A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces
aa r X i v : . [ m a t h . C A ] J un A Gneiting-like method for constructingpositive definite functions on metric spaces
V. S. Barbosa & V. A. Menegatto
This paper is concerned with the construction of positive definite functionson a cartesian product of quasi-metric spaces using generalized Stieltjes andcomplete Bernstein functions. The results we prove are aligned with a well-established method of T. Gneiting to construct space-time positive definitefunctions and its many extensions. For the right choice of the quasi-metricspaces, the models discussed in the paper lead to flexible, interpretable andeven computationally feasible classes of cross-covariance functions for multi-variate random fields adopted in statistics. Necessary and sufficient conditionsfor the strict positive definiteness of the models are provided when the spacesare metric.
Keywords: positive definite functions; generalized Stieltjes functions; Bernstein func-tions; Gneiting’s model; products of metric spaces2010 MSC: 42A82, 43A35
Let (
X, ρ ) be a quasi-metric space , that is, a nonempty set X endowed with a function ρ : X × X → [0 , ∞ ) (its quasi-distance ) satisfying ρ ( x, x ′ ) = ρ ( x ′ , x ) and ρ ( x, x ) = 0, x, x ′ ∈ X . Continuity on ( X, ρ ) can be defined as it is so on a metric space. Write D ρX toindicate the diameter-set of ( X, ρ ), i.e., D ρX = { ρ ( x, x ′ ) : x, x ′ ∈ X } . This paper is mainly concerned with radial positive definite functions on (
X, ρ ), that is,continuous functions f : D ρX → R satisfying n X j,k =1 c j c k f ( ρ ( x j , x k )) ≥ , (1.1)for n ≥
1, reals scalars c , . . . , c n , and points x , . . . , x n in X . Functions of this typeplay an important role in classical analysis, approximation theory, probability theory,nd statistics. Reference [29] covers what we will need in this paper about radial positivedefinite functions. The strict positive definiteness of a radial positive definite function f as above demands that the inequalities be strict when the x j are distinct and the c j arenot all zero. We will write f ∈ P D ( X, ρ ) and f ∈ SP D ( X, ρ ) to indicate that f is positivedefinite and strictly positive definite on ( X, ρ ), respectively.The two concepts just introduced extend to a product of finitely many quasi-metricspaces. However, we will formalize the extension only in the setting to be covered in thispaper. Unless stated otherwise, throughout the paper, (
X, ρ ), (
Y, σ ) and (
Z, τ ) will denotethree quasi-metric spaces while X × Y × Z will denote their cartesian product. Here, wewill not distinguish among the spaces X × Y × Z , X × ( Y × Z ) and ( X × Y ) × Z and will notdetach any special quasi-distance in them. A continuous function f : D X × D Y × D Z → R is said to be positive definite on X × Y × Z (the term radial will be abandoned), and wewrite f ∈ P D ( X × Y × Z, ρ, σ, τ ) if n X j,k =1 c j c k f ( ρ ( x j , x k ) , σ ( y j , y k ) , τ ( z j , z k )) ≥ , for n ≥
1, reals scalars c , . . . , c n , and points ( x , y , z ) . . . , ( x n , y n , z n ) in X × Y × Z . Afunction f in P D ( X × Y × Z, ρ, σ, τ ) is strictly positive definite if the inequalities aboveare strict when the ( x j , y j , z j ) are distinct and the c j are not all zero. In this case, we willuse the notation SP D ( X × Y × Z, ρ, σ, τ ).Two problems involving the concepts of positive definiteness and strict positive def-initeness are very common in the literature: to characterize
P D ( X, ρ ), SP D ( X, ρ ), P D ( X × Y, ρ, σ ), etc, for fixed choice of the spaces and to determine, explicitly, largefamilies of functions belonging to them that have some importance in applications.I. J. Schoenberg characterized in [26] the class
P D ( R n , ρ ) where ρ is the usual Eu-clidean distance. His result states that a continuous function f : [0 , ∞ ) → R belongs to P D ( R n , ρ ) if, and only if, f ( t ) = Z [0 , ∞ ) Ω n ( wt ) dµ ( w ) , t ≥ , where µ is a finite and positive measure on [0 , ∞ ) while Ω n ( x ) = Ω n ( ρ ( x, y ∈ S n − e ix · y over S n − . Here, · denotes the usual inner product in R n , S n − is the unit sphere in R n , if n ≥
2, while S = {− , } . He also characterized the class P D ( H , ρ ) where H is an infinite dimensional Hilbert space and ρ is the distance definedby its norm: a continuous function f : [0 , ∞ ) → R belongs to P D ( H , ρ ) if, and only if, t ∈ (0 , ∞ ) f ( t / ) is completely monotone. Recall that a function f : (0 , ∞ ) → R is completely monotone if it has derivatives of all orders and ( − n f ( n ) ( t ) ≥
0, for t > n = 0 , , . . . . Theorem 7.14 in [30] provides additional information regarding theclass P D ( H , ρ ). In order to obtain the classes SP D ( R n , ρ ), n ≥
2, and
SP D ( H , ρ ), one2eeds to eliminate the constant functions from SP D ( R n , ρ ) and P D ( H , ρ ), respectively.Characterizations for some of the classes P D ( L p ( A, µ ) , ρ ), where ( A, µ ) is a measure spaceand ρ is given through the p -norm of L p ( A, µ ) are presented in [29, Chapter 2].Schoenberg also provided characterizations for the classes
P D ( S d , ρ ), d ≥
