers. Then a necessary and sufficient condition for to have zeros arbitrarily close to any line parallel to the imaginary axis inside the stripis that for any with .”

From this result it follows that an open interval is included in if and only if geometrical principle (7) holds on . It is worth noticing that Moreno’s result (another version of this theorem which can be found in [8, Theorem 1]) is only applicable to open intervals and hence it does not identify the boundary points of . This characterization was used in [8] in order to establish a bound for the number of gaps of the sets defined in (4).

On the other hand, the topological properties of the set associated with the partial sums , , of the Riemann zeta function have been studied from different approaches. For example, an auxiliary function associated with was used in [17, Theorem 9] in order to establish conditions to decide whether a real number is in the set . This auxiliary function specifically arises from a known result of Avellar and Hale [9, Theorem 3.1] which leads us to analytical criteria in the case of exponential polynomials whose frequencies are not necessarily linearly independent over the rationals.

In this paper, by following the ideas of [17] and based on inequalities (7), we will get a practical characterization of the points of the set associated with an exponential polynomial of type (1). Also, in order to prove this pointwise geometrical principle and to obtain other preliminary properties, we will use Kronecker’s theorem which is stated under the following form:

(Hardy and Wright [25, p. 382]) “Let be a linearly independent set of non-null real numbers. For arbitrary real numbers , and , , there exist a real number and integers such that

In such a way, related to an exponential polynomial of type (1), in this paper we specifically prove the following:(i)Fix ; the closure of the set coincides with the image set of an auxiliary function, denoted by , associated with (see Proposition 3).(ii)A real number if and only if geometrical principle (35) holds on (see Theorem 6), which constitutes a practical pointwise characterization of the closure set of the real parts of the zeros of exponential polynomials whose frequencies are linearly independent over the rationals.(iii) is the union of, at most, disjoint nondegenerate closed intervals (see Theorem 9 and its demonstration which constitutes another shorter proof of [8, Theorem 9]).(iv) is invariant with respect to the moduli of the (see Corollary 10), which let us exert a control about the number of gaps of by means of an adequate selection of the moduli of the coefficients associated with .(v)If two exponential polynomials with two or three terms are written in their normalized forms and they have the same set , then the moduli of their coefficients are pairwise equal (see Proposition 12), which for these cases is a kind of converse assertion to the invariance result formulated in the previous point. This result cannot be extended to exponential polynomials with more than three terms (see Example 13).

2. An Auxiliary Function Associated with

Let be of form (1) with . Equation gives us the equality . Hence the zeros of for are given by Therefore, in this case, the set , defined in (4), is equal to the point and then, from now on, we will suppose that .

We next define an auxiliary function, associated with the exponential polynomials of form (1), which will be used in this paper in order to find some characterizations of the set defined in (4). This function is clearly inspired by [9, Theorem 3.1].

Definition 1. Let be an exponential polynomial of type (1). We define the auxiliary function associated with as

We next introduce the following notation which will be used to show the direct relation between an exponential polynomial of type (1) and the auxiliary function associated with it.

Notation 1. Given an exponential polynomial of type (1) and fixing , we consider the following sets of the complex plane:

Fixing , we now establish the direct inclusion between the sets above.

Lemma 2. Let be an exponential polynomial of type (1) and ; then

Proof. Fixing , consider that of type (1) and , then for some ; that is,On the other hand, from Definition 1, for any , we haveTherefore, by taking for , expression (14) becomes (13). Consequently, and the lemma is proved.

In fact, fixing , by using Kronecker’s theorem, we next prove that coincides with the closure of the set .

Proposition 3. Let be an exponential polynomial of type (1). Then, for every , it is verified that

Proof. Given , consider that , with , , and is an ordered set of real numbers linearly independent over the rationals. Fix . Notice that in virtue of Lemma 2. We are going to demonstrate that is dense in or, equivalently, given and there exists such that Indeed, if , then there exists a vector such thatFixing and , we next apply Kronecker’s Theorem [25, p. 382] with the following choice: Hence there exist some integers and such that that is,with , . Therefore, since from (17) and (20), we have which goes to when tends to because , for . Consequently, we conclude that

3. Characterizations of the Closure Set of the Real Projections of the Zeros

This section is devoted to showing some pointwise characterizations of the sets defined in (4). We previously need the next lemma (compare with [14, Lemma 3]).

