#### Abstract

Let be the family of continuous positive definite functions on . For an integer , a is called -divisible if there is such that . Some properties of infinite-divisible and -divisible functions may differ in essence. Indeed, if is infinite-divisible, then for each integer , there is an unique such that , but there is a -divisible such that the factor in is generally not unique. In this paper, we discuss about how rich can be the class for -divisible and obtain precise estimate for the cardinality of this class.

#### 1. Introduction

We start with some notations and definitions. Let , , , and be the families of integers, non-negative integers, real, and complex numbers, respectively. In the sequel, denotes the Banach algebra of bounded regular complex-valued Borel measures on with the convolution as multiplication. is equipped with the usual total variation norm of . The Fourier-Stieltjes transform of is given by

A function is said to be positive definite if for each and all , . Any such a function satisfies for all .

Positive definite functions on groups have a long history and have many applications in probability theory and areas such as stochastic processes [1], harmonic analysis [2], potential theory [3], and spectral theory [4]. See [5] for other applications and details. The analysis of the properties of positive definite functions has vast literature, and the above list is only a small sample.

We will denote by the family of continuous nontrivial () positive definite functions on . Note that if positive definite is continuous in a neighborhood of the origin, then it is uniformly continuous in (see, e.g., [4, Corollary 1.4.10]). Bochner’s theorem gives a description of in terms of the Fourier transform. Namely, according to this theorem (see, e.g., [6], p. 71]), a continuous function is positive definite if and only if there exists a nonnegative such that , . This statement implies, in particular, that for all . If, in addition, positive satisfies , then on the language of probability theory, such a and the function , , are called a probability measure and its characteristic function, respectively (see, e.g., [7, 8], p.p. 8-9]).

Recall that in the probability theory a random variable is called -divisible for certain , , if there exist independent and identically distributed random variables such that has the sane distribution as . In terms of the characteristic function of a real-valued random variable, this means that is -divisible if there exists such that for all . Next, is said to be infinite-divisible if it is -divisible for each , .

In the sequel, and denotes the families of -divisible and infinite divisible functions in , respectively. An early overview over divisibility of distributions is given in [9]. Until very recently, the vast majority of divisible positive definite functions or divisible distributions considered in the literature are also infinite-divisible. Important applications of -divisibility is in modelling, for example, bug populations in entomology [10] or in financial aspects of various insurance models [11, 12].

The motivation for our investigation are the following: (i) partly from the fact that properties of functions in has rich literature (see, e.g., [9, 13, 14]), but the -divisible functions have been studied much less; and (ii) partly from the fact that some properties of functions from and from may differ in essence. One of those properties is the following: if , then for each , , there is an unique such that , but there is -divisible such that the factor in is generally not unique. In this paper, we study the following problems: (i) how rich can be the class ; and (ii) what properties of determine the size of . We present several precise estimates for the cardinality of this class. Also, the main results are validated via illustrative examples.

More precise, for , and , we wish to study the family and the quantity , i.e., the cardinality of . Note that there exists such that the factor in Equation (5) will generally be not unique. It turns out that depends on and in some way also depends on the geometric structure of the zeros set of (see our Theorem 1 below). The essential support of is defined by . Combining Equation (3) with Equation (4), gives

Since functions are continuous on , it follows that is an open subset of . Therefore, can be represented as a finite or infinite but countable , where is the family of all open connected components of . In the sequel, denotes the cardinality of . According to relations (7), we see that either there is an such that or .

Theorem 1. *Let , , and let . Assume that
for some . Then
*

The following theorem shows that the estimate (9) is accurate.

Theorem 2. *For each , , each , and any open subset of which satisfies
there exists , such that and
*

We will present two examples of such that Equation (11) is satisfied. In order to make the examples easier to understand, we will consider only small values of . For , set . Note that if and only if (see, e.g., [15, 16, p. 282]). We start with the case where is a bounded subset of .

*Example 3. *For , , let
Then , since
Moreover, and
Now let us give an example of such that has unbounded components.

*Example 4. *For any and , , let
where was defined by Equation (13). Then , since
Moreover, and

Theorem 5. *Suppose that an open subset of satisfies
**Let be the family of open connected components of . Assume that there is such that
**Then, for any , , there exists such that and
*

#### 2. Preliminaries and Proofs

If , then the convolution is defined by for each Borel subset of . Note that

In particular, for any , where the convolution power is defined as the -fold iteration of the convolution of with itself.

