A series representation of the discrete fractional Laplace operator of arbitrary order
Tiffany Frugé Jones, Evdokiya Georgieva Kostadinova, Joshua Lee Padgett, Qin Sheng
aa r X i v : . [ m a t h . C A ] J a n A series representation of the discrete fractionalLaplace operator of arbitrary order
Tiffany Frugé Jones , Evdokiya Georgieva Kostadinova ,Joshua Lee Padgett , , and Qin Sheng , Department of Mathematics, University of Arizona,Tucson, Arizona 85721, USA, e-mail: [email protected] Center for Astrophysics, Space Physics, and Engineering Research,Baylor University, Waco, Texas 76798, USA, e-mail: [email protected] Department of Mathematical Sciences, University of Arkansas,Fayetteville, Arkansas 72701, USA, e-mail: [email protected] Center for Astrophysics, Space Physics, and Engineering Research,Baylor University, Waco, Texas 76798, USA Department of Mathematics, Baylor University,Waco, Texas, USA, e-mail: [email protected]
January 13, 2021
Abstract
Although fractional powers of non-negative operators have received much attentionin recent years, there is still little known about their behavior if real-valued exponentsare greater than one. In this article, we define and study the discrete fractional Laplaceoperator of arbitrary real-valued positive order. A series representation of the discretefractional Laplace operator for positive non-integer powers is developed. Its convergenceto a series representation of a known case of positive integer powers is proven as thepower tends to the integer value. Furthermore, we show that the new representationfor arbitrary real-valued positive powers of the discrete Laplace operator is consistentwith existing theoretical results.
Contents
The discrete Laplace operator of arbitrary order 7
Due to its wide array of applications in multi-physical sciences, the construction and approx-imation of fractional powers of the Laplace operator have been of great interest for nearly acentury (cf., e.g., [4, 28, 30, 36, 42] and references therein). Classically, only fractional powersof the order s ∈ (0 , are considered, and in this case, one can define the fractional Laplaceoperator applied to a smooth enough function in a natural way. Specifically, for d ∈ N , s ∈ (0 , let u : R d → R be a smooth function, and for every ε ∈ (0 , ∞ ) , x ∈ R d let B ε ( x ) be the d -dimensional ball of radius ε centered at x (with respect to the typical topology of R d ). Then for every x ∈ R d we may define the s -order fractional Laplace operator appliedto u at x as (cid:0) ( − ∆) s u (cid:1) ( x ) = c d,s lim ε → + (cid:20)Z R d \ B ε ( x ) u ( x ) − u ( y ) | x − y | d +2 s dy (cid:21) , (1.1)where c d,s ∈ [0 , ∞ ) is a known normalization constant.It is worth noting that the recent rapid increase in interest in the fractional Laplaceoperator is also due to the seminal work of Caffarelli and Silvestre [5]. In their work, itwas shown that one may study the non-local operator given by (1.1) via the Dirichlet-to-Neumann operator associated with a particular extension problem posed in R d × [0 , ∞ ) (albeit, one trades the non-locality for a problem which is either singular or degeneratedepending upon the value of s ∈ (0 , ). The employed Dirichlet-to-Neumann operator isa particular example of the Poincaré-Stecklov operator (cf., e.g., [26]). For a fixed domain,the Poincaré-Stecklov operator is known to map the boundary values of a harmonic functionto the normal derivative values of the same harmonic function on the same boundary. Wecan summarize the results of Caffarelli and Silvestre (cf., e.g., [5, Eq. (3.1)]) as follows. Let d ∈ N , s ∈ (0 , , let u : R d → R be a smooth function, and let v : [0 , ∞ ) × R d → R satisfyfor all x ∈ R d that v (0 , x ) = u ( x ) and for all t ∈ (0 , ∞ ) , x ∈ R d that (cid:16) ∂ ∂t v (cid:17) ( t, x ) + − st (cid:0) ∂∂t v (cid:1) ( t, x ) + (∆ x v )( t, x ) = 0 . (1.2)2hen there exists c ∈ [0 , ∞ ) such that for all x ∈ R d it holds that (cid:0) ( − ∆) s u (cid:1) ( x ) = c (cid:20) lim t → + t − s (cid:0) ∂∂t v (cid:1) ( t, x ) (cid:21) . (1.3)Interestingly, the constant c ∈ [0 , ∞ ) in (1.3) depends only upon the parameter s ∈ (0 , and not upon d ∈ N . More importantly, this demonstrates that one may trade out thehighly non-local problem given by (1.1) for the local problem given by (1.2) and (1.3).This technique has also been recently further generalized to cases of arbitrary non-negativeoperators defined on Banach spaces [2, 16, 31, 32, 34].While the above formulations (i.e., (1.1), (1.2), and (1.3)) may be used to provide insightsinto the continuous fractional Laplace operator with order s ∈ (0 , , they cannot be directlyused to provide any insight into the discrete case or the case where s ∈ (0 , ∞ ) . The discretecase is a natural consideration as it arises in the study of numerous physically relevantphenomena (cf., e.g., [22,23,35] and references therein) and also in an attempt to numericallyapproximate (1.1). The consideration of a truly discrete case—that is, the case which isthe fractional power of the discrete Laplace operator rather than a direct approximationof (1.1)—was originally studied by Ciaurri et al. [9]. By employing the basic languageof semigroups (e.g., a special case of Ciaurri et al. [8, Eq. (1)] combined with Padgett [34,Theorem 2.1]) Ciaurri et al. were able to develop the first series representation for the discretefractional Laplace operator of order s ∈ (0 , (cf. Definition 3.10, for clarity). Moreover,it was shown that this formulation did converge to the continuous case via adaptive meshrefinements (cf. Ciaurri et al. [8, Theorems 1.7 and 1.8]).The consideration of higher-order fractional Laplace operators has recently received anincrease in attention in continuous cases (cf., e.g., [7, 15, 17, 37, 43]), but to the authors’knowledge the only study in the discrete case has been carried out by Padgett et al. [35].Rectifying this aforementioned gap in theory is the primary goal of this article. In particular,we will develop a series representation of the discrete fractional Laplace operator of order s ∈ (0 , ∞ ) . This development is illustrated in Theorem 1.1, which is a partial descriptionof the main result of this article focused on the case of positive non-integer powers of thediscrete Laplace operator. Theorem 1.1.
Let m ∈ { , , , . . . } , s ∈ ( m − , m ) , let Z = { . . . , − , − , , , , . . . } , let R be the real number field, let ℓ ( Z ) be the set of all u : Z → R which satisfy that P k ∈ Z | u ( k ) | < ∞ , let − ∆ : ℓ ( Z ) → ℓ ( Z ) satisfy for all u ∈ ℓ ( Z ) , n ∈ Z that ( − ∆ u )( n ) = 2 u ( n ) − u ( n − − u ( n + 1) , and let v : [0 , ∞ ) × Z → R satisfy for all n ∈ Z that v (0 , n ) = (( − ∆) m − u )( n ) and for all t ∈ (0 , ∞ ) , n ∈ Z that (cid:16) ∂ ∂t v (cid:17) ( t, x ) + − s − m +1) t (cid:0) ∂∂t v (cid:1) ( t, x ) + (∆ v )( t, x ) = 0 . (1.4) Then Note that we define integer powers of − ∆ : ℓ ( Z ) → ℓ ( Z ) inductively. That is, we have for all k ∈ N , u ∈ ℓ ( Z ) , n ∈ Z that if k = 0 it holds that (( − ∆) k u )( n ) = − u ( n ) and if k ∈ N it holds that (( − ∆) k u )( n ) =( − ∆( − ∆) k − u )( n ) . i) there exists c ∈ [0 , ∞ ) such that for all n ∈ Z it holds that (cid:0) ( − ∆) s − m +1 ( − ∆) m − u (cid:1) ( n ) = (cid:0) ( − ∆) s u (cid:1) ( n ) = c (cid:20) lim t → + t − s − m +1) (cid:0) ∂∂t v (cid:1) ( t, n ) (cid:21) (1.5) and(ii) there exists K : Z → R , C ∈ [0 , ∞ ) such that for all n ∈ Z \{ } it holds that | K ( n ) | ≤ C | n | − (1+2 s ) and for all n ∈ Z it holds that K ( − n ) = K ( n ) and (cid:0) ( − ∆) s u (cid:1) ( n ) = X k ∈ Z K ( k ) (cid:0) u ( n ) − u ( n − k ) (cid:1) . (1.6)We now provide some clarifying remarks regarding the above result. Item (i) of Theo-rem 1.1 above is a direct consequence of combining Definition 3.11 and Padgett [34, The-orem 2.1] (applied for every n ∈ Z with s x s − m + 1 , A x ∆ , u x (( − ∆) m − u )( n ) , ( u ( t )) t ∈ [0 , ∞ ) x ( v ( t, n )) t ∈ [0 , ∞ ) in the notation of Padgett [34, Theorem 2.1]). See the be-ginning of Section 2 for an explanation of this “applied with” notation (i.e., the symbol“ x ”). The right-hand-side of (1.5) is not considered in detail, herein, as it is an elementaryconsequence of Padgett [34, Theorem 2.1]. Item (ii) of Theorem 1.1 follows directly fromLemma 4.4 and Lemma 5.1.The main result of this article is Theorem 5.4 in Section 5. This result provides a completedescription of the series representation of the discrete fractional Laplace operator. The mostsurprising implication of Theorem 5.4 is that the formula for the function K : Z → R inTheorem 1.1 depends only on the parameter s ∈ (0 , ∞ ) (cf. Definition 4.1 below). In fact,this function is continuous with respect to the parameter s for all s ∈ (0 , ∞ ) \{ , , , . . . } with the points s ∈ { , , , . . . } all being removable singularities of the function K : Z → R .Hence, we may extend the definition of K : Z → R to that of an analytic function (cf. (5.20)of Theorem 5.4).The remainder of this article is organized as follows. In Section 2 we recall several basicdefinitions and properties of sequence spaces and introduce the so-called logarithmic norm.Afterwards, in Section 3 we define the discrete Laplace operator of arbitrary real-valued pos-itive order. We do so by introducing the heat semigroup generated by the discrete Laplaceoperator and then defining higher-order powers via induction. In Section 4 we define a dis-crete fractional kernel function and provide a detailed investigation of its various quantitativeand qualitative properties. Thereafter, in Section 5 we construct a series representation forreal-valued positive powers of the discrete fractional Laplace operator by employing the re-sults from Sections 3 and 4. Finally, in