Lemma 4. Let be an exponential polynomial of type (1), a point in , and a real sequence such that Then .

Proof. In order to apply [12, Lemma, p. 73], we next prove the following two conditions that joint with the assumption of the existence of a sequence of points such that ; let us prove this lemma: (a) is bounded on its critical strip .(b)There exist positive numbers and such that on any segment of length of the line there is a point such that .Given , we first observe that where . Henceprovided that . Then is, in particular, bounded on its critical strip and we deduce (a). On the other hand, let be a real number such that and take . Then, since is an almost-periodic function [26, p. 101], there exists a real number such that every interval of length on the imaginary axis contains at least one translation number , associated with , satisfying for all . Thus, by taking , we have and, according to the choice of , it follows that . Therefore, , with , which proves (b). Consequently, the function has the properties needed to apply [12, Lemma, p. 73] and thus has zeros in any strip , for arbitrary . Then . This completes the proof.

The previous lemma provides a sufficient condition to get points of . It allows us to give a characterization of the sets by means of an ad hoc version of [9, Theorem 3.1], which is obtained through the auxiliary function analysed in the previous section.

Theorem 5. Let be an exponential polynomial of type (1). Then a real number if and only if there exists some vector such that , where is the auxiliary function associated with defined in (10).

Proof. For any zero of we havewhere . Assume that . Then there exists a sequence of zeros of such that and, from (27), we get that for each , which is equivalent to writingFor , since the sequence is on the unit circle, it is bounded, so there exists a subsequence convergent to a point for some . For , from the sequence , we can draw a subsequence convergent to a point for some and so successively for each . Then, by considering (28) for and taking the limit when , we obtain where .
Conversely, suppose that for some real number and a vector of . Thus, noticing that the components of the vector are linearly independent over the rationals, given the numbers ; and , with , by applying Kronecker’s theorem [25, p. 382] for each fixed , there exist and integers such that for each , we haveMultiplying by , (31) becomes which implies that Then we are led toNow, by Lemma 4, and then the theorem follows.

All of this results in a second pointwise characterization, through geometrical principle (35), of the closure set of the real parts of the zeros of exponential polynomials of form (1).

Theorem 6. Let be an exponential polynomial of type (1) with . Then a real number if and only if

Proof. Suppose that , then there exists some vector such that or, equivalently, . Therefore, and, by taking the modulus, we get Conversely, suppose that the positive real numbers , , satisfy inequalities (35). We index them in decreasing order so that is such that , such that , and so forth. Thus there is at least one -sided polygon whose sides have these lengths [12, p. 71]. That means that there exist real numbers verifying Consequently, by taking as the vector so that for , where denotes the principal argument of , we have Hence, from Theorem 5, .

Note that the proof of the result above includes a generalization of Euclid’s theorem on the inequality of the triangle, which leads to geometrical principle (35) that characterizes the sets .

Remark 7. Alternatively, as one of the referees pointed out, the necessity in Theorem 6 follows from the following fact: if (35) is not true then there exists, by continuity, a number such that the condition of type (35) is still false for the interval and thus, by the triangle inequality, there are no zeros with real part in this interval, which means that .
In any case, note that Theorem 6 has been proved without using [8] (in particular [8, Theorem 9]).

4. The Description of the Set

As it will be also shown below, the number of gaps that can have the set defined by , associated with an exponential polynomial of type (1) with , depends on the number of real solutions of the intermediate equations:More so, as the proof of the following lemma shows (compare with [8, Theorem3]), these equations above for the cases and provide us with the extremes of the critical interval , where and are defined in (2) and (3), respectively.

Lemma 8. Let be an exponential polynomial of type (1) with . The inequations and are satisfied for all in the critical interval . Moreover, there exist and such that the intervals and are both contained in .