1, where ρ is now the geodesic distance on S d . His result also included a characterization for the class P D ( S ∞ , ρ ), where S ∞ is the unit sphere in the real Hilbert space ℓ while ρ is its geodesicdistance ([25]). R. Gangolli ([9]) extended Schoenberg results to P D ( H, ρ ), where H isany compact two-point homogeneous space and ρ is its invariant Riemannian distance.After a normalization for the distances in these spaces is implemented, one can see thata continuous function f : [0 , π ] → R belongs to P D ( H, ρ ), if, and only if, f has a seriesrepresentation in the form f ( t ) = ∞ X k =0 a Hk P Hk (cos t ) , t ∈ [0 , π ] , where a Hk ≥ k and P ∞ k =0 a Hk P Hk (1) < ∞ . Here, P Hk is the monomial x k if H = S ∞ and a Jacobi polynomial of degree k that depends on the space H being used, otherwise.The classes SP D ( S d , ρ ), SP D ( S ∞ , ρ ), and SP D ( H, ρ ) were described in [3, 8] throughadditional conditions on the sets { k : a Hk > } .The same was done for the classes P D ( X × Y, ρ, σ ) and
SP D ( X × Y, ρ, σ ) for somechoices of (
X, ρ ) and (
Y, σ ). For the case where X and Y are compact two-point homo-geneous spaces with their respective Riemannian distances ρ and σ , the characterizationfor P D ( X × Y, ρ, σ ) appeared in [4, 13]: a continuous function f : [0 , π ] → R belongs to SP D ( X × Y, ρ, σ ) if, and only if, f has a series representation in the form f ( t, u ) = ∞ X k,l =0 a X,Yk,l P Xk (cos t ) P Yl (cos u ) , t, u ∈ [0 , π ] , with a X,Yk,l ≥ k and l and the series being convergent at ( t, u ) = (0 , SP D ( X × Y, ρ, σ ), a description can be found in [4, 12, 14, 15] and depends on additionalassumptions on the sets { k − l : a X,Yk,l > } . The cases in which ( X, ρ ) is the usualmetric space R n and Y is either a compact two-point homogeneous space or S ∞ wereconsidered recently: P D ( X × Y, ρ, σ ) was described in [6, 7, 11, 27] while a descriptionfor
SP D ( X × Y, ρ, σ ) can be inferred from [11].As for the explicit determination of large families in either
P D ( X, ρ ) or
SP D ( X, ρ ),the most efficient techniques make use of completely monotone functions and conditionallynegative definite functions on (
X, ρ ). A continuous function f : D σX → R is conditionallynegative definite on ( X, ρ ), and we write f ∈ CN D ( X, ρ ), if the quadratic forms in (1.1)are nonpositive when the coefficients c j satisfy P nj =1 c j = 0. Clearly, this notion can beextended to a cartesian product of quasi-metric spaces so that the symbol CN D ( X × Y, ρ, σ ) also makes sense. 3he following construction providing an efficient technique follows from Theorem 3.5in [20] along with Lemma 2.5 in [23]: if f is a bounded and completely monotone functionand g is a nonnegative valued function in CN D ( X, ρ ), then f ◦ g belongs to P D ( X, ρ ).Further, f ◦ g belongs to SP D ( X, ρ ) if, and only if, f is nonconstant and g ( t ) > g (0),for t ∈ D ρX \ { } . A quick analysis reveals that the following extension also holds: if f is a bounded and completely monotone function and g is a nonnegative valued functionin CN D ( X × Y, ρ, σ ), then f ◦ g belongs to P D ( X × Y, ρ, σ ). Further, f ◦ g belongsto SP D ( X × Y, ρ, σ ) if, and only if, f is nonconstant and g ( t, u ) > g (0 , t, u ) ∈ D ρX × D σY with t + u >
0. If we drop the boundedness of f , then the results above stillhold as long as we assume g is positive valued.Motivated by a celebrated result of Gneiting in [10], an interesting procedure to con-struct positive definite functions on a cartesian product of quasi-metric spaces was de-scribed in [21]. If f is a bounded and completely monotone function, g is a nonnegativevalued function in CN D ( X, ρ ) and h is a positive valued function in CN D ( Y, σ ), thenthe function F r given by F r ( t, u ) = 1 h ( u ) r f (cid:18) g ( t ) h ( u ) (cid:19) , ( t, u ) ∈ D ρX × D σY , (1.2)belongs to P D ( X × Y, ρ, σ ), as long as f is a bounded generalized Stieltjes function of order λ > r ≥ λ . Further, in the case in which ( X, ρ ) and (
Y, σ ) are metric spaces and X has at least two points, F r belongs to SP D ( X × Y, ρ, σ ) if, and only if, f is nonconstant, g ( t ) > g (0) for t ∈ D ρX \ { } , and h ( u ) > h (0) for u ∈ D σY \ { } . With some adaptationson the assumptions and specifying r accordingly, similar results can be expanded to thecase where f is an unbounded complete monotone function.In this paper, the target is to establish extensions of the criterion described in theprevious paragraph in order to produce functions in the classes P D ( X × Y × Z, ρ, σ, τ )and
SP D ( X × Y × Z, ρ, σ, τ ) that can be generalized to finitely many quasi-metric spaces.From a practical point of view, we envision the results we will prove here to be used inrandom fields evolving temporally over either a torus or a cylinder. On the other hand, wealso intend to prove mathematical results that either encompass or resemble some of themodels discussed in [1, 2] involving positive definiteness for the product of three metricspaces. The outline of the paper is as follows: in Section 2, we tackle the constructionof conditionally negative definite functions on a product of quasi-metric spaces, a notionrequired in the subsequent sections which is not frequently dealt with in the literature,except in some very particular cases. We will provide a simple technique to constructfunctions in
CN D ( X × Y, ρ, σ ) and another one in the specific case in which X is theusual metric space R n . In Section 3, we discuss a model to construct positive definiteand strictly positive definite functions in a product of three quasi-metric space givenby products of compositions of completely monotone functions and nonnegative valuedconditionally negative definite functions. In Section 4, we focus on extensions of the model41.2) to three quasi-metric spaces based on generalized Stieltjes functions of order λ > λ >
