# A Note onn-Divisible Positive Definite Functions

Authors: Saulius Norvidas
Journal: Journal of Mathematics
 (2022)

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## Abstract

LetPDℝ be the family of continuous positive definite functions on ℝ. For an integer n>1, a f∈PDℝ is called n-divisible if there is g∈PDℝ such that gn=f. Some properties of infinite-divisible and n-divisible functions may differ in essence. Indeed, if f is infinite-divisible, then for each integer n>1, there is an unique g such that gn=f, but there is a n-divisible f such that the factor g in gn=f is generally not unique. In this paper, we discuss about how rich can be the class g∈PDℝ:gn=f for n-divisible f∈PDℝ and obtain precise estimate for the cardinality of this class.


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## Body

## 1. Introduction

We start with some notations and definitions. Letℤ, ℕ, ℝ, and ℂ be the families of integers, non-negative integers, real, and complex numbers, respectively. In the sequel, Mℝ denotes the Banach algebra of bounded regular complex-valued Borel measures on ℝ with the convolution as multiplication. Mℝ is equipped with the usual total variation norm μ of μ∈Mℝ. The Fourier-Stieltjes transform of μ∈Mℝ is given by
(1)μ^x=∫ℝe−ixtdμt.A functionf:ℝ⟶ℂ is said to be positive definite if
(2)∑j,k=1mfxj−xkcjc¯k≥0,for each m∈ℕ and all c1,⋯,cm∈ℂ, x1,⋯,xm∈ℝ. Any such a function satisfies
(3)f−x=fx¯,for all x∈ℝ.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 byPDℝ the family of continuous nontrivial (≡0) positive definite functions on ℝ. Note that if positive definite f is continuous in a neighborhood of the origin, then it is uniformly continuous in ℝ (see, e.g., [4, Corollary 1.4.10]). Bochner’s theorem gives a description of f∈PDℝ in terms of the Fourier transform. Namely, according to this theorem (see, e.g., [6], p. 71]), a continuous function f:ℝ⟶ℂ is positive definite if and only if there exists a nonnegative μf∈Mℝ such that fx=μf^x, x∈ℝ. This statement implies, in particular, that
(4)fx≤f0=μf,for all x∈ℝ. If, in addition, positive μf∈Mℝ satisfies μf=1, then on the language of probability theory, such a μf and the function fx=μf^−x, x∈ℝ, 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 variableξ is called n-divisible for certain n∈ℕ, n>1, if there exist independent and identically distributed random variables ξ1,⋯,ξn such that ξ1+⋯+ξn has the sane distribution as ξ. In terms of the characteristic function f of a real-valued random variable, this means that f is n-divisible if there exists g∈PDℝ such that