Section 6, a number of useful concluding remarks areprovided. Continuing avenues of research based on the results developed in this article arealso outlined. In this section we review several basic concepts regarding sequence spaces and the logarithmicnorm. More specifically, in Subsection 2.1 we introduce the standard ℓ ( Z ) sequence space4nd an associated function which we denote the semi-inner product. In particular, Lemma 2.5demonstrates that the standard ℓ ( Z ) inner product coincides with our particular semi-innerproduct. Afterwards, in Subsection 2.2 we define the logarithmic norm and the so-calledupper-right Dini derivative. We then demonstrate a very useful property in Lemma 2.8regarding the upper-right Dini derivative of ℓ ( Z ) norms.It is worth noting that the contents of this section have been studied in various parts ofthe scientific literature (although rarely together and in this particular setting). The conceptof semi-inner products has been studied extensively in the literature; cf., e.g., [12, 19, 29].They were originally introduced in an effort to extend standard Hilbert space-type argumentsto the more general setting of normed vector spaces. Herein, we employ a slight abuse ofnotation as Definition 2.4 does not define a semi-inner product in the sense of Lumer (cf.,e.g., [29]). However, the object defined in Definition 2.4 does possess many of the desiredproperties of a semi-inner product and Lemma 2.5 demonstrates that no generality is lost byemploying this definition. It is also worth noting that we are not the only authors to employsuch notation; cf. e.g., Söderlind [40, Definition 5.1]. Moreover, it is worth mentioning thatLemma 2.8 appears in Jones et al. [20, Lemma 2.4] in a more general setting but we includeit below for clarity and completeness.Throughout this article, R and C stand for the usual real and complex number fields,respectively. Further, let i = √− ∈ C , let Z denote the set of integers, let N = { , , , . . . } ,let N = N ∪{ } , and for every z ∈ C let R ( z ) ∈ R denote the real part of the complex number z . In addition, we briefly mention a particular notation used throughout this article whichemphasizes how various outside results are applied. If, for example, a result is referencedwhich names a particular mathematical object X , then in order to state results about a familyof objects, herein, e.g., Y t , t ∈ R , we will write “applied for every t ∈ R with X x Y t in thenotation of . . . ” in order to clarify its use. We generalize this approach in the natural wayin the case if multiple mathematical objects are involved (cf., e.g., the proof of Lemma 3.7).Moreover, when carrying out mathematical induction on a variable, say n ∈ N , we will usethe notation “ N ∋ ( n − n ∈ N ” to emphasize and clarify both the inductive set andthe inductive variable (cf., e.g., the proof of Lemma 3.8 below). Definition 2.1 (Set of all sequences) . We denote by S the set of all functions with domain Z and codomain R . Definition 2.2 (The ℓ ( Z ) Hilbert space) . We denote by ℓ ( Z ) the set of all u ∈ S satisfyingthat P k ∈ Z | u ( k ) | < ∞ (cf. Definition 2.1). Furthermore, we denote by k·k : S → [0 , ∞ ] thefunction which satisfies for all u ∈ S that k u k = P k ∈ Z | u ( k ) | . Definition 2.3 (The ℓ ( Z ) inner product) . We denote by h· , ·i : ℓ ( Z ) × ℓ ( Z ) → R thefunction which satisfies for all u, v ∈ ℓ ( Z ) that h u, v i = P k ∈ Z u ( k ) v ( k ) (cf. Definition 2.2). Definition 2.4 (Semi-inner product) . We denote by [ · , · ] : ℓ ( Z ) × ℓ ( Z ) → R the functionwhich satisfies for all u, v ∈ ℓ ( Z ) that [ u, v ] = (cid:20) lim ε → + k v + εu k − k v k ε (cid:21) k v k (2.1)5cf. Definition 2.2). Lemma 2.5.
Let u, v ∈ ℓ ( Z ) (cf. Definition 2.2). Then(i) it holds that h u, u i = k u k and(ii) it holds that h u, v i = [ u, v ] (cf. Definitions 2.3 and 2.4).Proof of Lemma 2.5. First, observe that item (i) follows immediately from Definitions 2.2and 2.3. Next, note that item (i), Definition 2.4, and the fact that h· , ·i : ℓ ( Z ) × ℓ ( Z ) → R is a symmetric bilinear form assure that [ u, v ] = " lim ε → + k v + εu k − k v k ε k v k = " lim ε → + k v + εu k − k v k ε · k v + εu k + k v k k v + εu k + k v k k v k = " lim ε → + k v + εu k − k v k ε (cid:0) k v + εu k + k v k (cid:1) k v k = " lim ε → + h v + εu, v + εu i − h v, v i ε (cid:0) k v + εu k + k v k (cid:1) k v k (2.2) = " lim ε → + h v, v i + 2 ε h u, v i + ε h u, u i − h v, v i ε (cid:0) k v + εu k + k v k (cid:1) k v k = " lim ε → + ε h u, v i + ε h u, u i ε (cid:0) k v + εu k + k v k (cid:1) k v k = " lim ε → + h u, v i + ε h u, u ik v + εu k + k v k k v k = " h u, v i k v k k v k = h u, v i . This establishes item (ii). The proof of Lemma 2.5 is thus complete.
Definition 2.6 (Logarithmic norm) . We denote by µ ( A ) ∈ R , A : ℓ ( Z ) → ℓ ( Z ) , thefunction which satisfies for all A : ℓ ( Z ) → ℓ ( Z ) that µ ( A ) = sup v ∈ ℓ ( Z ) k v k =0 h Av, v ik v k (2.3)(cf. Definitions 2.2 and 2.3). Definition 2.7 (Upper-right Dini derivative) . For every v : [0 , ∞ ) → R we denote by D + t v ( t ) ∈ [ −∞ , ∞ ] , t ∈ [0 , ∞ ) , the function which satisfies for all t ∈ [0 , ∞ ) that D + t v ( t ) = lim sup ε → + v ( t + ε ) − v ( t ) ε . (2.4) It is well known that h· , ·i : ℓ ( Z ) × ℓ ( Z ) → R is a symmetric bilinear form as this follows immediatelyfrom Definition 2.3. emma 2.8. It holds for all differentiable v : [0 , ∞ ) → ℓ ( Z ) , t ∈ [0 , ∞ ) that D + t k v ( t ) k = (cid:20) h ddt v ( t ) , v ( t ) ik v ( t ) k (cid:21) k v ( t ) k (2.5) (cf. Definitions 2.2, 2.3, and 2.7).Proof of Lemma 2.8. Throughout this proof let v : [0 , ∞ ) → ℓ ( Z ) , let t ∈ [0 , ∞ ) , and assumewithout loss of generality that k v ( t ) k = 0 (cf. Definition 2.2). Note that the hypothesis that v is differentiable and Taylor’s theorem (cf., e.g., Cartan et al. [6, Theorem 5.6.3]) yield thatthere exists δ t ( ε ) ∈ ℓ ( Z ) , ε ∈ R , such that for all ε ∈ R with | ε | sufficiently small(A) it holds that v ( t + ε ) = v ( t ) + ε ddt v ( t ) + | ε | δ t ( ε ) and(B) it holds that lim ε → δ t ( ε ) = 0 .Combining items (A) and (B) with Definition 2.7 and item (ii) of Lemma 2.5 hence showsthat D + t k v ( t ) k = lim sup ε → + k v ( t ) + ε ddt v ( t ) + | ε | δ t ( ε ) k − k v ( t ) k ε = lim sup ε → + k v ( t ) + ε ddt v ( t ) k − k v ( t ) k ε = lim ε → + k v ( t ) + ε ddt v ( t ) k − k v ( t ) k ε (2.6) = (cid:20) lim ε → + k v ( t ) + ε ddt v ( t ) k − k v ( t ) k ε (cid:21) k v ( t ) k k v ( t ) k = (cid:10) ddt v ( t ) , v ( t ) (cid:11) k v ( t ) k k v ( t ) k (cf. Definition 2.3). The proof of Lemma 2.8 is thus complete.We close Subsection 2.2 with a brief discussion of Definition 2.6. For every A : ℓ ( Z ) → ℓ ( Z ) let k A k op = inf { c ∈ [0 , ∞ ] : ∀ v ∈ ℓ ( Z ) it holds that k Av k ≤ c k v k } (cf. Defini-tion 2.2). Then Definitions 2.4 and 2.6 and the Rayleigh quotient theorem (cf., e.g., [13,Theorem A.26]) imply that for every A : ℓ ( Z ) → ℓ ( Z ) with k A k op ∈ [0 , ∞ ) , A = A ∗ , andnonempty pure point spectrum it holds that µ ( A ) = sup v ∈ ℓ ( Z ) k v k =0 h Av, v ik v k = max { λ ∈ R : ∃ v ∈ ℓ ( Z ) with k v k = 0 and Av = λv } (2.7)(e.g., µ ( A ) is the maximal eigenvalue of A ). This fact will prove quite useful in the proof ofLemma 3.7 in Subsection 3.1. In this section we introduce the discrete fractional Laplace operator and define the notionof real-valued positive powers of this operator. First, in Subsection 3.1 we define the dis-crete Laplace operator as well as introduce and study its associated discrete heat semigroup.Proposition 3.2 is presented in order to clarify the fact that positive integer powers of the7iscrete Laplace operator map elements of ℓ ( Z ) into ℓ ( Z ) (cf. Definition 2.2). The asso-ciated discrete heat semigroup is shown to be a strongly continuous contraction semigroupvia the tools developed in Subsection 2.2. Note that Definition 3.1 is provided for clarity, asthe evaluation of the Gamma function with arguments whose real parts are negative occursfrequently throughout the remainder of this article.Afterwards, in Subsection 3.2 we provide a series representation of positive integer powersof the discrete Laplace operator. The result in Lemma 3.8 is well-known in the literature,but its proof is included for completeness (cf., e.g., Kelley and Peterson [21, Eq. (2.1)]). Theseries representation presented in Lemma 3.8 will be a crucial component in proving themain result of this article (cf. Theorem 5.4).In Subsection 3.3 we define arbitrary real-valued positive powers of the discrete Laplaceoperator (cf. Definition 3.11). This is accomplished by first defining the case when the posi-tive real-valued power is bounded above by one (cf. Definition 3.10). We then define higher-order positive real-valued powers via an inductive procedure. This definition is shown to bewell-defined in ℓ ( Z ) in Lemma 3.9; i.e., it is shown that the discrete fractional Laplace op-erator maps ℓ ( Z ) into ℓ ( Z ) . At this point, we wish to again emphasize that Definition 3.11is not a direct “discretization” of the pointwise formula for the continuous case (cf., e.g.,(1.1) for the case where s ∈ (0 , ), but rather the fractional power of the discrete Laplaceoperator. Definition 3.1 (Gamma function) . Let X = { z ∈ C : R ( z ) ∈ (0 , ∞ ) } and let ˜Γ : X → C bethe function which satisfies for all z ∈ X that ˜Γ( z ) = R ∞ x z − exp( − x ) dx . Then we denote by Γ : C → C the function which satisfies for all z ∈ X that Γ( z ) = ˜Γ( z ) , for all z ∈ C with R ( z ) ∈ ( −∞ , \{ . . . , − , − , } that Γ( z ) = ˜Γ( z + |⌊ R ( z ) ⌋| )( z + |⌊ R ( z ) ⌋| − z + |⌊ R ( z ) ⌋| − · . . . · z , (3.1)and for all z ∈ C with R ( z ) ∈ { . . . , − , − , } that / Γ( z ) = 0 . Proposition 3.2.