0. In Section 6, we address someexamples that can serve as applications of the main results proved in the paper.
C N D ( X × Y , ρ, σ ) Results that deliver large classes of functions in
CN D ( X × Y, ρ, σ ) are rare in the literature.Here, we will present two methods that hold in general and another one that holds in thespecific case where X is the usual metric space R n . Two of them depend upon Bernsteinfunctions (see [24, Chapter 3]) the notion of which we now recall. A function f : (0 , ∞ ) → R is a Bernstein function if it has derivatives of all orders and ( − n − f ( n ) ( t ) ≥
0, for t > n = 1 , , . . . . A Bernstein function f has an integral representation in the form f ( w ) = a + bw + Z (0 , ∞ ) (1 − e − sw ) dµ ( s ) , w ≥ , where a, b ≥ µ is a positive measure on (0 , ∞ ) satisfying Z (0 , ∞ ) (1 ∧ s ) dµ ( s ) < ∞ . A Bernstein function f can be continuously extended to 0 by setting f (0) = lim w → + f ( w ).It is well known that if f is a Bernstein function and g is a nonnegative positive valuedfunction in CN D ( X, ρ ), then f ◦ g belongs to CN D ( X, ρ ). Theorem 2.1 provides a gen-eralization of this fact.
Theorem 2.1.
Let f be a Bernstein function. If g is a nonnegative valued function in CN D ( X, ρ ) and h is a nonnegative valued function in CN D ( Y, σ ) , then the function φ given by φ ( t, u ) = f ( g ( t ) + h ( u )) , ( t, u ) ∈ D ρX × D σY . belongs to CN D ( X × Y, ρ, σ ) .Proof. Assume g and h are as in the statement of the lemma. Let n be a positive integer, c , . . . , c n real numbers satisfying P nj =1 c j = 0, and ( x , y ) , . . . , ( x n , y n ) points in X × Y .Direct calculation shows that n X j,k =1 c j c k f ( g ( ρ ( x j , x k )) + h ( σ ( y j , y k ))) = b n X j,k =1 c j c k [ g ( ρ ( x j , x k )) + h ( σ ( y j , y k ))] − Z [0 , ∞ ) n X j,k =1 c j c k e − sg ( ρ ( x j ,x k )) − sh ( σ ( y j ,y k )) dµ ( s ) . x ∈ (0 , ∞ ) e − x is bounded and completely monotone and the matrix[ − sg ( ρ ( x j , x k )) − sh ( σ ( y j , y k ))] nj,k =1 is almost positive semi-definite, then Lemma 2.5 in[23] implies that n X j,k =1 c j c k e − sg ( ρ ( x j ,x k )) − sh ( σ ( y j ,y k )) ≥ . Thus, n X j,k =1 c j c k f ( g ( ρ ( x j , x k )) + h ( σ ( y j , y k ))) ≤ , and the proof is complete.Here are some examples of functions in CN D ( X × Y, ρ, σ ) provided by Theorem 2.1.The functions g and h need to be as in the statement of the theorem: φ ( t, u ) = f ( t ) + g ( u ) , φ ( t, u ) = f ( t ) + g ( u ) + p f ( t ) + g ( u ) ,φ ( t, u ) = 1 − e − f ( t ) − g ( u ) , and φ ( t, u ) = ln(1 + f ( t ) + g ( u )) . The second method we want present is based on positive valued Bernstein functionsand holds when one of the spaces is the usual R n . Theorem 2.2.
Assume R n is endowed with its usual Euclidean distance ρ . If ( Y, σ ) isa quasi-metric space, f is a positive valued Bernstein function and h is a positive valuedfunction in CN D ( Y, σ ) , then ( t, u ) ∈ [0 , ∞ ) × D ρY f ( t /h ( u )) belongs to CN D ( R n × Y, ρ, σ ) .Proof. Theorem 3.7 in [24] shows that a function f : (0 , ∞ ) → (0 , ∞ ) is a Bernsteinfunction if, and only if, e − wf is completely monotone for all w >
0. Hence, if f and h areas in the statement of the theorem, then the Bernstein-Widder’s Theorem leads to therepresentation e − wf ( x ) = Z [0 , ∞ ) e − sx dµ wf ( s ) , x ≥ , w > , for some finite and positive measure µ uf on [0 , ∞ ). Consequently, e − wf ( t /h ( u )) = Z [0 , ∞ ) e − st /h ( u ) dµ wf ( s ) , ( t, u ) ∈ [0 , ∞ ) × D σY , w > . Since Theorem 3.2-( i ) in [22] shows that( t, u ) ∈ [0 , ∞ ) × D σY e − st /h ( u ) P D ( R n × Y, ρ, σ ), we may infer that so does( t, u ) ∈ [0 , ∞ ) × D σY e − wf ( t /h ( u )) , w > . By Theorem 2.2 in [5], it follows that( t, u ) ∈ [0 , ∞ ) × D σY f ( t /h ( u ))belongs to CN D ( R n × Y, ρ, σ ).At last, we will provide a method to construct functions in
CN D ( X × Y, ρ, σ ) viageneralized Stieltjes functions. A function f is a generalized Stieltjes function of order λ >
0, and we will write S λ , it it can be represented in the form f ( w ) = C f + D f w λ + Z (0 , ∞ ) w + s ) λ dµ f ( s ) , w > , (2.3)where C f = lim w →∞ f ( w ), D f ≥
0, and µ f is a positive measure on (0 , ∞ ) such that Z (0 , ∞ ) s ) λ dµ f ( s ) < ∞ . It is not hard to see that a generalized Stieltjes function f of order λ is bounded if, andonly if, D f = 0 and Z (0 , ∞ ) s λ dµ f ( s ) < ∞ . The set of all bounded functions from S λ will be written as S bλ . Examples and additionalproperties of functions in both S λ and S bλ can be found in [18, 19, 21, 24, 28] and referencesquoted in there. It is known that every function in S λ is completely monotone. Theorem 2.3.