The Lebesgue space can be identified with the closed ideal in of measures absolutely continuous with respect to the Lebesgue measure on . Namely, if , then is associated with the measure for each Borel subset of . Hence . In particular, if , where and is such that and on , then is called the probability density function of , or the probability density for short.

We define the inverse Fourier transform by

. Then the inversion formula holds for suitable .

*Proof of Theorem 1. *The conditions (7) and (8) imply that there exists a sequence of real numbers
such that
where , , and for . Note that in Equation (27) also might be . Let . Then it is immediate that and
for all . Fix any in Equation (28). Since is an open connected component of , we have there are two continuous functions such that
for all . Using the identity , it follows from Equations (29) and (30) that, for each for , there exists some integer in such that
for all . Therefore, for any , we have
where is the indicator function of the set and denotes the positive th root of positive number , . We claim that . Indeed, Equation (4) implies that and are positive numbers. Then Equation (30) implies that . Combining this with Equation (31), yields the claim. Next, applying the property Equation ((3)()), we get that for all . Therefore, we conclude from Equation (32) that
for all . Finally, keeping in mind that each , , may take any value in , we obtain from Equation (33) the estimate Equation (9). Theorem 1 is proved.

*Remark 6. *Of course, we are not claiming that each of th possible functions in Equation (33) belongs to .

*Remark 7. *In the proof of Theorem 1 we concerned with the so-called problem of phase retrieval (see the equality Equation (29)), i.e., the problem of the recovery of a measure given the amplitude of its Fourier transform . This problem is well known in various fields of science and engineering, including crystallography, nuclear magnetic resonance and optics (see, for example, survey [17]).

*Proof of Theorem 2. *Note that as in the proof of Theorem 1, in light of Equation (10), we see that there exists a sequence of real numbers Equation (27) such that
Let us split our proof into two cases. First we consider the case when all in Equation (34) are finite intervals. Denote by the minimal length of , , i.e.,

Let . Assume, in addition, that is real-valued on and

For example, we can take , where the truncated power function was defined by Equation (11). Next, for each , we take any sequence of real numbers such that

In addition, we assume that for all . Then we define the function . Now Equations (37) and (38) imply that is supported on and . Since is real-valued on and satisfies Equation (36), we conclude that is even and positive on . Therefore, the function Equation (39) and the function are strictly positive on and on , respectively.

Let us define the function such that it is supported on and . To this end, we take any sequence of real numbers such that for . Next, for an arbitrary sequence of positive numbers , we define . Of course, Equations (41) and (42) imply that is supported on and .

Finally, given any fixed sequence of positive numbers , we set

We claim that and . First, as real-valued function satisfies Equation (36), it follows from Equations (34)–(38) and from Equations (41)–(42) that is continuous on and

Second, since is continuous and compactly supported, it follows that . Therefore, the inverse Fourier transform of is well-defined. Hence

Let us fix the previously chosen positive numbers and . Then we increase if necessary, the value of in such a way that

Bochner’s theorem shows that for all , since . Combining Equation (36) with (48), we see that is nonnegative on and . In addition, we conclude (see, e.g., [15, p. 409]) that . Thus, applying the Fourier transform to and using again Bohner’s theorem, we see that is continuous nontrivial positive definite, i.e., . This proves our claim.

Define . We claim that satisfies the hypotheses of Theorem 2. Let us first prove that . Indeed, from Equation (27) we see that the essential support of , defined by Equation (44) can be represented as the union of a family of pairwise disjoint sets , . Therefore, . We have already proven that . On the other hand, it is well known that for each and any , it follows that . Thus, . Combining this fact with Equation (41), we conclude that and .

Second, we will prove that the function defined by Equation (50) has the property Equation (11). To this end, let denote the group . Given , define . We claim that . By the same argument as before for the function defined by Equation (44), we see that . Therefore, is well-defined and

. Now using Equation (48), we get that for all and . Hence, by Bochner’s theorem it follows that . Next, . Combining this representation with Equations (49) and (50), we see that , which yields our claim.