It holds for all s ∈ N , u ∈ ℓ ( Z ) that X n ∈ Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s X k =0 ( − k − s (cid:18) sk (cid:19) u ( n − s + k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∞ (3.2) (cf. Definitions 2.2 and 3.1).Proof of Proposition 3.2. Throughout this proof let s ∈ N , u ∈ ℓ ( Z ) . Observe that thefact that u ∈ ℓ ( Z ) and Definition 2.2 ensure that P k ∈ Z | u ( k ) | < ∞ . This, the triangleinequality, the fact that s ∈ N , and the fact Jensen’s inequality implies that for all r, m ∈ N , Note that ⌊·⌋ : Z → R satisfies for all x ∈ R that ⌊ x ⌋ = max { n ∈ Z : n ≤ x } . Note that for all n ∈ N , k ∈ { , , . . . , n } it holds that (cid:0) nk (cid:1) = Γ(1+ n ) / (Γ(1+ k )Γ(1+ n − k )) (cf. Definition 3.1). , v , . . . , v m ∈ [0 , ∞ ) it holds that [ P mk =1 v k ] r ≤ m max { r − , } P mk =1 v rk assure that X n ∈ Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s X k =0 ( − k − s (cid:18) sk (cid:19) u ( n − s + k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X n ∈ Z " s X k =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) sk (cid:19) u ( n − s + k ) (cid:12)(cid:12)(cid:12)(cid:12) (3.3) ≤ X n ∈ Z " s s X k =0 (cid:18) sk (cid:19) | u ( n − s + k ) | = 2 s s X k =0 (cid:18) sk (cid:19)"X n ∈ Z | u ( n − s + k ) | < ∞ . The proof of Proposition 3.2 is thus complete.
Definition 3.3 (Discrete Laplace operator) . We denote by ∆ : ℓ ( Z ) → ℓ ( Z ) the functionwhich satisfies for all u ∈ ℓ ( Z ) , n ∈ Z that (∆ u )( n ) = u ( n − − u ( n ) + u ( n + 1) (3.4)(cf. Definition 2.2). Definition 3.4 (Identity operator) . We denote by I : S → S the operator which satisfies forall u ∈ S , n ∈ Z that ( I u )( n ) = u ( n ) (cf. Definition 2.1). Proposition 3.5.
Let u ∈ ℓ ( Z ) (cf. Definition 2.2). Then it holds for all z ∈ [0 , ∞ ) that X n ∈ Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X k ∈ Z exp( − z ) " X j ∈ N z j + n − k Γ(1 + j )Γ( j + n − k + 1) u ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∞ (3.5) (cf. Definition 3.1).Proof of Proposition 3.5. Throughout this proof let I k : [0 , ∞ ) → R , k ∈ Z , satisfy for all z ∈ [0 , ∞ ) , k ∈ Z that I k ( z ) = X j ∈ N (cid:0) z / (cid:1) j + k Γ(1 + j )Γ( j + k + 1) (3.6)(cf. Definition 3.1). Observe that (3.6) and Olver et al. [33, Eq. 10.27.1] (applied for every k ∈ Z with I k x I k , z x z in the notation of Olver et al. [33, Eq. 10.27.1]) assure that forall z ∈ [0 , ∞ ) it holds that X k ∈ Z " X j ∈ N z j + k Γ(1 + j )Γ( j + k + 1) = X k ∈ Z I k (2 z ) (3.7) = I (2 z ) + X k ∈ N (cid:0) I k (2 z ) + I − k (2 z ) (cid:1) = I (2 z ) + 2 X k ∈ N I k (2 z ) . Combining this, (3.6), the triangle inequality, Minkowski’s inequality, and Olver et al. [33, Eq.10.35.5] (applied with I k x I k , z x z in the notation of Olver et al. [33, Eq. 10.35.5]) ensuresthat for all z ∈ [0 , ∞ ) it holds that X n ∈ Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X k ∈ Z exp( − z ) " X j ∈ N z j + n − k Γ(1 + j )Γ( j + n − k + 1) u ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ∈ Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X k ∈ Z exp( − z ) I k (2 z ) u ( n − k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.8) ≤ exp( − z ) X k ∈ Z | I k (2 z ) | X n ∈ Z | u ( n − k ) | ! / = exp( − z ) "X k ∈ Z I k (2 z ) k u k = exp( − z ) " I (2 z ) + 2 X k ∈ N I k (2 z ) k u k = exp( − z ) (cid:2) exp(2 z ) (cid:3) k u k = k u k < ∞ . The proof of Proposition 3.5 is thus complete.
Definition 3.6 (Discrete heat semigroup) . We denote by S z (∆) : ℓ ( Z ) → ℓ ( Z ) , z ∈ [0 , ∞ ) ,the function which satisfies for all z ∈ [0 , ∞ ) , u ∈ ℓ ( Z ) , n ∈ Z that (cid:0) S z (∆) u (cid:1) ( n ) = X k ∈ Z exp( − z ) " X j ∈ N z j + n − k Γ(1 + j )Γ( j + n − k + 1) u ( k ) (3.9)(cf. Definitions 2.2, 3.1, 3.3, and 3.6). Lemma 3.7.
Let u ∈ ℓ ( Z ) (cf. Definition 2.2). Then(i) it holds that S z (∆) : ℓ ( Z ) → ℓ ( Z ) , z ∈ [0 , ∞ ) , is a strongly continuous semigroup,(ii) it holds for all z ∈ [0 , ∞ ) , n ∈ Z that ddz ( S z (∆) u )( n ) = (∆ S z (∆) u )( n ) ,(iii) it holds that µ (∆) ∈ ( −∞ , , and(iv) it holds for all z ∈ [0 , ∞ ) that k S z (∆) u k ≤ exp( zµ (∆)) k u k ≤ k u k (cf. Definitions 2.6, 3.3, and 3.6).Proof of Lemma 3.7. First, note that S z (∆) : ℓ ( Z ) → ℓ ( Z ) , z ∈ [0 , ∞ ) , (cf. Definitions 3.3and 3.6) is a strongly continuous semigroup if(A) it holds that S (∆) = I (cf. Definition 3.4),(B) it holds for all t, z ∈ [0 , ∞ ) that S t + z (∆) = S t (∆) S z (∆) , and(C) it holds for all v ∈ ℓ ( Z ) that lim z → + k S z (∆) v − v k = 0 (cf., e.g., Jones et al. [20, Definition 2.6]). Observe that Definition 3.6, the fact that for all v ∈ ℓ ( Z ) it holds that sup n ∈ Z | v ( n ) | < ∞ , and, e.g., Ciaurri et al. [8, Proposition 1] (appliedfor every v ∈ ℓ ( Z ) with f x v , W t x S z (∆) in the notation of Ciaurri et al. [8, Proposition1]) show that S z (∆) : ℓ ( Z ) → ℓ ( Z ) , z ∈ [0 , ∞ ) satisfies items (A), (B), and (C). Thisestablishes item (i). Next, note that Definition 3.6 and, e.g., Ciaurri et al. [8, Proposition 2](applied for every n ∈ Z with f x u , ( u ( n, t )) t ∈ [0 , ∞ ) x (( S z (∆) u )( n )) z ∈ [0 , ∞ ) in the notationof Ciaurri et al. [8, Proposition 2]) establish item (ii). In addition, observe that Definition 2.4implies that if for all v ∈ ℓ ( Z ) with k v k = 0 it holds that h ∆ v, v i ∈ ( −∞ , then it holds10hat µ (∆) ∈ ( −∞ , . To that end, note that Definition 2.3 guarantees that for all v ∈ ℓ ( Z ) it holds that h ∆ v, v i = X k ∈ Z (cid:2) (∆ v )( k ) (cid:3) v ( k ) = X k ∈ Z (cid:0) v ( k − − v ( k ) + v ( k + 1) (cid:1) v ( k )= X k ∈ Z v ( k − v ( k ) − X k ∈ Z v ( k ) + X k ∈ Z v ( k + 1) v ( k )= X k ∈ Z v ( k − v ( k ) − X k ∈ Z v ( k ) − X k ∈ Z v ( k − + X k ∈ Z v ( k ) v ( k − (3.10) = − X k ∈ Z (cid:2) v ( k − − v ( k − v ( k ) + v ( k ) (cid:3) = − X k ∈ Z (cid:2) v ( k − − v ( k ) (cid:3) . This demonstrates that for all v ∈ ℓ ( Z ) with k v k = 0 it holds that h ∆ v, v i ∈ ( −∞ , . Thisand Definition 2.6 establish item (iii). Moreover, observe that item (i), item (iii), and Joneset al. [20, Lemma 2.8] (applied with x x u , A x ∆ , ( T t ( A )) t ∈ [0 , ∞ ) x ( S z (∆)) z ∈ [0 , ∞ ) in thenotation of Jones et al. [20, Lemma 2.8]) hence prove item (iv). The proof of Lemma 3.7 isthus complete. Lemma 3.8.