Let f be a function in S bλ , g a nonnegative valued function in CN D ( X, ρ ) ,and h a function in CN D ( Y, σ ) . If the function F r in (1.2) is bounded from above by M > , then M − F r belongs to CN D ( X × Y, ρ, σ ) .Proof. This follows from Theorem 2.4-( i ) in [21] where it is proved that F r belongs to P D ( X × Y, ρ, σ ). P D ( X × Y × Z, ρ, σ, τ ) In this section, we will present models that may belong to either
P D ( X × Y × Z, ρ, σ, τ ) or
SP D ( X × Y × Z, ρ, σ, τ ) based upon compositions of completely monotone functions andconditionally negative definite functions. This methodology, and also the others to come7n Sections 4 and 5, presupposes the existence of conditionally negative definite functionson a product of quasi-metric spaces as discussed in Section 2.The Schur Product Theorem implies that if f and f are completely monotone func-tions and g and h are positive valued functions in CN D ( X, ρ ) and
CN D ( Y × Z, σ, τ ),respectively, then the function F given by F ( t, u, v ) = f ( g ( t )) f ( h ( u, v )) , ( t, u, v ) ∈ D ρX × D σY × D τZ , (3.4)belongs to P D ( X × Y × Z, ρ, σ, τ ). And, if f and f are bounded, we can even assume g and h are nonnegative valued. Theorem 3.1 provides a setting in which the strict positivedefiniteness of the model can be granted. Theorem 3.1.
Assume ( X, ρ ) , ( Y, σ ) and ( Z, τ ) are metric spaces. Let f and f benonconstant completely monotone functions and g and h positive valued functions in CN D ( X, ρ ) and CN D ( Y × Z, σ, τ ) , respectively. The following assertions concerning thefunction F given by (3.4) are equivalent: ( i ) F belongs to SP D ( X × Y × Z, ρ, σ, τ )( ii ) g ( t ) > g (0) , for t ∈ D ρX \ { } , and h ( u, v ) > h (0 , , for ( u, v ) ∈ D σY × D τZ , with u + v > .Proof. If g ( t ) = g (0) for some t ∈ D ρX \ { } , we can pick two distinct points ( x , y , z )and ( x , y , z ) in X × Y × Z with ρ ( x , x ) = t , y = y , and z = z in order to obtainthe singular matrix[ F ( ρ ( x j , x k ) , σ ( y j , y k ) , τ ( z j , z k ))] j,k =1 = [ f ( g (0)) f ( h (0 , j,k =1 . If h ( u, v ) = h (0 ,
0) for some ( u, v ) ∈ D σY × D τZ with u + v >
0, we can pick two distinctpoints ( x , y , z ) and ( x , y , z ) in X × Y × Z with x = x , σ ( y , y ) = u , and τ ( z , z ) = v in order to obtain the very same singular matrix. In either case, F cannot belong to SP D ( X × Y × Z, ρ, σ, τ ) and the implication ( i ) ⇒ ( ii ) follows. As for the converse, firstwe invoke the Bernstein-Widder Theorem to write f ( g ( t )) f ( h ( u, v )) = Z [0 , ∞ ) (cid:20)Z [0 , ∞ ) e − g ( t ) s − h ( u, v ) s ′ dµ ( s ) (cid:21) dµ ( s ′ ) , where µ and µ are (not necessarily finite) positive measures on [0 , ∞ ). Recalling theproof of Theorem 2.1, we know already that the functions( t, u, v ) ∈ D ρX × D σY × D τZ e − g ( t ) s − h ( u, v ) s ′ , s, s ′ > , belong to P D ( X × Y × Z, ρ, σ, τ ). Hence, so do the functions( t, u, v ) ∈ D ρX × D σY D τZ Z [0 , ∞ ) e − g ( t ) se − h ( u, v ) s ′ dµ ( s ) , s ′ > . (3.5)8f f is nonconstant, F will belong to SP D ( X × Y × Z, ρ, σ, τ ) if we can show that thefunctions in (3.5) belong to
SP D ( X × Y × Z, ρ, σ, τ ). However, if f is nonconstant, it ispromptly seen that F will belong to SP D ( X × Y × Z, ρ, σ, τ ) as long as can show thatthe functions ( t, u, v ) ∈ D ρX × D σY × D τZ e − g ( t ) s − h ( u, v ) s ′ , s, s ′ > , belong to SP D ( X × Y × Z, ρ, σ ). So, in order to complete the proof, we will show that,under the assumptions in ( ii ), the matrices h e − g ( ρ ( x j , x k )) s − h ( σ ( y j , y k ) , τ ( z j , z k )) s ′ i nj,k =1 are positive definite whenever s, s ′ > x , y , z ) , . . . , ( x n , y n , z n ) are distinct pointsin X × Y × Z . If n = 1, there is nothing to be proved. If n ≥
2, according to Lemma 2.5in [23], the aforementioned positive definiteness will hold if, and only if, g (0) s + h (0) s ′ < g ( ρ ( x j , x k )) s + h ( σ ( y j , y k ) , τ ( z j , z k )) s ′ , j = k. (3.6)If x j = x k , then ρ ( x j , x k ) > g implies that g ( ρ ( x j , x k )) > g (0). If y j = y k , then σ ( y j , y k ) > h implies that h ( σ ( y j , y k ) , τ ( z j , z k )) >h (0 , z j = z k . Thus, in any case, (3.6) holds.The model given by (3.4) has a considerable drawback: the variables u and w areseparated from t . Since separability is usually not present in models that come fromapplications, the results in the next sections may be interpreted as an attempt to providemodels with no such inconvenience. Here, we will extend and analyze the model (1.2) for three quasi-metric spaces. Sincethere is more than one way to do it, we will begin with one possible extension of (1.2)and will establish a basic necessary condition for its strict positive definiteness.