(5)fx=gnx,for all x∈ℝ. Next, f∈PDℝ is said to be infinite-divisible if it is n-divisible for each n∈ℕ, n≥2.In the sequel,PDnℝ and PD∞ℝ denotes the families of n-divisible and infinite divisible functions in PDℝ, 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 n-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 inPD∞ℝ has rich literature (see, e.g., [9, 13, 14]), but the n-divisible functions have been studied much less; and (ii) partly from the fact that some properties of functions from PD∞ℝ and from PDnℝ may differ in essence. One of those properties is the following: if f∈PD∞ℝ, then for each n∈ℕ, n>1, there is an unique g∈PDℝ such that gn=f, but there is n-divisible f such that the factor g in gn=f is generally not unique. In this paper, we study the following problems: (i) how rich can be the class g∈PDℝ:gn=f; and (ii) what properties of f determine the size of g∈PDℝ:gn=f. We present several precise estimates for the cardinality of this class. Also, the main results are validated via illustrative examples.More precise, forn∈ℕ, n>1 and f∈PDnℝ, we wish to study the family
(6)Dnf=g∈PDℝ:gn=fand the quantity cardDnf, i.e., the cardinality of Dnf. Note that there exists f∈PDnℝ\PD∞ℝ such that the factor g in Equation (5) will generally be not unique. It turns out that cardDnf depends on n and in some way also depends on the geometric structure of the zeros set Nf=x∈ℝ:fx=0 of f∈PDnℝ (see our Theorem 1 below). The essential support Sf of f∈PDℝ is defined by Sf=ℝ\Nf. Combining Equation (3) with Equation (4), gives
(7)0∈Sfand−Sf=Sf.Since functionsf∈PDℝ are continuous on ℝ, it follows that Sf is an open subset of ℝ. Therefore, Sf can be represented as a finite or infinite but countable Sf=⋃j∈ΣEj, where Ejj∈Σ is the family of all open connected components of Sf. In the sequel, compSf denotes the cardinality of Σ. According to relations (7), we see that either there is an k∈ℕ such that compSf=2k−1 or compSf=∞.Theorem 1.
Letn∈ℕ, n≥2, and let f∈PDnℝ. Assume that
(8)compSf=2k−1,for some k∈ℕ. Then
(9)cardDnf≤nk−1.The following theorem shows that the estimate (9) is accurate.Theorem 2.
For eachn∈ℕ, n≥2, each k∈ℕ, and any open subset Eofℝ which satisfies
(10)0∈E,−E=EandcompE=2k−1,there exists f∈PDnℝ, such that Sf=E and
(11)cardDnf=nk−1.We will present two examples off∈PDnℝ such that Equation (11) is satisfied. In order to make the examples easier to understand, we will consider only small values of k. For α>0, set
(12)Λαx=max1−x;0α,x∈ℝ. Note that Λα∈PDℝ if and only if α≥1 (see, e.g., [15, 16, p. 282]). We start with the case where Sf is a bounded subset of ℝ.Example 3.
Forn∈ℕ, n>1, let
(13)f1x=Λnx+123nΛ1x−π−1+Λ1x−2π+1n+Λ1x+π+1+Λ1x+2π−1n+124nΛ1x−11+Λ1x−12+Λ1x−13+Λ1x−14n+Λ1x+11+Λ1x+12+Λ1x+13+Λ1x+14n.
ThencompSf1=5, since
(14)Sf1=−15,−10∪−2π,−π∪−1,1∪π,2π∪10,15.
Moreover,f1∈PDnℝ and
(15)cardDnf1=n5.
Now let us give an example off∈PDnℝ such that Sf has unbounded components.Example 4.
For anya>15 and n∈ℕ, n>1, let
(16)f2x=f1x+122n∑r=1∞12rΛ1x−a−rn+∑r=1∞12rΛ1x+a+rn,where f1 was defined by Equation (13). Then compSf2=7, since
(17)Sf2=−∞,−a∪−15,−10∪−2π,−π∪−1,1∪π,2π∪10,15∪a,∞.
Moreover,f2∈PDnℝ and
(18)cardDnf2=n7.Theorem 5.
Suppose that an open subsetE of ℝ satisfies
(19)0∈E,−E=EandcompE=∞.
LetEjj∈Σ be the family of open connected components of E. Assume that there is σ>0 such that
(20)infj∈Σsupa,b∈Eja−b=2σ.
Then, for anyn∈ℕ, n≥2, there exists f∈PDnℝ such that Sf=E and
(21)cardDnf=∞.