It holds for all s ∈ N , u ∈ ℓ ( Z ) , n ∈ Z that (cid:0) ( − ∆) s u (cid:1) ( n ) = s X k =0 ( − k − s (cid:18) sk (cid:19) u ( n − s + k ) (3.11) (cf. Definitions 2.2 and 3.11).Proof of Lemma 3.8. We prove (3.11) by induction on s ∈ N . For the base case s = 1 observethat Definition 3.3 establishes (3.11). This proves (3.11) in the case s = 1 . For the inductionstep N ∋ ( s − s ∈ N ∩ [2 , ∞ ) , let s ∈ N ∩ [2 , ∞ ) and assume for all s ∈ { , , . . . , s − } , u ∈ ℓ ( Z ) , n ∈ Z that (3.11) holds. Note that the fact that s ∈ N ∩ [2 , ∞ ) implies that forall u ∈ ℓ ( Z ) , n ∈ Z it holds that (cid:0) ( − ∆) s u (cid:1) ( n ) = (cid:0) − ∆(( − ∆) s − u ) (cid:1) ( n ) . (3.12)This, the induction hypothesis, and Definition 3.3 ensure that for all u ∈ ℓ ( Z ) , n ∈ Z itholds that (cid:0) ( − ∆) s u (cid:1) ( n ) = (cid:18) − ∆ (cid:18) s − P k =0 ( − k − ( s − (cid:0) s − k (cid:1) u ( · − ( s −
1) + k ) (cid:19)(cid:19) ( n )= 2 s − P k =0 ( − k − s +1 (cid:0) s − k (cid:1) u ( n − s + 1 + k ) − s − P k =0 ( − k − s +1 (cid:0) s − k (cid:1) u (( n − − s + 1 + k ) − s − P k =0 ( − k − s +1 (cid:0) s − k (cid:1) u (( n + 1) − s + 1 + k ) s − P k =0 ( − k − s +1 (cid:0) s − k (cid:1) u ( n − s + 1 + k ) − s − P k =0 ( − k − s +1 (cid:0) s − k (cid:1) u ( n − s + k ) − s − P k =0 ( − k − s +1 (cid:0) s − k (cid:1) u ( n + 2 − s + k ) (3.13) = 2 s − P k =1 ( − k − s (cid:0) s − k − (cid:1) u ( n − s + k ) + s − P k =0 ( − k − s (cid:0) s − k (cid:1) u ( n − s + k )+ s P k =2 ( − k − s (cid:0) s − k − (cid:1) u ( n − s + k )= (cid:20) s − P k =1 ( − k − s (cid:0) s − k − (cid:1) u ( n − s + k ) + s − P k =0 ( − k − s (cid:0) s − k (cid:1) u ( n − s + k ) (cid:21) + (cid:20) s − P k =1 ( − k − s (cid:0) s − k − (cid:1) u ( n − s + k ) + s P k =2 ( − k − s (cid:0) s − k − (cid:1) u ( n − s + k ) (cid:21) . Combining this and the fact that for all n ∈ N , k ∈ { , , . . . , n − } it holds that (cid:0) nk (cid:1) = (cid:0) n − k (cid:1) + (cid:0) n − k − (cid:1) assures that for all u ∈ ℓ ( Z ) , n ∈ Z it holds that (cid:0) ( − ∆) s u (cid:1) ( n ) = ( − s − u ( n + s −
1) + ( − − s u ( n − s ) + s − P k =1 ( − k − s (cid:0) s − k (cid:1) u ( n − s + k )+ ( − − s u ( n + 1 − s ) + ( − s u ( n + s )+ s − P k =2 ( − k − s (cid:0) s − k − (cid:1) u ( n − s + k )= ( − s − u ( n + s −
1) + ( − − s u ( n − s ) + ( − − s u ( n + 1 − s )+ ( − s u ( n + s ) + ( − − s (cid:0) s − (cid:1) u ( n − s + 1)+ ( − s − (cid:0) s − s − (cid:1) u ( n − s + (2 s − (3.14) + s − P k =2 ( − k − s h(cid:0) s − k (cid:1) + (cid:0) s − k − (cid:1)i u ( n − s + k )= ( − s − u ( n + s −
1) + ( − − s u ( n − s ) + ( − − s u ( n + 1 − s )+ ( − s u ( n + s ) + ( − − s (cid:0) s − (cid:1) u ( n − s + 1)+ ( − s − (cid:0) s − s − (cid:1) u ( n + s −
1) + s − P k =2 ( − k − s (cid:0) sk (cid:1) u ( n − s + k ) . Next, observe that the fact that for all n ∈ N , k ∈ { , , . . . , n − } it holds that (cid:0) nk (cid:1) = (cid:0) n − k (cid:1) + (cid:0) n − k − (cid:1) and the fact that for all n ∈ N , k ∈ { , , . . . , n } it holds that (cid:0) nk (cid:1) = (cid:0) nn − k (cid:1) show that (cid:0) s − s − (cid:1) = (cid:0) s − s − (cid:1) + (cid:0) s − s − (cid:1) = (cid:0) s s − (cid:1) (3.15)and (cid:0) s − (cid:1) = (cid:0) s − (cid:1) + (cid:0) s − (cid:1) = (cid:0) s (cid:1) . (3.16)Combining (3.14), (3.15), and (3.16) hence implies that for all u ∈ ℓ ( Z ) , n ∈ Z it holds that (cid:0) ( − ∆) s u (cid:1) ( n ) = ( − − s (cid:0) s (cid:1) u ( n − s ) + ( − s (cid:0) s s (cid:1) u ( n + s ) + ( − − s (cid:0) s (cid:1) u ( n − s + 1) ( − s − (cid:0) s s − (cid:1) u ( n + s −
1) + s − P k =2 ( − k − s (cid:0) sk (cid:1) u ( n − s + k ) (3.17) = s P k =0 ( − k − s (cid:0) sk (cid:1) u ( n − s + k ) . Induction therefore establishes (3.11). The proof of Lemma 3.8 is thus complete.
Lemma 3.9.
Let m ∈ N , s ∈ ( m − , m ) . Then it holds for all u ∈ ℓ ( Z ) that (cid:13)(cid:13)(cid:13)(cid:13) − ( s − m + 1)) Z ∞ z − s + m − (cid:2) S z (∆) − I (cid:3)(cid:0) ( − ∆) m − u (cid:1) dz (cid:13)(cid:13)(cid:13)(cid:13) < ∞ (3.18) (cf. Definitions 2.2, 3.1, 3.3, 3.4, and 3.6).Proof of Lemma 3.9. Throughout this proof let m ∈ N , s ∈ ( m − , m ) . We claim that forall z ∈ (0 , ∞ ) , u ∈ ℓ ( Z ) it holds that (cid:13)(cid:13) ( S z (∆) − I ) u (cid:13)(cid:13) ≤ Z z exp (cid:0) ( z − w ) µ (∆) (cid:1) k ∆ u k dw (3.19)(cf. Definitions 2.2, 2.6, 3.3, 3.4, and 3.6). Note that the fact that h· , ·i : ℓ ( Z ) × ℓ ( Z ) → R is a symmetric bilinear form and Definition 3.6 ensure that for all u ∈ ℓ ( Z ) , z ∈ (0 , ∞ ) itholds that ddz (cid:13)(cid:13) ( S z (∆) − I ) u (cid:13)(cid:13) = h ddz ( S z (∆) − I ) u, ( S z (∆) − I ) u ik ( S z (∆) − I ) u k (3.20) = h ∆ S z (∆) u, ( S z (∆) − I ) u ik ( S z (∆) − I ) u k = h ∆( S z (∆) − I ) u, ( S z (∆) − I ) u ik ( S z (∆) − I ) u k + h ∆ u, ( S z (∆) − I ) u ik ( S z (∆) − I ) u k ≤ " sup v ∈ ℓ ( Z ) h ∆ v, v ik v k + h ∆ u, ( S z (∆) − I ) u ik ( S z (∆) − I ) u k = µ (∆) + h ∆ u, ( S z (∆) − I ) u ik ( S z (∆) − I ) u k (cf. Definition 2.3). This, the Cauchy-Swartz inequality, and the fact that item (i) ofLemma 3.7 implies that for all u ∈ ℓ ( Z ) it holds that lim z → + k ( S z (∆) − I ) u k = 0 as-sure that for all z ∈ (0 , ∞ ) , u ∈ ℓ ( Z ) it holds that (cid:13)(cid:13) ( S z (∆) − I ) u (cid:13)(cid:13) ≤ Z z exp (cid:0) ( z − w ) µ (∆) (cid:1) (cid:12)(cid:12) h ∆ u, ( S w (∆) − I ) u i (cid:12)(cid:12) k ( S w (∆) − I ) u k dw (3.21) ≤ Z z exp (cid:0) ( z − w ) µ (∆) (cid:1) k ∆ u k k ( S w (∆) − I ) u k k ( S w (∆) − I ) u k dw = Z z exp (cid:0) ( z − w ) µ (∆) (cid:1) k ∆ u k dw. Combining this and the fact that Proposition 3.2 (applied with s x in the notation ofProposition 3.2) ensures that for all u ∈ ℓ ( Z ) it holds that k ∆ u k < ∞ proves (3.19). Next,observe that (3.19), Lemma 3.8, and Jensen’s inequality guarantee that for all u ∈ ℓ ( Z ) itholds that (cid:13)(cid:13)(cid:13)(cid:13) − ( s − m + 1)) Z ∞ z − s + m − (cid:2) S z (∆) − I (cid:3)(cid:0) ( − ∆) m − u (cid:1) dz (cid:13)(cid:13)(cid:13)(cid:13) | Γ( − ( s − m + 1)) | Z ∞ z − s + m − (cid:13)(cid:13)(cid:2) S z (∆) − I (cid:3)(cid:0) ( − ∆) m − u (cid:1)(cid:13)(cid:13) dz (3.22) ≤ k ( − ∆) m u k | Γ( − ( s − m + 1)) | Z ∞ z − s + m − (cid:20)Z z exp (cid:0) ( z − w ) µ (∆) (cid:1) dw (cid:21) dz (cf. Definition 3.1). In addition, note that item (iii) of Lemma 3.7 and integration by partsshow that ≤ Z ∞ z − s + m − (cid:20) − exp (cid:0) zµ (∆) (cid:1) − µ (∆) (cid:21) dz = Z z − s + m − (cid:20) − exp (cid:0) zµ (∆) (cid:1) − µ (∆) (cid:21) dz + Z ∞ z − s + m − (cid:20) − exp (cid:0) zµ (∆) (cid:1) − µ (∆) (cid:21) dz ≤ Z z − s + m − (cid:20) − exp (cid:0) zµ (∆) (cid:1) − µ (∆) (cid:21) dz + Z ∞ z − s + m − dz = lim w → + (cid:20) z − s + m − − s + m − (cid:21)(cid:20) − exp (cid:0) zµ (∆) (cid:1) − µ (∆) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) z = w − Z (cid:20) z − s + m − − s + m − (cid:21) exp( zµ (∆)) dz + lim w →∞ z − s + m − − s + m − (cid:12)(cid:12)(cid:12)(cid:12) wz =1 (3.23) = (cid:20) − s + m − (cid:21)(cid:20) µ (∆) − exp (cid:0) µ (∆) (cid:1) − µ (∆) (cid:21) − Z (cid:20) z − s + m − − s + m − (cid:21) exp( zµ (∆)) dz ≤ (cid:20) − s + m − (cid:21)(cid:20) µ (∆) − exp (cid:0) µ (∆) (cid:1) − µ (∆) (cid:21) + Z (cid:20) z − s + m − s − m + 1 (cid:21) dz = (cid:20) − s + m − (cid:21)(cid:20) µ (∆) − exp (cid:0) µ (∆) (cid:1) − µ (∆) (cid:21) + 1( − s + m )( s − m + 1) < ∞ . Combining (3.22), (3.23), and the fact that Proposition 3.2 assures that for all u ∈ ℓ ( Z ) itholds that k ( − ∆) m u k < ∞ hence proves (3.18). The proof of Lemma 3.9 is thus complete. Definition 3.10 (Discrete fractional Laplace operator for s ∈ (0 , ) . Let s ∈ (0 , . Thenwe denote by ( − ∆) s : ℓ ( Z ) → ℓ ( Z ) the function which satisfies for all u ∈ ℓ ( Z ) , n ∈ Z that (cid:0) ( − ∆) s u (cid:1) ( n ) = 1Γ( − s ) Z ∞ z − s − (cid:2) S z (∆) − I (cid:3) u ( n ) dz (3.24)(cf. Definitions 2.2, 3.1, 3.3, 3.4, and 3.6). Definition 3.11 (Discrete fractional Laplace operator for s ∈ (0 , ∞ ) ) . Let s ∈ (0 , ∞ ) . Thenwe denote by ( − ∆) s : ℓ ( Z ) → ℓ ( Z ) the function which satisfies for all u ∈ ℓ ( Z ) , n ∈ Z that (cid:0) ( − ∆) s u (cid:1) ( n ) = (cid:0) ( − ∆) s −⌊ s ⌋ ( − ∆) ⌊ s ⌋ u (cid:1) ( n ) (3.25) = ((cid:0) ( − ∆) s u (cid:1) ( n ) : s ∈ N − ( s −⌊ s ⌋ )) R ∞ z − ( s −⌊ s ⌋ ) − (cid:2) S z (∆) − I (cid:3)(cid:0) ( − ∆) ⌊ s ⌋ u (cid:1) ( n ) dz : s ∈ (0 , ∞ ) \ N (cf. Definitions 2.2, 3.1, 3.3, 3.4, 3.6, and 3.10).14 The discrete fractional kernel
In this section we introduce a kernel function which will allow us to conveniently provide aseries representation of (3.25) in Definition 3.11. Proposition 4.2 and Lemma 4.3 are prelim-inary results which allow us to prove Lemma 4.4—a result which outlines useful propertiesexhibited by the kernel defined in Definition 4.1. It is worth noting that Proposition 4.2 isa well-known result and that Lemma 4.3 is a generalization of [9, Lemma 9.2 (a)], whichwas only proven in the case where s ∈ (0 , . Proposition 4.5 and Lemma 4.7 are the mainresults of this section and allow us to prove Lemma 5.1 in Subsection 5.1. Definition 4.1 (Fractional kernel) . We denote by K s : Z → R , s ∈ R , the function whichsatisfies for all k ∈ Z , m ∈ N , s ∈ ( m − , m ) that K s ( k ) = − Z \{ } ( k ) 4 s Γ( / + s )Γ( | k | − s ) √ π Γ( − s )Γ( | k | + 1 + s ) (4.1)(cf. Definition 3.1). Proposition 4.2.