Theorem 4.1.
Let f belong to S λ , g a positive valued function in CN D ( X, ρ ) and h apositive valued function in CN D ( Y × Z, σ, τ ) . For r ≥ λ , set G r ( t, u, v ) = 1 h ( u, v ) r f (cid:18) g ( t ) h ( u, v ) (cid:19) , ( t, u, v ) ∈ D ρX × D σY × D τZ . (4.7) The following assertions hold: ( i ) G r belongs to P D ( X × Y × Z, ρ, σ, τ ) . ii ) If G r belongs to SP D ( X × Y × Z, ρ, σ, τ ) , then g ( t ) > g (0) , for t ∈ D ρX \ { } , and h ( u, v ) > h (0 , , for ( u, w ) ∈ D σY × D τZ with u + v > .Proof. Inserting the integral representation for f in (4.7) leads to the formula G r ( t, u, v ) = C f h ( u, v ) r + D f g ( t ) λ h ( u, v ) r − λ + 1 h ( u, v ) r − λ Z (0 , ∞ ) g ( t ) + sh ( u, v )] λ dµ f ( s ) . In order to prove ( i ), it suffices to show that each one of the three summands above belongto P D ( X × Y × Z, ρ, σ, τ ). Once the functions w ∈ (0 , ∞ ) w α , α = λ, r, r − λ, are completely monotone, some of the basic results quoted at the introduction of the papershow that t ∈ D ρX g ( t ) − λ belongs to P D ( X, ρ ) while ( u, w ) ∈ D σY × D τZ h ( u, w ) α , α = r, r − λ , belong to P D ( Y × Z, σ, τ ). Hence, it is easily seen that all the functions ( t, u, v ) ∈ D ρX × D σY × D τZ g ( t ) − λ and ( t, u, v ) ∈ D ρX × D σY × D τZ h ( u, v ) α , α = r, r − λ , belongto P D ( X × Y × Z, ρ, σ, τ ). The fact that
P D ( X × Y × Z, ρ, σ, τ ) is closed under productsis all that is needed in order to see that ( t, u, v ) ∈ D ρX × D σY × D τZ g ( t ) − λ h ( u, v ) r − λ belongs to P D ( X × Y × Z, ρ, σ, τ ) as well. It remains to show that the third summandbelongs to
P D ( X × Y × Z, ρ, σ, τ ). Since w ∈ (0 , ∞ ) e − w is completely monotone,the same reasoning reveals that ( t, u, v ) ∈ D ρX × D σY × D τZ exp( − wg ( t ) − wh ( u, v ))belongs to P D ( X × Y × Z, ρ, σ, τ ) for w >
0. The fact that integration with respect to anindependent parameter does not affect positive definiteness and the elementary identityΓ( λ )( s + t ) λ = Z ∞ e − sw e − tww λ − dw, s, t > . (4.8)now implies that all the functions( t, u, v ) ∈ D ρX × D σY × D τZ g ( t ) + sh ( u, v )] λ , s > , belong to P D ( X × Y × Z, ρ, σ, τ ). But, since
P D ( X × Y × Z, ρ, σ, τ ) is closed underproducts, we now see that( t, u, v ) ∈ D ρX × D σY × D τZ h ( u, v ) r − λ Z (0 , ∞ ) g ( t ) + sh ( u, v )] λ dµ f ( s ) , s > , also do. The proof of ( ii ) needs to be done by contradiction. If g ( t ) = g (0), for some t ∈ D ρX \ { } , by picking two distinct points ( x , y , z ) and ( x , y , z ) in X × Y × Z suchthat ρ ( x , x ) = t , y = y , and z = z , we obtain the singular matrix[ G r ( ρ ( x j , x k ) , σ ( y j , y k ) , τ ( z j , z k ))] j,k =1 = (cid:20) h (0 , r f (cid:18) g (0) h (0 , (cid:19)(cid:21) j,k =1 h ( u, v ) = h (0 , u, v ) ∈ D σY × D τZ with u + v >
0, we can take two distinct points( x , y , z ) and ( x , y , z ) in X × Y × Z such that x = x , σ ( y , y ) = u , and τ ( z , z ) = v in order to obtain the very same singular matrix.Henceforth, we will say a quasi-metric space is nontrivial if it contains at least twopoints. Theorem 4.2 provides additional necessary conditions for the strict positive definiteof the model in Theorem 4.1 in some specific cases. Theorem 4.2.