## 2. Preliminaries and Proofs

Ifν,μ∈Mℝ, then the convolution ν∗μ is defined by
(22)ν∗μE=∫ℝνE−xdμxfor each Borel subset E of ℝ. Note that
(23)ν∗μ^=ν^·μ^.In particular, for anyn∈ℕ,
(24)μ∗n^=μ∧n,where the convolution power μ∗n is defined as the n-fold iteration of the convolution of μ with itself.The Lebesgue spaceL1ℝ can be identified with the closed ideal in Mℝ of measures absolutely continuous with respect to the Lebesgue measure dx on ℝ. Namely, if φ∈L1ℝ, then φ is associated with the measure
(25)μφE=∫Eφtdtfor each Borel subset E of ℝ. Hence φ^x=∫ℝe−itxφtdt. In particular, if μ=φtdt, where φ∈L1ℝ and φ is such that φL1ℝ=1 and φ≥0 on ℝ, then φ is called the probability density function of μ, or the probability density for short.We define the inverse Fourier transform by(26)ψ˘t=12π∫ℝe−itxψxdx,t∈ℝ. Then the inversion formula ψ˘^=ψ holds for suitable ψ∈L1ℝ.Proof of Theorem 1.
The conditions (7) and (8) imply that there exists a sequence of real numbers
(27)0<b0<a1<b1<a2<b2<⋯<ak−1<bk−1,such that
(28)Sf=⋃j=−1−k−1Ej⋃E0⋃⋃j=1k−1Ej,where E0=−b0,b0, Ej=aj,bj, and E−j=−Ej=−bj,−aj for j=1,2,⋯,k−1. Note that in Equation (27) also might be bk−1=+∞. Let g∈Dnf. Then it is immediate that Sf=Sg and
(29)gxn=fxfor all x∈Sf. Fix any Ej⊂Sf in Equation (28). Since Ej is an open connected component of Sf, we have there are two continuous functions uf,j,ug,j:Ej⟶−π,π such that
(30)fx=fxeiuf,jxandgx=gxeiug,jxfor all x∈Ej. Using the identity gn=f, it follows from Equations (29) and (30) that, for each for j∈−k−1,⋯,k−1, there exists some integer mj in 0,⋯,n−1 such that
(31)n·ug,jx=uf,jx+2πmjfor all x∈Ej. Therefore, for any x∈Sf, we have
(32)gx=∑j=−k−1k−1gxχEjxeiug,j=∑j=−k−1k−1fx+1/nχEjxeiuf,j+2πmj/n,where χEj is the indicator function of the set Ej and fx+1/n denotes the positive nth root of positive number fx, x∈Sf. We claim that m0=0. Indeed, Equation (4) implies that f0 and g0 are positive numbers. Then Equation (30) implies that uf,00=ug,00=0. Combining this with Equation (31), yields the claim. Next, applying the property Equation ((3)()), we get that m−j=−mj for all j∈0,1,⋯,k−1. Therefore, we conclude from Equation (32) that
(33)gx=fx+1/nχE0x+∑j=1k−1fx+1/nχEj−xe−iuf,jx+2πmj/n+χEjxeiuf,jx+2πmj/n,for all x∈Sg. Finally, keeping in mind that each mj, j=1,2,⋯,k−1, may take any value in 0,1,⋯,n−1, we obtain from Equation (33) the estimate Equation (9). Theorem 1 is proved.Remark 6.
Of course, we are not claiming that each ofnk−1th possible functions in Equation (33) belongs to PDℝ.Remark 7.
In the proof of Theorem1 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 μ given the amplitude f of its Fourier transform f=μ^. 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 Theorem1, in light of Equation (10), we see that there exists a sequence of real numbers Equation (27) such that
(34)E=⋃j=−1−k−1Ej⋃E0⋃⋃j=1k−1Ej.
Let us split our proof into two cases. First we consider the case when allEj in Equation (34) are finite intervals. Denote by 2σ the minimal length of Ej, j=−k−1,⋯,k−1, i.e.,
(35)2σ=min2b0;min1≤j≤k−1bj−aj.Letφ∈PDℝ. Assume, in addition, that φ is real-valued on ℝ and