It holds for all a, b ∈ R , λ ∈ (0 , ∞ ) with ≤ a < b < ∞ that min { λ, } ≤ b λ − a λ b λ − ( b − a ) ≤ max { λ, } . (4.2) Proof of Proposition 4.2.
First, note that for all a, b ∈ R with ≤ a < b < ∞ it holds that ≤ a / b < (4.3)Observe that (4.3) ensures that for all λ ∈ (0 , , a, b ∈ R with ≤ a < b < ∞ it holds that ≤ a / b ≤ (cid:0) a / b (cid:1) λ < . This assures that for all λ ∈ (0 , , a, b ∈ R with ≤ a < b < ∞ itholds that b λ − a λ b λ − ( b − a ) = 1 − ( a / b ) λ − a / b ≤ . (4.4)Combining the mean value theorem and (4.4) hence guarantee that for all λ ∈ (0 , , a, b ∈ R with ≤ a < b < ∞ it holds that there exists c ∈ ( a / b , such that b λ − a λ b λ − ( b − a ) = 1 − ( a / b ) λ − a / b = λc λ − ≥ λ. (4.5)Next, note that (4.3) demonstrates that for all λ ∈ [1 , ∞ ) , a, b ∈ R with ≤ a < b < ∞ it holds that ≤ (cid:0) a / b (cid:1) λ ≤ a / b < . This shows that for all λ ∈ [1 , ∞ ) , a, b ∈ R with ≤ a < b < ∞ it holds that b λ − a λ b λ − ( b − a ) = 1 − ( a / b ) λ − a / b ≥ . (4.6) Let A ⊆ R . Then it holds for all x ∈ A that A ( x ) = 1 and for all x ∈ R \ A that A ( x ) = 0 . λ ∈ [1 , ∞ ) , a, b ∈ R with ≤ a < b < ∞ it holds that there exists d ∈ ( a / b , such that b λ − a λ b λ − ( b − a ) = 1 − ( a / b ) λ − a / b = λd λ − ≤ λ. (4.7)Combining (4.4), (4.5), (4.6), and (4.7) thus establishes (4.2). The proof of Proposition 4.2is thus complete. Lemma 4.3.
It holds for all m ∈ N , s ∈ ( m − , m ) that there exists C ∈ R such that forall k ∈ Z with | k | ∈ [ m, ∞ ) it holds that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ( | k | − s )Γ( | k | + 1 + s ) − | k | s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C | k | s (4.8) (cf. Definition 3.1).Proof of Lemma 4.3. Throughout this proof let m ∈ N , s ∈ ( m − , m ) and without loss ofgenerality let k ∈ Z with k ∈ [ m, ∞ ) . Note that the triangle inequality assures that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ( | k | − s )Γ( | k | + 1 + s ) − | k | s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ( | k | − s )Γ( | k | + 1 + s ) − | k − s | s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | k − s | s − | k | s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (4.9) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ( k − s )Γ( k + 1 + s ) − k − s ) s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k − s ) s − k s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Next, observe that Proposition 4.2 (applied with λ x s , a x / k , b x / ( m − s ) in thenotation of Proposition 4.2) ensures that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k − s ) s − k s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = " (cid:12)(cid:12) ( k − s ) − (1+2 s ) − k − (1+2 s ) (cid:12)(cid:12) ( k − s ) − s (cid:2) ( k − s ) − − k − (cid:3) ( k − s ) − s (cid:2) ( k − s ) − − k − (cid:3) ≤ max { s, } ( k − s ) s " k − s − k = 1 + 2 s ( k − s ) s " k − ( k − s )( k − s ) k = (1 + 2 s ) sk ( k − s ) s . (4.10)This and the fact that k ∈ [ m, ∞ ) show that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k − s ) s − k s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (1 + 2 s ) s ( k − s ) s ≤ " sup k ∈ [ m, ∞ ) k s ( k − s ) s (1 + 2 s ) sk s . (4.11)In addition, note that, e.g., Tricomi and Erdélyi [41, Eq. (15), page 140] (applied with z x k , α x − s , β x s in the notation of Tricomi and Erdélyi [41, Eq. (15), page 140]) guaranteesthat Γ( k − s )Γ( k + 1 + s ) = 1Γ(1 + 2 s ) Z ∞ exp( − ( k − s ) v ) (cid:0) − exp( − v ) (cid:1) s dv. (4.12)16his, the fact that for all r ∈ [0 , ∞ ) it holds that R ∞ exp( − rv ) v s dv = Γ(1 + 2 s ) r − (1+2 s ) ,and Jensen’s inequality prove that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ( k − s )Γ( k + 1 + s ) − k − s ) s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s ) Z ∞ exp( − ( k − s ) v ) (cid:0) − exp( − v ) (cid:1) s dv − k − s ) s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (4.13) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s ) Z ∞ exp( − ( k − s ) v ) h(cid:0) − exp( − v ) (cid:1) s − v s i dv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ s ) Z ∞ exp( − ( k − s ) v ) v s (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) − exp( − v ) v (cid:17) s − (cid:12)(cid:12)(cid:12)(cid:12) dv. Moreover, observe that Proposition 4.2 (applied with λ x s , a x (1 − exp( − v )) / v , b x in thenotation of Proposition 4.2) ensures that for all v ∈ (0 , ∞ ) it holds that − (cid:16) − exp( − v ) v (cid:17) s ≤ max { s, } (cid:20) − − exp( − v ) v (cid:21) . (4.14)Combining this, (4.13), the fact that for all v ∈ (0 , ∞ ) it holds that v − (1 − exp( − v )) < v / ,the fact that for all r ∈ [0 , ∞ ) it holds that R ∞ exp( − rv ) v s dv = Γ(2 + 2 s ) r − (2+2 s ) , andthe fact that Definition 3.1 implies that for all z ∈ (0 , ∞ ) it holds that Γ(1 + z ) = z Γ( z ) assures that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ( k − s )Γ( k + 1 + s ) − k − s ) s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ max { s, } Γ(1 + 2 s ) Z ∞ exp( − ( k − s ) v ) v s (cid:12)(cid:12)(cid:12)(cid:12) − (cid:16) − exp( − v ) v (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) dv (4.15) ≤ max { s, / } Γ(1 + 2 s ) Z ∞ exp( − ( k − s ) v ) v s dv = max { s, / } Γ(2 + 2 s )Γ(1 + 2 s ) 1( k − s ) s = max { s, / } (1 + 2 s ) 1( k − s ) s ≤ " sup k ∈ [ m, ∞ ) k s ( k − s ) s max { s, / } (1 + 2 s ) k s . Combining this, (4.9), (4.11), and the fact that sup k ∈ [ m, ∞ ) k s ( k − s ) − (2+2 s ) ∈ R henceproves (4.8). The proof of Lemma 4.3 is thus complete. Lemma 4.4.
Let m ∈ N , s ∈ ( m − , m ) . Then(i) it holds for all k ∈ Z that K s ( − k ) = K s ( k ) and(ii) it holds that there exists C ∈ R such that for all k ∈ Z with | k | ∈ Z \{ } it holds that | K s ( k ) | ≤ C | k | s (4.16)17 cf. Definition 4.1).Proof of Lemma 4.4. First, note that Definition 4.1 ensures that for all k ∈ Z it holds that K s ( − k ) = − Z \{ } ( − k ) 4 s Γ( / + s )Γ( |− k | − s ) √ π Γ( − s )Γ( |− k | + 1 + s ) (4.17) = − Z \{ } ( k ) 4 s Γ( / + s )Γ( | k | − s ) √ π Γ( − s )Γ( | k | + 1 + s ) = K s ( k ) (cf. Definitions 3.1 and 4.1). This establishes item (i). Next, observe that for all k ∈ Z ∩ ( − m, m ) with k = 0 it holds that | K s ( k ) | ≤ s Γ( / + s ) | Γ( | k | − s ) |√ π | Γ( − s ) | Γ( | k | + 1 + s ) = 4 s Γ( / + s ) | Γ( | k | − s ) |√ π | Γ( − s ) | Γ( | k | + 1 + s ) · | k | s | k | s ≤ s Γ( / + s ) m s √ π | Γ( − s ) | " sup j ∈ Z ∩ ( − m,m ) | Γ( | j | − s ) | Γ( | j | + 1 + s ) | k | s . (4.18)This, the fact that s ∈ (0 , ∞ ) \ N implies that sup j ∈ Z ∩ ( − m,m ) | Γ( | j |− s ) | / Γ( | j | +1+ s ) ∈ R , andLemma 4.3 establish item (ii). The proof of Lemma 4.4 is thus complete. Proposition 4.5.