Let f belong to S λ , g a positive valued function in CN D ( X, ρ ) and h a positive valued function in CN D ( Y × Z, σ, τ ) . The following assertion holds for thefunction G r as in (4.7) : ( i ) If ( X, ρ ) is nontrivial and G r belongs to SP D ( X × Y × Z, ρ, σ, τ ) , then either D f > or µ f is not the zero measure.Further, in the case in which r = λ and D f > , the following additional assumptionholds: ( ii ) If either ( Y, σ ) or ( Z, τ ) is nontrivial and G λ belongs to SP D ( X × Y × Z, ρ, σ, τ ) ,then either C f > or µ f is not the zero measure.Proof. If (
X, ρ ) is nontrivial, D f = 0 and µ f is the zero measure, then we can take twodistinct points ( x , y , z ) and ( x , y , z ) in X × Y × Z with y = y and z = z in orderto obtain the singular matrix[ G r ( ρ ( x j , x k ) , σ ( y j , y k ) , τ ( z i , z j ))] j,k =1 = (cid:20) C f h (0 , r (cid:21) j,k =1 . Similarly, if either (
Y, σ ) or (
Z, τ ) is nontrivial, r = λ , C f = 0 < D f and µ f is the zeromeasure, then we can take two distinct points ( x , y , z ) and ( x , y , z ) in X × Y × Z with x = x in order to obtain the singular matrix[ G λ ( ρ ( x j , x k ) , σ ( y j , y k ) , τ ( z j , z k ))] j,k =1 = (cid:20) D f g (0) λ (cid:21) j,k =1 , In either case, we may infer that G r cannot belong to SP D ( X × Y × Z, ρ, σ, τ ).Theorem 4.3 achieves a necessary and sufficient condition for the strict positive defi-niteness of G r in the case in which r > λ and D f > f . Theorem 4.3.
Assume ( X, ρ ) , ( Y, σ ) and ( Z, τ ) are metric spaces. Let f belong to S λ , g a positive valued function in CN D ( X, ρ ) and h a positive valued function in CN D ( Y × Z, σ, τ ) . If D f > and r > λ , then the following assertions for G r as in (4.7) areequivalent: i ) G r belongs to SP D ( X × Y × Z, ρ, σ, τ ) . ( ii ) g ( t ) > g (0) , for t ∈ D ρX \ { } , and h ( u, v ) > h (0 , , for ( u, v ) ∈ D σY × D τZ with u + v > .Proof. In view of Theorem 4.1-( ii ), only the implication ( ii ) ⇒ ( i ) needs to be proved.Assume D f > r > λ , and also the two assumptions on g and h quoted in ( ii ). Theorem3.1 coupled with arguments justified in the proof of Theorem 4.1 reveal that( t, u, v ) ∈ D ρX × D σY × D τZ D f g ( t ) λ h ( u, v ) r − λ belongs to SP D ( X × Y × Z, ρ, σ, τ ). As the other two summands appearing in the equationdefining G r ( t, u, v ) belong to P D ( X × Y × Z, ρ, σ, τ ), the result follows.Next, we provide a necessary and sufficient condition for strict positive definiteness inthe case in which D f = 0, and r ≥ λ . Theorem 4.4.
Assume ( X, ρ ) , ( Y, σ ) and ( Z, τ ) are metric spaces. Let f belong to S λ , g a positive valued function in CN D ( X, ρ ) and h a positive valued function in CN D ( Y × Z, σ, τ ) . If ( X, ρ ) is nontrivial, D f = 0 , and r ≥ λ , then the following assertions for G r as in (4.7) are equivalent: ( i ) G r belongs to SP D ( X × Y × Z, ρ, σ, τ ) . ( ii ) f is nonconstant, g ( t ) > g (0) , for t ∈ D ρX \ { } , and h ( u, w ) > h (0 , , for ( u, w ) ∈ D σY × D τZ with u + w > .Proof. Assume (
X, ρ ) is nontrivial, D f = 0, and r ≥ λ . If G r belongs to SP D ( X × Y × Z, ρ, σ, τ ), Theorem 4.2-( i ) shows that µ f is nonzero. In particular, f is nonconstant. Onthe other hand, Theorem 4.1-( ii ) reveals that the other two conditions in ( ii ) hold as well.Thus, ( i ) implies ( ii ). Conversely, if f is nonconstant, the assumption D f = 0 implies thatthe measure µ f is nonzero. That being said, ( i ) will follow if we can prove that( t, u, v ) ∈ D ρX × D σY × D τZ Z (0 , ∞ ) g ( t ) + sh ( u, v )] λ dµ f ( s )belongs to SP D ( X × Y × Z, ρ, σ, τ ) under the other two assumptions in ( ii ). Indeed, since( t, u, v ) ∈ D ρX × D σY × D τZ h ( u, v ) λ − r belongs to P D ( X × Y × Z, ρ, σ, τ ) and h (0 , > i ). Since µ f is nonzero, it suffices toshow that ( t, u, v ) ∈ D ρX × D σY × D τZ g ( t ) + sh ( u, v )] λ SP D ( X × Y × Z, ρ, σ, τ ), for s >
0. Reporting to (4.8), what needs to beproved is that the functions( t, u, v ) ∈ D ρX × D σY × D τZ e − g ( t ) w − h ( u, v ) sw, w, s > , belong to SP D ( X × Y × Z, ρ, σ, τ ). But that follows by the same trick employed at theend of the proof of Theorem 3.1.The proof of Theorem 4.4 justifies the following complement of Theorem 4.3.
Theorem 4.5.
Assume ( X, ρ ) , ( Y, σ ) , and ( Z, τ ) are metric spaces. Let f belong to S λ , g a positive valued function in CN D ( X, ρ ) , and h a positive valued function in CN D ( Y × Z, σ, τ ) . If D f > , r = λ , and µ f is nonzero, then the following assertions for G r as in(4.7) are equivalent: ( i ) G λ belongs to SP D ( X × Y × Z, ρ, σ, τ ) . ( ii ) g ( t ) > g (0) , for t ∈ D ρX \ { } , and h ( u, v ) > h (0 , , for ( u, v ) ∈ D σY × D τZ with u + w > . It remains to consider the case in which D f > r = λ and µ f = 0. However, Theorem4.2-( ii ) shows that, essentially, what needs to be analyzed is the case where C f D f > r = λ and µ f = 0 and also imposing the non triviality of some of the spaces involved. Inthis case G r takes the form G λ ( t, u, v ) = C f h ( u, v ) λ + D f g ( t ) λ , ( t, u, v ) ∈ D ρX × D σY × D τZ , with C f D f >
0. So far, this case remains open with respect to strict positive definiteness.
Remark 4.6.