(36)Sφ=−σ,σ.For example, we can takeφx=Λαx/σ, where the truncated power function Λα was defined by Equation (11). Next, for each j∈1,2,⋯,k−1, we take any sequence of real numbers τj,ss=1mj such that
(37)aj+σ=τj,1<τj,2<⋯<τj,m=bj−σ.In addition, we assume that(38)τj,s+1−τj,s<2σfor all s=1,⋯,mj. Then we define the function
(39)ujx=∑s=1mjφx−τj,s,x∈ℝ. Now Equations (37) and (38) imply that uj is supported on aj,bj=Ej¯ and Suj=Ej. Since φ∈PDℝ is real-valued on ℝ and satisfies Equation (36), we conclude that φ is even and positive on Sφ. Therefore, the function Equation (39) and the function
(40)uj−x=∑s=1mjφx+τj,sare strictly positive on Ej and on E−j=−Ej, respectively.Let us define the functionu0 such that it is supported on −b0,b0=E0¯ and Su0=E0. To this end, we take any sequence θii=0r of real numbers such that
(41)−b0+σ=θr<⋯θ1<θ0=0,(42)θi−θi+1<2σ,for i=0,⋯,l−1. Next, for an arbitrary sequence of positive numbers ωii=0l, we define
(43)u0x=∑i=0lωiφx−θi+φx+θi,x∈ℝ. Of course, Equations (41) and (42) imply that u0 is supported on −b0,b0=E0¯ and Su0=E0.Finally, given any fixed sequence of positive numbersαjj=1k−1, we set
(44)ux=u0x+∑j=1k−1αjujx+uj−x,(45)x∈ℝ.We claim thatu∈PDℝ and Su=E. First, as real-valued function φ∈PDℝ satisfies Equation (36), it follows from Equations (34)–(38) and from Equations (41)–(42) that u is continuous on ℝ and
(46)Su=⋃j=−kkEj=E.Second, sinceu is continuous and compactly supported, it follows that u∈L1ℝ. Therefore, the inverse Fourier transform of u is well-defined. Hence
(47)u˘t=u˘0t+∑j=1k−1αjφ˘t∑s=1mj2cosτj,s·t=2φ˘t∑i=0lωicosθi·t+∑j=1k−1αj∑s=1mjcosτj,s·t=2φ˘tω0+∑i=1lωicosθi·t+∑j=1k−1αj∑s=1mjcosτj,s·t.Let us fix the previously chosen positive numbersω1,⋯,ωk and α1,⋯,αk−1. Then we increase if necessary, the value of ω0 in such a way that
(48)ω0>∑i=1lωi+∑j=1k−1αjmj.Bochner’s theorem shows thatφ˘t≥0 for all t∈ℝ, since φ∈PDℝ. Combining Equation (36) with (48), we see that u˘ is nonnegative on ℝ and u˘≢0. In addition, we conclude (see, e.g., [15, p. 409]) that u˘∈L1ℝ. Thus, applying the Fourier transform to u˘ and using again Bohner’s theorem, we see that u is continuous nontrivial positive definite, i.e., u∈PDℝ. This proves our claim.Define(49)fx=u0nx+∑j=1k−1αjnujnx+ujn−x,x∈ℝ. We claim that f satisfies the hypotheses of Theorem 2. Let us first prove that f∈PDℝ. Indeed, from Equation (27) we see that the essential support Su of u, defined by Equation (44) can be represented as the union ⋃j=−k−1k−1Ej of a family of pairwise disjoint sets Ej=Suj, j=−k−1⋯,k−1. Therefore,
(50)unx=u0nx+∑j=1k−1αjnujnx+ujn−x,x∈ℝ. We have already proven that u∈PDℝ. On the other hand, it is well known that for each n∈ℕ and any ζ∈PDℝ, it follows that ζn∈PDℝ. Thus, un∈PDℝ. Combining this fact with Equation (41), we conclude that f∈PDℝ and Sf=Su=⋃j=−k−1k−1Ej=E.Second, we will prove that the functionf defined by Equation (50) has the property Equation (11). To this end, let ℤn denote the group ℤn=ℤ/nℤ≅0,1,⋯,n−1. Given Λ=p1,p2,⋯,pk−1∈ℤnk−1, define
(51)gΛx=u0x+∑j=1k−1αjei2πpj/nujx+e−i2πpj/nuj−x,x∈ℝ. We claim that gΛ∈Dnf. By the same argument as before for the function u defined by Equation (44), we see that gΛ∈L1ℝ. Therefore, gΛ˘ is well-defined and