It holds for all m ∈ N , s ∈ ( m − , m ) , k ∈ Z that K s ( k ) = Z \{ } ( k ) ( − k +1 Γ(2 s + 1)Γ(1 + s + k )Γ(1 + s − k ) (4.19) (cf. Definitions 3.1 and 4.1).Proof of Proposition 4.5. Throughout this proof let m ∈ N , s ∈ ( m − , m ) and without lossof generality let k ∈ N (cf. item (i) of Lemma 4.4). Observe that Definition 4.1 and theLegendre duplication formula (cf., e.g., [1, Eq. (6.1.18), Page 256]) ensure that K s ( k ) = − s Γ( / + s )Γ( k − s ) √ π Γ( − s )Γ( k + 1 + s ) = − s Γ( / + s )Γ( k − s ) √ π Γ( − s )Γ( k + 1 + s ) · Γ( s )Γ( s )= − s (cid:2) − s √ π Γ(2 s ) (cid:3) Γ( k − s ) √ π Γ( − s )Γ( s )Γ( k + 1 + s ) = − s )Γ( k − s )Γ( − s )Γ( s )Γ( k + 1 + s ) . (4.20)This and the fact that Definition 3.1 implies that for all z ∈ C with R ( z ) ∈ R \{ . . . , − , − , } it holds that z Γ( z ) = Γ(1 + z ) assure that K s ( k ) = − s )Γ( k − s )Γ( − s )Γ( s )Γ( k + 1 + s ) · ss = Γ(2 s + 1)Γ( k − s )Γ(1 − s )Γ( s )Γ( k + 1 + s ) . (4.21)Next, note that the Euler reflection formula (cf., e.g., [1, Eq. (6.1.17), Page 256]) guaranteesthat Γ( s )Γ(1 − s ) = ( − k +1 Γ( k − s )Γ(1 + s − k ) . (4.22)Combining (4.21) and (4.22) hence yields (4.19). The proof of Proposition 4.5 is thus com-plete. 18 roposition 4.6. Let s ∈ (0 , ∞ ) \ N . Then it holds for all m ∈ N that Γ( m − s )2 s Γ( m + s ) + m − X k =1 Γ( k − s )Γ( k + 1 + s ) = − Γ( − s )2Γ(1 + s ) (4.23) (cf. Definition 3.1).Proof of Proposition 4.6. We prove (4.23) by induction on m ∈ N . For the base case m = 1 note that the fact that Definition 3.1 ensures that for all z ∈ C with R ( z ) ∈ R \{ . . . , − , − , } it holds that z Γ( z ) = Γ( z + 1) guarantees that Γ(1 − s )2 s Γ(1 + s ) + X k =1 Γ( k − s )Γ( k + 1 + s ) = Γ(1 − s )2 s Γ(1 + s ) = − s Γ( − s )2 s Γ(1 + s ) = − Γ( − s )2Γ(1 + s ) . (4.24)This establishes (4.23) in the case m = 1 . For the induction step N ∋ ( m − m ∈ N ∩ [2 , ∞ ) , let m ∈ N ∩ [2 , ∞ ) and assume for all m ∈ { , , . . . , m − } that (4.23) holds.Observe that the induction hypothesis shows that for all m ∈ N ∩ [2 , ∞ ) it holds that Γ( m − s )2 s Γ( m + s ) + m − X k =1 Γ( k − s )Γ( k + 1 + s )= (cid:20) Γ( m − s )2 s Γ( m + s ) + Γ(( m − − s )Γ(( m −
1) + 1 + s ) (cid:21) + ( m − − X k =1 Γ( k − s )Γ( k + 1 + s ) (4.25) = Γ( m − s )2 s Γ( m + s ) + Γ( m − − s )Γ( m + s ) + − Γ( m − − s )2 s Γ( m − s ) + − Γ( − s )2Γ(1 + s ) . Next, note that the fact that Definition 3.1 ensures that for all z ∈ C with R ( z ) ∈ R \{ . . . , − , − , } it holds that z Γ( z ) = Γ( z + 1) demonstrates that for all m ∈ N ∩ [2 , ∞ ) it holds that Γ( m − s )2 s Γ( m + s ) + − Γ( m − − s )2 s Γ( m − s ) = Γ( m − s )2 s Γ( m + s ) + − ( m − s )Γ( m − − s )2 s ( m − s )Γ( m − s )= Γ( m − s ) − ( m − s )Γ( m − − s )2 s Γ( m + s ) = ( m − − s )Γ( m − s ) − ( m − s )Γ( m − s )2 s ( m − − s )Γ( m + s )= (cid:20) ( m − − s ) − ( m − s )2 s ( m − − s ) (cid:21) Γ( m − s )Γ( m + s ) = − Γ( m − s )( m − − s )Γ( m + s ) . (4.26)Moreover, observe that Definition 3.1 assures that for all z ∈ C with R ( z ) ∈ R \{ . . . , − , − , } it holds that z Γ( z ) = Γ( z + 1) demonstrates that for all m ∈ N ∩ [2 , ∞ ) it holds that − Γ( m − s )( m − − s )Γ( m + s ) + Γ( m − − s )Γ( m + s ) = − Γ( m − s ) + ( m − − s )Γ( m − − s )( m − − s )Γ( m + s )= − Γ( m − s ) + Γ( m − s )( m − − s )Γ( m + s ) = 0 . (4.27)Combining (4.25), (4.26), and (4.27) therefore proves (4.23). The proof of Proposition 4.6 isthus complete. 19 emma 4.7. It holds for all m ∈ N , s ∈ ( m − , m ) that X k ∈ Z K s ( k ) = 4 s Γ( / + s ) √ π Γ(1 + s ) (4.28) (cf. Definitions 3.1 and 4.1).Proof of Lemma 4.7. First, note that item (ii) of Lemma 4.4 ensures that for all m ∈ N , s ∈ ( m − , m ) it holds that P k ∈ Z K s ( k ) ∈ R . Next, observe that Definition 4.1 and item (i)of Lemma 4.4 assure that for all m ∈ N , s ∈ ( m − , m ) it holds that X k ∈ Z K s ( k ) = − s Γ( / + s ) √ π Γ( − s ) "X k ∈ Z Z \{ } ( k )Γ( | k | − s )Γ( | k | + 1 + s ) (4.29) = − · s Γ( / + s ) √ π Γ( − s ) "X k ∈ N Γ( k − s )Γ( k + 1 + s ) . In addition, note that, e.g., Artin [3, Eq. (2.13)] (applied for every m ∈ N , s ∈ ( m − , m ) , k ∈ N ∩ [ m, ∞ ) with x x k − s , y x s in the notation of Artin [3, Eq. (2.13)]) impliesthat for all m ∈ N , s ∈ ( m − , m ) , k ∈ N ∩ [ m, ∞ ) it holds that Γ( k − s )Γ( k + 1 + s ) = 1Γ(1 + 2 s ) (cid:20) Γ( k − s )Γ(1 + 2 s )Γ( k + 1 + s ) (cid:21) = 1Γ(1 + 2 s ) Z (1 − z ) (1+2 s ) − z ( k − s ) − dz = 1Γ(1 + 2 s ) Z (1 − z ) s z k − s − dz. (4.30)This and Fubini’s theorem guarantee that for all m ∈ N , s ∈ ( m − , m ) it holds that X k ∈ N Γ( k − s )Γ( k + 1 + s ) = m − X k =1 Γ( k − s )Γ( k + 1 + s ) + ∞ X k = m Γ( k − s )Γ( k + 1 + s )= m − X k =1 Γ( k − s )Γ( k + 1 + s ) + ∞ X k = m (cid:20) s ) Z (1 − z ) s z k − s − dz (cid:21) = m − X k =1 Γ( k − s )Γ( k + 1 + s ) + 1Γ(1 + 2 s ) Z (1 − z ) s z − s − " ∞ X k = m z k dz (4.31) = m − X k =1 Γ( k − s )Γ( k + 1 + s ) + 1Γ(1 + 2 s ) Z (1 − z ) s z m − s − " ∞ X k =0 z k dz = m − X k =1 Γ( k − s )Γ( k + 1 + s ) + 1Γ(1 + 2 s ) Z (1 − z ) s − z m − s − dz. Combining this, the fact that Definition 3.1 ensures that for all z ∈ C with R ( z ) ∈ R \{ . . . , − , − , } it holds that z Γ( z ) = Γ( z + 1) , and, e.g., Artin [3, Eq. (2.13)] (applied20or every m ∈ N , s ∈ ( m − , m ) with x x m − s , y x s in the notation of Artin [3, Eq.(2.13)]) demonstrates that for all m ∈ N , s ∈ ( m − , m ) it holds that X k ∈ N Γ( k − s )Γ( k + 1 + s ) = m − X k =1 Γ( k − s )Γ( k + 1 + s ) + 1Γ(1 + 2 s ) (cid:20) Γ( m − s )Γ(2 s )Γ( m + s ) (cid:21) = m − X k =1 Γ( k − s )Γ( k + 1 + s ) + 12 s Γ(2 s ) (cid:20) Γ( m − s )Γ(2 s )Γ( m + s ) (cid:21) (4.32) = m − X k =1 Γ( k − s )Γ( k + 1 + s ) + Γ( m − s )2 s Γ( m + s ) . Combining this and Proposition 4.6 proves that for all m ∈ N , s ∈ ( m − , m ) it holds that X k ∈ Z K s ( k ) = − · s Γ( / + s ) √ π Γ( − s ) " m − X k =1 Γ( k − s )Γ( k + 1 + s ) + Γ( m − s )2 s Γ( m + s ) (4.33) = − · s Γ( / + s ) √ π Γ( − s ) (cid:20) − Γ( − s )2Γ(1 + s ) (cid:21) = 4 s Γ( / + s ) √ π Γ(1 + s ) . The proof of Lemma 4.7 is thus complete.
In this section we prove the main result of this article. First, in Subsection 5.1 we pro-vide a series representation of the real-valued non-integer powers of the discrete Laplaceoperator (cf. Lemma 5.1). This representation employs the fractional kernel introduced inDefinition 4.1 and its proof hinges upon the results developed in Section 4. It is particularlyinteresting to note that the representation obtained in Lemma 5.1 coincides with the repre-sentation presented in Ciaurri et al. [9, Theorem 1.1] (the case where m = 1 ) and Padgettet al. [35, Theorem 2] (the case where m = 2 ).In Subsection 5.2 we demonstrate that the representation presented in Lemma 5.1 holdsfor s ∈ N if we consider the limiting values of the discrete kernel function. The main result ofthe article, Theorem 5.4, follows immediately from the combination of Lemmas 5.1 and 5.3.In particular, Theorem 5.4 demonstrates that all real-valued positive powers of the discreteLaplace operator may be represented with the same series (or, at least, as the limit of thisseries). Therefore, it is the case that the discrete fractional Laplace operator is, in somesense, a perturbation of the standard positive integer power case.While it is not the purpose of this article to discuss such issues, we wish to emphasizethe importance of the last sentence in the previous paragraph. The fact that the discretefractional Laplace operator’s series representation coincides with the series representation forpositive integer powers provides a framework to endow fractional calculus with potentiallyenlightening physical interpretations. A particularly lacking feature of the fractional calculusis the lack of meaningful physical interpretations in many situations, which has been one21f the primary limiting factor in its widespread application. However, Theorem 5.4 allowsus to view the fractional powers as “transitional phases” between the positive integer cases,loosely speaking. Thus, we may use the known physical intuition for positive integer powersto provide a deeper understanding of the positive non-integer cases. Lemma 5.1.