All the theorems proved so far can be re-stated and demonstrated for themodel H r ( t, u, v ) = 1 g ( t ) r f (cid:18) h ( u, v ) g ( t ) (cid:19) , ( t, u, v ) ∈ D ρX × D σY × D τZ , f ∈ S λ . with r , g and h as before. The obvious adjustments and the details on that will be left tothe readers. In this section, we will point how to extend the results proved in Section 4 to modelsdefined by functions coming from the class B λ of generalized complete Bernstein functions f order λ >
0, that is, functions f having a representation in the form f ( w ) = A f + B f w λ + Z (0 , ∞ ) (cid:18) ww + s (cid:19) λ dν f ( s ) , x > , (5.9)where A f , B f ≥ ν f is a positive measure on (0 , ∞ ) for which Z (0 , ∞ ) s ) λ dν f ( s ) < ∞ . The class B is more common in the literature. Functions in it may receive differentnames depending where they are used: operator monotone functions, L¨owner functions,Pick functions, Nevanlinna functions, etc. Many examples of functions in B λ can be foundscattered in [24].As we shall see below, the proofs of the results to be enunciated in this section arevery similar to those of the theorems proved in Section 3. For that reason, most of thedetails will be omitted.We begin with a version of Theorem 4.1 for models generated by functions in B λ . Theorem 5.1.
Let f belong to B λ , g a positive valued function in CN D ( X, ρ ) , and h apositive valued function in CN D ( Y × Z, σ, τ ) . For r ≥ λ , set I r ( t, u, v ) = 1 g ( t ) r f (cid:18) g ( t ) h ( u, v ) (cid:19) , ( t, u, v ) ∈ D ρX × D σY × D τZ . (5.10) The following assertions hold: ( i ) I r belongs to P D ( X × Y × Z, ρ, σ, τ ) . ( ii ) If I r belongs to SP D ( X × Y, ρ, σ ) , then g ( t ) > g (0) , for t ∈ D ρX \ { } , and h ( u, v ) >h (0 , , for ( u, v ) ∈ D σY × D τZ with u + v > .Proof. It suffices to use the formula I r ( t, u, v ) = A f g ( t ) r + B f h ( u, v ) λ g ( t ) r − λ + 1 g ( t ) r − λ Z (0 , ∞ ) g ( t ) + sh ( u, v )] λ dν f ( s )that derives from the integral representation for f and to mimic the proof of Theorem4.1.Theorem 4.2 takes the following form. Theorem 5.2.
Let f belong to B λ , g a positive valued function in CN D ( X, ρ ) , and h a positive valued function in CN D ( Y × Z, σ, τ ) . The following assertion holds for thefunction I r in (5.10) : i ) If either ( Y, σ ) or ( Z, τ ) is nontrivial and I r belongs to SP D ( X × Y × Z, ρ, σ, τ ) ,then either B f > or ν f is not the zero measure.In the case in which r = λ and D f > , the following additional assumption holds: ( ii ) If ( X, ρ ) is nontrivial and I λ belongs to SP D ( X × Y × Z, ρ, σ, τ ) , then either A f > or ν f is not the zero measure. As for the strict positive definiteness of the models being considered in this section,the following three results settle an if, and only if, condition.
Theorem 5.3.
Assume ( X, ρ ) , ( Y, σ ) , and ( Z, τ ) are metric spaces. Let f belong to B λ , g a positive valued function in CN D ( X, ρ ) , and h a positive valued function in CN D ( Y × Z, σ, τ ) . If B f > and r > λ , then the following assertions for I r as in (5.10) areequivalent: ( i ) I r belongs to SP D ( X × Y × Z, ρ, σ, τ ) . ( ii ) g ( t ) > g (0) , for t ∈ D ρX \ { } , and h ( u, v ) > h (0 , , for ( u, v ) ∈ D σY × D τZ with u + v > . Theorem 5.4.
Assume ( X, ρ ) , ( Y, σ ) , and ( Z, τ ) are metric spaces. Let f belong to B λ , g a positive valued function in CN D ( X, ρ ) , and h a positive valued function in CN D ( Y × Z, σ, τ ) . If either ( Y, σ ) or ( Z, τ ) is nontrivial, B f = 0 , and r ≥ λ , then the followingassertions for I r as in (5.10) are equivalent: ( i ) I r belongs to SP D ( X × Y × Z, ρ, σ, τ ) . ( ii ) f is nonconstant, g ( t ) > g (0) , for t ∈ D ρX \ { } , and h ( u, v ) > h (0 , , for ( u, v ) ∈ D σY × D τZ , u + v > . Theorem 5.5.
Assume ( X, ρ ) , ( Y, σ ) , and ( Z, τ ) are metric spaces. Let f belong to B λ , g a positive valued function in CN D ( X, ρ ) , and h a positive valued function in CN D ( Y × Z, σ, τ ) . If B f > , r = λ , and ν f is nonzero, then the following assertions for I r as in(5.10) are equivalent: ( i ) I λ belongs to SP D ( X × Y × Z, ρ, σ, τ ) . ( ii ) g ( t ) > g (0) , for t ∈ D ρX \ { } , and h ( u, v ) > h (0 , , for ( u, v ) ∈ D σY × D τZ with u + v > . Remark 5.6.