(52)gΛ˘t=2φ˘tω0+∑i=1lωicosθi·t+∑j=1k−1αj∑s=1mjcosτj,s·t+2πpjn,t∈ℝ. Now using Equation (48), we get that gΛ˘t≥0 for all t∈ℝ and gΛ˘≢0. Hence, by Bochner’s theorem it follows that gΛ∈PDℝ. Next,
(53)gΛnx=u0nx+∑j=1k−1αjnei2πpj/nujxn+e−i2πpj/nuj−xn=u0nx+∑j=1k−1αjnujxn+uj−xn,x∈ℝ. Combining this representation with Equations (49) and (50), we see that gΛ∈Dnf, which yields our claim.Finally, again using the fact thatSgΛ is the union of a family of pairwise disjoint sets Suj, j=−k−1,⋯,k−1, we conclude from Equation (51) that gΛ≡gΛ1 for some Λ,Λ1∈ℤnk−1, if and only if Λ=Λ1. This proves Equation (11) in the case where each Ej in Equation (33) is a finite interval.Now we consider the second case withbk−1=∞, i.e., if in Equation (33) we have Ek−1=ak−1,∞. Using the same φ∈PD satisfying Equation (36), we define the functions u0 and uj, j=−k−2,⋯,−1,1,⋯,k−2 by Equations (43) and (39)–(40), respectively. For j=k−1, let us take an arbitrary sequence of positive numbers γrr=1∞ such that
(54)∑r=1∞γr=1.Then we define(55)uk−1x=∑r=1∞γrφx−ak−rσ,x∈ℝ. Obviously, suppuk−1=Ek−1¯ and Suk−1=Ek−1. Next, for the function u, defined by Equation (44), it follows from Equation (47) that
(56)u˘t=2φ˘tω0+∑i=1lωicosθi·t+∑j=1k−2αj∑s=1mjcosτj,s·t+αk−1∑r=1∞γrcosak−1+rσt.Again, for fixed positive numbersω1,⋯ωk, α1,⋯,αk−1, and γ1,γ2,⋯, we take the value of ω0 in such a way that
(57)ω0>∑i=1lωi+∑j=1k−2αjmj+αk−1.Combining Equation (36) with Equation (57), we conclude from Equation (56) that u is nonnegative on ℝ and u≡0. Thus, u∈PDℝ. Finally, we claim that the function f defined by Equation (49) also satisfies the hypotheses of Theorem 2 in our case with Ek−1=ak−1,∞. 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 areu0, u1, u2 and α1, α2 such that the function in Equation (13) coincides with the function defined by Equation (50). Indeed, set φ=Λ1, j=3, m1=2, m2=4 and
(58)τ11=π+1,τ12=2π−1,τ21=11,τ22=12,τ23=13,τ24=14.Sinceσ defined by Equation (35) is equal now to 1, then τjs defined above satisfies Equations (37) and (38). Next, set
(59)v1x=Λ1x−π−1+Λ1x−2π+1,v2x=Λ1x−11+Λ1x−12+Λ1x−13+Λ1x−14.Forv0=Λ1, α1=1/8 and α2=1/16, let v be given by
(60)vx=Λ1x+18v1x+v1−x+116v2x+v2−x.It is easily seen thatΛ1, v1 and v2 are supported on a family of pairwise disjoint sets. Therefore,
(61)vnx=Λnx+18v1nx+v1n−x+116v2nx+v2n−x,since Λ1n=Λn. Next
(62)Svn=Sv=−15,−10∪−2π,−π∪−1,1∪π,2π∪10,15.The functionvn 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 thatv∈PDℝ. Indeed,
(63)v˘t=2Λ˘1t1+∑j=12αj∑s=1mjcosτjs·t=2Λ˘1t1+18cosτ11t+cosτ12t+116cosτ21t+cosτ22t+cosτ23t+cosτ24t≥Λ˘1t≥0for all t∈ℝ, since Λ1∈PDℝ. Therefore, Bochner’s theorem shows that v∈PDℝ.By repeating the finally part of the proof of Theorem 2, we complete the proof of Example3.Proof of Example 4.
This example concerns the case that was considered in the second part of the proof of Theorem2, i.e., when Sf contains two unbounded components −∞,−a and a,∞. 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 γr=1/2r, r=1,2,⋯. We conclude from Equations (13) and (16) that