It holds for all m ∈ N , s ∈ ( m − , m ) , u ∈ ℓ ( Z ) , n ∈ Z that (cid:0) ( − ∆) s u (cid:1) ( n ) = X k ∈ Z K s ( n − k ) (cid:0) u ( n ) − u ( k ) (cid:1) (5.1) (cf. Definitions 2.2, 3.11, and 4.1).Proof of Lemma 5.1. Throughout this proof let u ∈ ℓ ( Z ) , let v : Z → ℓ ( Z ) satisfy for all n ∈ Z that v ( n ) = ( − ∆ u )( n ) , and let A s ∈ R , s ∈ (0 , ∞ ) , satisfy for all s ∈ (0 , ∞ ) that A s = 4 s Γ( / + s ) √ π Γ(1 + s ) (5.2)(cf. Definitions 3.1 and 3.3). Observe that Padgett et al. [35, Theorem 1] establishes (5.1)in the case that m = 1 , s ∈ (0 , . Next, note that Definition 3.11 and the fact that forall m ∈ N ∩ [2 , ∞ ) , n ∈ Z it holds that (( − ∆) m u )( n ) = (( − ∆) m − ( − ∆) u )( n ) (i.e., we areinvoking the fact that standard function composition is associative on its domain) ensurethat for all m ∈ N ∩ [2 , ∞ ) , s ∈ ( m − , m ) , n ∈ Z it holds that (cid:0) ( − ∆) s u (cid:1) ( n ) = (cid:0) ( − ∆) s − ( m − (( − ∆) m − u ) (cid:1) ( n ) = (cid:0) ( − ∆) s − ( m − (( − ∆) m − ( − ∆ u )) (cid:1) ( n )= (cid:0) ( − ∆) s − ( m − (( − ∆) m − v ) (cid:1) ( n ) = (cid:0) ( − ∆) s − v (cid:1) ( n ) . (5.3)We now claim that for all m ∈ N ∩ [2 , ∞ ) , s ∈ ( m − , m ) , n ∈ N it holds that (cid:0) ( − ∆) s u (cid:1) ( n ) = X k ∈ Z K s ( n − k ) (cid:0) u ( n ) − u ( k ) (cid:1) . (5.4)We prove (5.4) by induction on m ∈ N ∩ [2 , ∞ ) . For the base case m = 2 observe that Padgettet al. [35, Theorem 2] establishes (5.4). For the induction step N ∩ [2 , ∞ ) ∋ ( m − m ∈ N ∩ [3 , ∞ ) , let m ∈ N ∩ [3 , ∞ ) and assume for all m ∈ { , , . . . , m − } , s ∈ ( m − , m ) , n ∈ Z that (5.4) holds. Observe that the induction hypothesis, (5.2), (5.3), (5.4), and Lemma 4.7demonstrate that for all s ∈ ( m − , m ) , n ∈ Z it holds that (cid:0) ( − ∆) s u (cid:1) ( n ) = (cid:0) ( − ∆) s − v (cid:1) ( n ) = P k ∈ Z K s − ( n − k ) (cid:0) v ( n ) − v ( k ) (cid:1) (5.5) = v ( n ) P k ∈ Z K s − ( n − k ) − P k ∈ Z K s − ( n − k ) v ( k ) = A s − v ( n ) − P k ∈ Z K s − ( n − k ) v ( k ) . This, Definition 4.1, the fact that for all n ∈ Z it holds that v ( n ) = ( − ∆ u )( n ) , and item (i)of Lemma 4.4 show that for all s ∈ ( m − , m ) , n ∈ Z it holds that (cid:0) ( − ∆) s u (cid:1) ( n ) = A s − v ( n ) − P k ∈ Z K s − ( n − k ) v ( k ) A s − (cid:0) u ( n ) − u ( n − − u ( n + 1) (cid:1) − P k ∈ Z K s − ( n − k ) (cid:0) u ( k ) − u ( k − − u ( k + 1) (cid:1) = A s − (cid:0) u ( n ) − u ( n − − u ( n + 1) (cid:1) − P k ∈ Z (cid:2) K s − ( k ) − K s − ( k − − K s − ( k + 1) (cid:3) u ( n − k )= A s − (cid:0) u ( n ) − u ( n − − u ( n + 1) (cid:1) − (cid:2) K s − (0) − K s − ( − − K s − (1) (cid:3) u ( n ) (5.6) − (cid:2) K s − (1) − K s − (0) − K s − (2) (cid:3) u ( n − − (cid:2) K s − ( − − K s − ( − − K s − (0) (cid:3) u ( n + 1) − P k ∈ Z \{− , , } (cid:2) K s − ( k ) − K s − ( k − − K s − ( k + 1) (cid:3) u ( n − k )= (cid:2) A s − + 2 K s − (1) (cid:3) u ( n ) − (cid:2) A s − + 2 K s − (1) − K s − (2) (cid:3)(cid:0) u ( n −
1) + u ( n + 1) (cid:1) − P k ∈ Z \{− , , } (cid:2) K s − ( k ) − K s − ( k − − K s − ( k + 1) (cid:3) u ( n − k ) . Next, note that (5.2), the fact that Definition 3.1 implies that for all z ∈ C with R ( z ) ∈ R \{ . . . , − , − , } it holds that Γ( z +1) = z Γ( z ) , and the fact that − s ∈ ( −∞ , guaranteethat for all s ∈ ( m − , m ) it holds that K s − (1) = − Z \{ } (1) 4 s − Γ( / + ( s − | | − ( s − √ π Γ( − ( s − | | + 1 + ( s − − s − Γ( s − / )Γ(2 − s ) √ π Γ(1 − s )Γ(1 + s )= − s − Γ( s − / )(1 − s )Γ(1 − s ) √ π Γ(1 − s ) s Γ( s ) = 4 s − Γ( s − / )( s − √ πs Γ( s ) (5.7)and K s − (2) = − Z \{ } (2) 4 s − Γ( / + ( s − | | − ( s − √ π Γ( − ( s − | | + 1 + ( s − − s − Γ( s − / )Γ(3 − s ) √ π Γ(1 − s )Γ(2 + s )= − s − Γ( s − / )(2 − s )(1 − s )Γ(1 − s ) √ π Γ(1 − s )(1 + s ) s Γ( s ) = 4 s − Γ( s − / )(2 − s )( s − √ π (1 + s ) s Γ( s ) . (5.8)Observe that (5.2), (5.7), and the fact that Definition 3.1 implies that for all z ∈ C with R ( z ) ∈ R \{ . . . , − , − , } it holds that Γ( z + 1) = z Γ( z ) hence ensure that for all s ∈ ( m − , m ) it holds that A s − + 2 K s − (1) = 2 (cid:20) s − Γ( s − / ) √ π Γ( s ) (cid:21) + 2 (cid:20) s − Γ( s − / )( s − √ πs Γ( s ) (cid:21) = 2 · s − Γ( s − / ) √ π Γ( s ) (cid:20) s − s (cid:21) = 2 · s − Γ( s − / ) √ π Γ( s ) (cid:20) s − s (cid:21) (5.9) = 4 s ( s − / )Γ( s − / ) √ πs Γ( s ) = 4 s Γ( / + s ) √ π Γ(1 + s ) = A s .
23n addition, observe that Proposition 4.5 and the fact that Definition 3.1 implies that for all z ∈ C with R ( z ) ∈ R \{ . . . , − , − , } it holds that Γ( z + 1) = z Γ( z ) demonstrate that forall k ∈ Z \{− , , } , s ∈ ( m − , m ) it holds that K s − ( k ) − K s − ( k − − K s − ( k + 1)= − − k Γ(2 s − s + k )Γ( s − k ) − ( s + k − − k Γ(2 s − s − k )Γ( s + k )Γ( s − k ) − ( s − k − − k Γ(2 s − s + k )Γ( s + k )Γ( s − k ) (5.10) = ( − k Γ(2 s − s + k )Γ( s − k ) (cid:20) − − s + k − s − k − s − k − s + k (cid:21) = ( − k Γ(2 s − s + k + 1)Γ( s + k − (cid:2) s − s (cid:3) = ( − k +1 Γ(2 s + 1)Γ(1 + s + k )Γ(1 + s − k ) = K s ( k ) . Moreover, note that Definition 4.1, (5.2), (5.7), (5.8), and the fact that Definition 3.1 impliesthat for all z ∈ C with R ( z ) ∈ R \{ . . . , − , − , } it holds that Γ( z + 1) = z Γ( z ) assure thatfor all s ∈ ( m − , m ) it holds that A s − + 2 K s − (1) − K s − (2)= 4 s − Γ( s − / ) √ π Γ( s ) + 2 (cid:20) s − Γ( s − / )( s − √ πs Γ( s ) (cid:21) − s − Γ( s − / )(2 − s )( s − √ π (1 + s ) s Γ( s )= 4 s − Γ( s − / ) √ π Γ( s ) " (cid:18) s − s (cid:19) + ( s − s − s (1 + s ) = 4 s − Γ( s − / ) √ π Γ( s ) " s − / ) s + 1 (5.11) = 4 s Γ( s + / ) √ π Γ( s )( s + 1) = 4 s Γ( s + / ) √ π Γ( s )( s + 1) · s Γ( − s ) s Γ( − s ) = − s Γ( s + / )Γ(1 − s ) √ π Γ( − s )Γ(2 + s ) = K s (1) . Combining (5.6), (5.9), (5.10), (5.11), and item (i) of Lemma 4.4 therefore yields that for all s ∈ ( m − , m ) , n ∈ Z it holds that (cid:0) ( − ∆) s u (cid:1) ( n ) = (cid:2) A s − + 2 K s − (1) (cid:3) u ( n ) − (cid:2) A s − + 2 K s − (1) − K s − (2) (cid:3)(cid:0) u ( n −
1) + u ( n + 1) (cid:1) − P k ∈ Z \{− , , } (cid:2) K s − ( k ) − K s − ( k − − K s − ( k + 1) (cid:3) u ( n − k ) (5.12) = A s u ( n ) − K s (1) (cid:0) u ( n −
1) + u ( n + 1) (cid:1) − P k ∈ Z \{− , , } K s ( k ) u ( n − k )= A s u ( n ) − P k ∈ Z \{ } K s ( k ) u ( n − k ) = P k ∈ Z K s ( n − k ) (cid:0) u ( n ) − u ( k ) (cid:1) . Induction hence establishes (5.4). The proof of Lemma 5.1 is thus complete.
Lemma 5.2.