Proof. By defining the real functionsthe number of changes of the sign of its coefficients, say and , is . Then if and are the numbers of zeros of and , respectively, by Pólya and Szegő’s result [27, p. 46], and are even nonnegative integers. Hence, and we denote by and the unique simple zeros of and , respectively. Then, since , we deduce that for all and Analogously, we can deduce that Therefore, according to Theorem 6, and must coincide with and . Now, from continuity applied on the above strict inequalities we can determine two real numbers and such that any of the corresponding intervals and satisfies the inequalities (35). Furthermore, and are nonpositive real numbers when is in the critical interval. The lemma is proved.

We are now ready to give the description of the set associated with an exponential polynomial of type (1). This leads us to another shorter proof of [8, Theorem9].

Theorem 9. Given an exponential polynomial of type (1) with , is either or the union of at most disjoint nondegenerate closed intervals. In the latter case, the gaps of are exclusively produced by those equations (40) having two solutions.

Proof. Note that, fixing , if we suppose that some of inequations (35) are not satisfied, then we deduce from Theorem 6 that . Thus, in view of Lemma 8, we next focus our attention on the intermediate inequations: Let us define the real functions:Since the number of changes of sign of their coefficients is , by Pólya and Szegő’s result [27, p. 46], each equation can have solutions, solution when the zero of is of second order, or (distinct) solutions, , when has two simple zeros. Furthermore, since is clearly positive for sufficiently large, then is uniquely satisfied in the open intervals of the form (i.e., when has two distinct solutions). Now, by defining the set we have, by continuity, if with . Define also the set then the complementary set of is that is, it consists of at most disjoint open intervals and consequently consists of at most nondegenerate closed intervals.

As a consequence of the theorem above, the set has no isolated point for any exponential polynomial of type (1) with . The following important result is another easy consequence of this characterization theorem.

Corollary 10. Given an exponential polynomial of type (1), is invariant with respect to the moduli of its coefficients .

Example 11. Consider the exponential polynomials: whose frequencies are linearly independent over the rationals. From Corollary 10, first notice that . In this respect, the distribution of their zeros with imaginary part between and can be observed in Figure 1, which was performed using Maple.

Furthermore, by using Theorem 9, we can exactly determine the boundary points of the closure set of the real parts of their zeros. In fact, the gaps of are produced when we find two solutions in some of the following equations:Equation (50) has no solutions. Equality (51) is reached in and . Finally, and are the solutions of (52). Also, from Lemma 8, we calculate the extremes of the critical interval which are and . Therefore, the disjoint nondegenerate closed intervals of are , , and .

5. On the Converse of the Result of Invariance

Given an exponential polynomial of form (1), by dividing by the first term, it is not restrictive to consider, for our purposes, its underlying exponential polynomial: where and for .

We are now going to study whether the converse of Corollary 10 is true; that is, given two exponential polynomials and of type (1) with the same set of frequencies and so that , are the moduli of the coefficients (in the normalized form and ) the same? It is an elementary check that the answer is affirmative when and have two terms. We next prove that the answer is also affirmative when our exponential polynomials have three terms (the case ).

Proposition 12. Let and be two exponential polynomials of type (1) and let and be their normalized forms. Thus if and only if for each .

Proof. If for then, by taking Corollary 10 into account, we easily get . Conversely suppose that ; then in particular we have that and . Hence, we deduce from the proof of Lemma 8 that Equivalently, Therefore, it is an elementary check that and the result follows.

However, as the following example shows, Proposition 12 cannot be extended to any integer number .

Example 13. Let and be the normalized exponential polynomials associated with two exponential polynomials and of type (1) and with the same frequencies, where , and , and Notice that for . Nevertheless, it is verified that . Indeed, , , and have been chosen so that the extremes of the critical interval verify and or, equivalently, they satisfy More so, (40) have no solution for both exponential polynomials and thus , where and . That is, has no gaps. In this respect, the distribution of the zeros of and (which are the same as and , resp.) whose imaginary part is between and can be observed in Figure 2.

Competing Interests

The author declares that he has no competing interests.

Acknowledgments

The author thanks the anonymous referees for the suggestion to extend the state of the art and their helpful comments. He is also grateful to T. Vidal, G. Mora, and E. Dubon for many helpful discussions. The research was partially supported by Generalitat Valenciana under Project GV/2015/035.

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Copyright © 2016 J. M. Sepulcre. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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