(64)k=4,σ=1,m1=2,m2=4,ω0=1,ω1=0,α1=18,α1=116andα3=14.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 Theorem2. Therefore, we sketch the proof only. From Equations (19) and (20) we have that there exits an infinite sequence
(65)0<b0<a1<b1<a2<b2<⋯<∞such that
(66)E=⋃j∈ℤEj,where E0=−b0,b0 and Ej=aj,bj=−E−j for each j∈ℕ. Moreover, from Equation (20) we see that
(67)min2b0;infj∈ℕbj−aj=2σ>0.Letφ∈PDℝ be the same function satisfying Equation (36). For j=0 and for j∈ℕ, we define the functions u0 and uj by Equations (39) and (43)., respectively. Note that the sequences τjss=1mj and θii=0l satisfy Equations (37)–(38) and (40)–(42), respectively. For any j∈ℕ, let us define
(68)αj=12jmjEj,where Ej is the length of Ej, i.e., Ej=supa−b:a,b∈Ej. Note that the condition (20) guarantees that αjj∈ℕ is a well-defined sequence of positive numbers. Set
(69)ux=u0x+∑j∈ℕαjujx+uj−x,x∈ℝ. Combining Equations (4), (39), and (43) with (68), we get
(70)ux≤2φ0∑i=0lωiIE0x+∑j∈ℕ12jEjIEjxfor any x∈ℝ. Here, IDx denotes the indicator function of a subset D⊂ℝ. Hence,
(71)φL1ℝ≤2φ02b0∑i=0lωi+1.Therefore,φ∈L1ℝ and
(72)ut=2φtω0+∑i=1lωicosθi·t+∑j∈ℕαj∑s=1mjcosτjs·t.Again, for fixed positive numbersω1,⋯ωk, we take the value of ω0 in such a way that
(73)ω0>∑i=1lωi+12σ.Combining this condition with Equation (67) and Equation (68), we conclude that the function u˘ in Equation (61) is nonnegative on ℝ and u˘≢0. Thus, u∈PDℝ. Define
(74)fx=u0nx+∑j∈ℕαjnujnx+ujn−x,x∈ℝ. Next, for Λ=p1,p2,⋯∈ℤ∞, set
(75)gΛx=u0x+∑j∈ℕαjei2πpj/nujx+e−i2πpj/nuj−x,x∈ℝ. We claim that: (i) f∈PDℝ, (ii) E defined by Equation (66) satisfies Equations (19) and (20), and (iii) gΛ∈Dnf. The proofs of these claims is exactly the same as in the case of functions f and gΛ defined by Equations (49) and (51), respectively.Finally, again using the fact thatSf=Su and therefore, SgΛ=Su are the unit of a family of pairwise disjoint sets Suj, j∈ℤ, where u−jx=uj−x, j∈ℕ, we see from Equation (75) that gΛ≡gΛ1 for some Λ,Λ1∈ℤ∞, if and only if Λ=Λ1. This proves Equation (21) and therefore completes the proof our theorem.


## 3. Conclusion

We study then-divisible functions in PDℝ, where PDℝ denotes the family of continuous positive definite functions on the real line ℝ. While there is rich literature on infinite-divisible functions in PDℝ, for an integer n>1, properties of n-divisible functions from PDℝ have been studied much less. Surprisingly, it appears that some properties of infinite-divisible and n-divisible functions f∈PDℝ 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 f∈PDℝ is infinite-divisible, then it is well-known that, for each integer n>1, there is an unique g∈PDℝ, such that gn=f. On the other hand, there is n-divisible f such that the factor g in gn=f is generally not unique. For n-divisible f∈PDℝ, we study the following questions: (i) how rich can be the class Dnf=g∈PDℝ:gn=f; and (ii) what properties of f determine the size of Dnf. We present several precise estimates for the cardinality of Dnf. Also, the main results are validated via illustrative examples.



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