Let s ∈ N . Then it holds for all k ∈ Z that lim z → s K z ( k ) = { , ,...,s } ( k ) ( − k +1 Γ(2 s + 1)Γ(1 + s + k )Γ(1 + s − k ) (5.13) (cf. Definitions 3.1 and 4.1). roof of Lemma 5.2. First, note that Definition 4.1 and Proposition 4.5 ensure that for all k ∈ Z it holds that lim z → s K z ( k ) = lim z → s Z \{ } ( k ) ( − k +1 Γ(2 s + 1)Γ(1 + s + k )Γ(1 + s − k ) (5.14)(cf. Definitions 3.1 and 4.1). Combining this and the fact that Definition 3.1 ensures that forall z ∈ { . . . , − , − , } it holds that / Γ( z ) = 0 establishes (5.13). The proof of Lemma 5.2 isthus complete. Lemma 5.3.
It holds for all s ∈ N , u ∈ ℓ ( Z ) , n ∈ Z that (cid:0) ( − ∆) s u (cid:1) ( n ) = s X k =0 ( − k − s (cid:18) sk (cid:19) u ( n − s + k ) = lim z → s "X k ∈ Z K z ( n − k ) (cid:0) u ( n ) − u ( k ) (cid:1) (5.15) (cf. Definitions 2.2, 3.11, and 4.1).Proof of Lemma 5.3. First, note that Definition 3.1 and the fact that for all a ∈ N , b ∈{ , , . . . , a } it holds that (cid:0) ab (cid:1) = (cid:0) aa − b (cid:1) assure that for all s ∈ N , u ∈ ℓ ( Z ) , n ∈ Z it holdsthat s X k =0 ( − k − s (cid:18) sk (cid:19) u ( n − s + k ) = s X k = − s ( − k (cid:18) ss + k (cid:19) u ( n − k )= s X k = − s (cid:20) ( − k Γ(2 s + 1)Γ(1 + s + k )Γ(1 + s − k ) (cid:21) u ( n − k ) (5.16) = (cid:20) Γ(2 s + 1)Γ(1 + s )Γ(1 + s ) (cid:21) u ( n ) − s X k =1 (cid:20) ( − k +1 Γ(2 s + 1)Γ(1 + s + k )Γ(1 + s − k ) (cid:21)(cid:0) u ( n − k ) + u ( n + k ) (cid:1) (cf. Definitions 2.2 and 3.1). Next, observe that Lemma 4.7 ensures that for all z ∈ (0 , ∞ ) \ N , u ∈ ℓ ( Z ) , n ∈ Z it holds that X k ∈ Z K z ( n − k ) (cid:0) u ( n ) − u ( k ) (cid:1) = u ( n ) X k ∈ Z K z ( n − k ) − X k ∈ Z K z ( n − k ) u ( k )= (cid:20) z Γ( / + z ) √ π Γ(1 + z ) (cid:21) u ( n ) − X k ∈ Z K z ( n − k ) u ( k ) (5.17)(cf. Definition 4.1). This, Definition 3.1, Definition 4.1, items (i) and (ii) of Lemma 4.4,Lemma 5.2, and, e.g., Rudin [38, Theorem 7.17] guarantee that for all s ∈ N , u ∈ ℓ ( Z ) , n ∈ Z it holds that lim z → s "X k ∈ Z K s ( n − k ) (cid:0) u ( n ) − u ( k ) (cid:1) = (cid:20) lim z → s z Γ( / + z ) √ π Γ(1 + z ) (cid:21) u ( n ) − lim z → s "X k ∈ Z K z ( n − k ) u ( k ) = (cid:20) s Γ( / + s ) √ π Γ(1 + s ) (cid:21) u ( n ) − lim z → s "X k ∈ N K z ( k ) (cid:0) u ( n − k ) + u ( n + k ) (cid:1) (cid:20) s Γ( / + s ) √ π Γ(1 + s ) (cid:21) u ( n ) − X k ∈ N h lim z → s K z ( k ) i(cid:0) u ( n − k ) + u ( n + k ) (cid:1) (5.18) = (cid:20) s Γ( / + s ) √ π Γ(1 + s ) (cid:21) u ( n ) − X k ∈ N (cid:20) { , ,...,s } ( k ) ( − k +1 Γ(2 s + 1)Γ(1 + s + k )Γ(1 + s − k ) (cid:21)(cid:0) u ( n − k ) + u ( n + k ) (cid:1) = (cid:20) s Γ( / + s ) √ π Γ(1 + s ) (cid:21) u ( n ) − s X k =1 (cid:20) ( − k +1 Γ(2 s + 1)Γ(1 + s + k )Γ(1 + s − k ) (cid:21)(cid:0) u ( n − k ) + u ( n + k ) (cid:1) . In addition, note that the fact that Definition 3.1 implies that for all z ∈ C with R ( z ) ∈ R \{ . . . , − , − , } it holds that z Γ( z ) = Γ(1 + z ) and the Legendre duplication formula (cf.,e.g., [1, Eq. (6.1.18), Page 256]) demonstrate that for all s ∈ N it holds that s Γ( / + s ) √ π Γ(1 + s ) = 4 s Γ( / + s ) √ π Γ(1 + s ) · s Γ( s ) s Γ( s ) = 4 s (cid:2) − s √ πs Γ(2 s ) (cid:3) √ π Γ( s )Γ(1 + s ) = Γ(2 s + 1)Γ(1 + s )Γ(1 + s ) . (5.19)Combining this, (5.16), (5.18), Lemma 3.8, and Definition 3.11 proves (5.15). The proof ofLemma 5.3 is thus complete. Theorem 5.4.
It holds for all s ∈ (0 , ∞ ) , u ∈ ℓ ( Z ) , n ∈ Z that (cid:0) ( − ∆) s u (cid:1) ( n ) = lim z → s "X k ∈ Z K z ( n − k ) (cid:0) u ( n ) − u ( k ) (cid:1) (5.20) (cf. Definitions 2.2, 3.11, and 4.1).Proof of Theorem 5.4. Note that combining Lemmas 5.1 and 5.3 establishes (5.20). Theproof of Theorem 5.4 is thus complete.
In this article we developed novel results regarding real-valued positive fractional powers ofthe discrete Laplace operator. In particular, we defined a discrete fractional Laplace operatorfor arbitrary real-valued positive powers (cf. Definition 3.11) and then developed its seriesrepresentation (cf. Theorem 5.4). This latter task was primarily accomplished through thedevelopment of two sets of results. First, we constructed the series representation for positiveinteger powers of the discrete Laplace operator (cf. Lemmas 3.8 and 5.3). Next, we developedseries representations for positive non-integer powers of the discrete Laplace operator (cf.Lemma 5.1). The main result of the article (cf. Theorem 5.4) is obtained by showing thatthe series representations obtained in each of the previous steps in fact coincide.The main results developed—i.e., the results of Section 5—required numerous prelimi-nary results from various areas of mathematics. The results in Section 2 are of a functionalanalysis flavor and allow for a beautiful description of important properties of strongly contin-uous semigroups. These results were combined with results from discrete harmonic analysis26n Section 3 in order to define and study the discrete fractional Laplace operator. Sincethe presented definition of this operator (cf. Definition 3.11) employs a so-called semigrouplanguage, it was imperative that all novel mathematical objects are determined to be welldefined in ℓ ( Z ) (cf. Definition 2.2). Finally, Section 4 provides a detailed study of the pro-posed fractional kernel function (cf. Definition 4.1) which is necessary for the developmentof the coefficients of the series representations presented in Section 5. Therein, it is shownthat the proposed fractional kernel function is well-defined, symmetric, and continuous forall s ∈ (0 , ∞ ) \ N . It is later shown in Lemma 5.2 that the values s ∈ N are in fact removablesingularities.As a final remark, we wish to emphasize the importance of the presented results. Dueto the rapidly growing interest in problems related to fractional calculus, there is a needto determine the validity of including fractional operators into existing models. The studyof the discrete fractional Laplace operator, or its continuous counterpart, for the case when s ∈ (0 , is well-understood and often used in physical sciences. In this setting, the operatormay be used to model super-diffusive phenomena [35]. Moreover, there have been rigorousstudies of the operator in this parameter regime which demonstrate that such considerationsare well-defined and well-behaved. As such, it is natural to attempt to extend these studiesto the case when s ∈ (1 , ∞ ) , as well. The current article has demonstrated that such ex-tensions are indeed well-defined in the discrete case. In addition, Theorem 5.4 shows thatone may potentially use the existing understanding of the case of positive integer powers ofthe discrete Laplace operator to provide some much needed physical intuition to the discretefractional Laplace operator. However, there are still numerous unanswered questions regard-ing important properties of these operators and we will outline a few such open problemsand research directions in Subsection 6.2 below. First and foremost, there is a need to continue the analytical work presented herein in orderto obtain a better understanding of the discrete fractional Laplace operator. In this article,we have considered the setting where all objects are defined in ℓ ( Z ) , however, this is notalways the appropriate setting for physically relevant problems. As such, we intend to extendour study to the situation where the underlying function spaces have less regularity (e.g.,Hölder spaces) and develop standard regularity estimates. We also intend to develop similarseries representations for the case of real-valued negative exponents. Such representations arehighly important for studying fractional Poisson-like problems, as they provide representa-tions of the solution to these problems. Finally, we hope to develop an understanding of thespectral properties of the discrete fractional Laplace operator. While it is well-known thatthe discrete Laplace operator (cf. Definition 3.3) has purely absolutely continuous spectra(cf., e.g., Dutkay and Jorgensen [14]), to the authors’ knowledge this has not been rigorouslyproven in the case of the discrete fractional Laplace operator. Demonstrating this to be thecase is of utmost importance and will have far-reaching implications in mathematics andphysics (for clarification, see the techniques outlined in Liaw [27]).The proposed discrete fractional Laplace operator is also of interest due to its importancein the physical sciences. An example of interest for future research is transport in turbulentplasmas. It has been experimentally observed that heat and particle transport in turbulent27lasmas is non-local (i.e., anomalous ) in nature (cf., e.g., [11, 18, 39]). Comparison be-tween transport models using the fractional Laplace operator and experimental results havedemonstrated that electron transport in turbulent fusion plasmas is characterized by frac-tional exponents in the range s ∈ (0 . , , which indicates super-diffusive behavior [24, 25].Moreover, a generalized approach to modeling anomalous diffusive transport in turbulentplasmas employs diffusion-type equations where fractional derivatives occur in both spaceand time (cf., e.g., [10]). Using the series representations presented herein, we intend to showthat the fractional derivative in time can be incorporated into the spatial derivative, whichcan greatly simplify such equations. Acknowledgments
The second author acknowledges funding by the National Science Foundation (NSF 1903450)and the Department of Energy (DE-SC0021284). The third author acknowledges fundingby the National Science Foundation (NSF 1903450). The fourth author would like to thankthe College of Arts and Sciences at Baylor University for partial support through a researchleave award.
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