All the theorems proved so far in this section can be re-stated and provedfor the model J r ( t, u, v ) = 1 h ( u, v ) r f (cid:18) h ( u, v ) g ( t ) (cid:19) , ( t, u, v ) ∈ D ρX × D σY × D τZ , f ∈ B λ , with r , g and h as before and with some small adjustments. Once again, we leave theproofs to the interested readers. 15 Some concrete realizations
This section contains some illustrations of the theorems proved in Section 4. All of themcan be adapted in order to become applications of the theorems presented in Section 5,but that will be left to the readers.
Example 6.1.
Let X be the unit sphere S d in R d +1 endowed with is usual geodesicdistance ρ d and let Y = [0 , π/
2] and Z = R n both endowed with their usual Euclideandistances σ and τ respectively. The function g given by the formula g ( t ) = 3 − cos t, t ∈ [0 , π ] , belongs to CN D ( S d , ρ d ) while results proved in [17] points that, if s ∈ (0 , h given by h ( u, v ) = 1 + sin u + v s , ( u, v ) ∈ [0 , π/ × [0 , ∞ ) , belongs to CN D ( Y × Z, σ, τ ). It is also easily seen that g ( t ) > g (0) for all t ∈ (0 , π ] and h ( u, v ) > h (0 ,
0) for ( u, v ) ∈ [0 , π/ × [0 , ∞ ) with u + v >
0. Under the setting of eitherTheorem 4.3 or Theorem 4.4, the model G r ( t, u, v ) = 1[1 + sin u + v s ] r f (cid:18) − cos t u + v s (cid:19) , ( t, u, v ) ∈ [0 , π ] × [0 , π/ × [0 , ∞ ) , defines a function G r in SP D ( X, Y, Z, ρ d , σ, τ ), whenever f comes from S λ . A similarconclusion holds for the model H r ( t, u, v ) = 1[3 − cos t ] r f (cid:18) u + v s − cos t (cid:19) , ( t, u, v ) ∈ [0 , π ] × [0 , π/ × [0 , ∞ ) , under the setting in Remark 4.6. These examples can be expanded a little bit, by letting Z be a Hilbert space and τ the distance induced by its norm, keeping all the rest thesame. As a matter of fact, we can let ( Z, τ ) be a quasi-metric space which is isometricallyembedded in an infinite dimensional Hilbert space.
Example 6.2.
Here we consider X = R endowed with its Euclidean norm ρ . On the otherhand, we let Y = S d and Z = S d ′ , both endowed with their geodesic distances σ d and τ d ′ . Since t ∈ [0 , π ] t belongs to both CN D ( Y, σ d ) and CN D ( Z, τ d ′ ), then the mapping h : [0 , π ] → R given by h ( u, v ) = c + u + v defines a positive valued function that belongs to CN D ( Y × Z, σ d , τ d ′ ), whenever c is a positive constant. In addition, h ( u, v ) > c = h (0 , u + v >
0. On the other hand, g : [0 , ∞ ) → R given by g ( t ) = t s , t ≥
0, belongsto
CN D ( X, ρ ), as long as s ∈ (0 , c + g is a positive valued function that belongs16o CN D ( X, ρ ) for which g ( t ) > c = g (0) for t >
0. With this in mind, it is now clear thatunder the setting of either Theorem 4.3 or Theorem 4.4, the model G r ( t, u, v ) = 1[ c + u + v ] r f (cid:18) c + t s c + u + v (cid:19) , ( t, u, v ) ∈ [0 , ∞ ) × [0 , π ] × [0 , π ] , defines a function G r in SP D ( X, Y, Z, ρ, σ d , τ d ′ ), as long as f comes from S λ . The inter-ested reader can implement quite more complicated examples along the same lines byusing the characterization of functions in CN D ( S d , σ d ) obtained in [20] and the manyconcrete examples of functions in CN D ( R , ρ ) listed in [17]. Example 6.3.
Here we let Y = R n endowed with its usual Euclidean distance σ while( X, ρ ) and (
Z, τ ) are general metric spaces. We also let f be a positive valued Bernsteinfunction, g a positive valued function in CN D ( X, ρ ) and h a positive valued function in CN D ( Z, τ ). Theorem 2.2 shows that( u, v ) ∈ [0 , ∞ ) × D τZ f ( u /h ( v ))is a positive valued function in CN D ( Y × Z, σ, τ ). Thus, under the setting of eitherTheorem 4.3 or Theorem 4.4, the model G r ( t, u, v ) = 1 f ( u /h ( v )) r f (cid:18) g ( t ) f ( u /h ( v )) (cid:19) , ( t, u, v ) ∈ D ρX × [0 , ∞ ) × D τZ , defines a function G r in SP D ( X, Y, Z, ρ, σ, τ ), whenever f comes from S λ . Two importantparticular examples of the setting just described deserve to be detached:- Let X = R d and Z = R k , both endowed with their Euclidean distances and pick g ( t ) = f ( t ), t ≥ h ( v ) = f ( v ), v ≥
0, in which f and f are positive valuedBernstein functions. Under the setting of either Theorem 4.3 or Theorem 4.4, we mayinfer that G r ( t, u, v ) = 1 f ( u /f ( v )) r f (cid:18) f ( t ) f ( u /f ( v )) (cid:19) , ( t, u, v ) ∈ [0 , ∞ ) × [0 , ∞ ) × [0 , ∞ ) . defines a function G r in SP D ( X, Y, Z, ρ, σ, τ ), whenever f comes from S λ . By letting k = 1, f ( t ) = t , t ≥
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