Finally, again using the fact that is the union of a family of pairwise disjoint sets , , we conclude from Equation (51) that for some , if and only if . This proves Equation (11) in the case where each in Equation (33) is a finite interval.

Now we consider the second case with , i.e., if in Equation (33) we have . Using the same satisfying Equation (36), we define the functions and , by Equations (43) and (39)–(40), respectively. For , let us take an arbitrary sequence of positive numbers such that

Then we define . Obviously, and . Next, for the function , defined by Equation (44), it follows from Equation (47) that

Again, for fixed positive numbers , , and , we take the value of in such a way that

Combining Equation (36) with Equation (57), we conclude from Equation (56) that is nonnegative on and . Thus, . Finally, we claim that the function defined by Equation (49) also satisfies the hypotheses of Theorem 2 in our case with . The proof of this claim is exactly the same as that of the first case. Therefore, we skip the details of this proof. Theorem 2 is proved.

*Proof of Example 3. *We claim that there are , , and , such that the function in Equation (13) coincides with the function defined by Equation (50). Indeed, set , , , and

Since defined by Equation (35) is equal now to , then defined above satisfies Equations (37) and (38). Next, set

For , and , let be given by

It is easily seen that , and are supported on a family of pairwise disjoint sets. Therefore, since . Next

The function coincides with the function in Equation (13) and is defined by the same rules as in Equation (50). Our claim is proved.

Now, it is enough to show that . Indeed, for all , since . Therefore, Bochner’s theorem shows that .

By repeating the finally part of the proof of Theorem 2, we complete the proof of Example 3.

*Proof of Example 4. *This example concerns the case that was considered in the second part of the proof of Theorem 2, i.e., when contains two unbounded components and . Also, as in the proof of Theorem 2, is enough to show that Equations (54) and (57) are satisfied. Indeed, Equation (54) is clear, since , . We conclude from Equations (13) and (16) that

Therefore, a simple calculation shows that Equation (57) is also satisfied. This completes the proof.

*Proof of Theorem 5. *We will prove this theorem using essential the same techniques as in the proof of Theorem 2. Therefore, we sketch the proof only. From Equations (19) and (20) we have that there exits an infinite sequence
such that
where and for each . Moreover, from Equation (20) we see that

Let be the same function satisfying Equation (36). For and for , we define the functions and by Equations (39) and (43)., respectively. Note that the sequences and satisfy Equations (37)–(38) and (40)–(42), respectively. For any , let us define where is the length of , i.e., . Note that the condition (20) guarantees that is a well-defined sequence of positive numbers. Set . Combining Equations (4), (39), and (43) with (68), we get for any . Here, denotes the indicator function of a subset . Hence,

Therefore, and

Again, for fixed positive numbers , we take the value of in such a way that

Combining this condition with Equation (67) and Equation (68), we conclude that the function in Equation (61) is nonnegative on and . Thus, . Define . Next, for , set

. We claim that: (i) , (ii) defined by Equation (66) satisfies Equations (19) and (20), and (iii) . The proofs of these claims is exactly the same as in the case of functions and defined by Equations (49) and (51), respectively.

Finally, again using the fact that and therefore, are the unit of a family of pairwise disjoint sets , , where , , we see from Equation (75) that for some , if and only if . This proves Equation (21) and therefore completes the proof our theorem.

#### 3. Conclusion

We study the -divisible functions in , where denotes the family of continuous positive definite functions on the real line . While there is rich literature on infinite-divisible functions in , for an integer , properties of -divisible functions from have been studied much less. Surprisingly, it appears that some properties of infinite-divisible and -divisible functions may differ in essence. In this paper, we examine one such property, which has not yet been discussed in detail in the literature. More precisely, if is infinite-divisible, then it is well-known that, for each integer , there is an unique , such that . On the other hand, there is -divisible such that the factor in is generally not unique. For -divisible , we study the following questions: (i) how rich can be the class ; and (ii) what properties of determine the size of . We present several precise estimates for the cardinality of . Also, the main results are validated via illustrative examples.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The author declares that they have no competing interests.

#### Acknowledgments

This research was funded by Vilnius University as part of the project: “Analysis and Application of Probabilistic and Deterministic Models”.