A Gneiting-like method for constructing positive definite functions on metric spaces

This paper is concerned with the construction of positive definite functions on a cartesian product of quasi-metric spaces using generalized Stieltjes and complete Bernstein functions. The results we prove are aligned with a well-established method of T. Gneiting to construct space-time positive definite functions and its many extensions. For the right choice of the quasi-metric spaces, the models discussed in the paper lead to flexible, interpretable and even computationally feasible classes of cross-covariance functions for multivariate random fields adopted in statistics. Necessary and sufficient conditions for the strict positive definiteness of the models are provided when the spaces are metric.


Introduction
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 ρ X to indicate the diameter-set of (X, ρ), i.e., This paper is mainly concerned with radial positive definite functions on (X, ρ), that is, continuous functions f : D ρ X → R satisfying n j,k=1 c j c k f (ρ(x j , x k )) ≥ 0, (1.1) needs 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 space and 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 2 while ρ is its geodesic distance ( [25]). R. Gangolli ([9]) extended Schoenberg results to P D(H, ρ), where H is any 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 that a continuous function f : [0, π] → R belongs to P D(H, ρ), if, and only if, f has a series representation in the form where a H k ≥ 0 for all k and ∞ k=0 a H k P H k (1) < ∞. Here, P H k 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] through additional conditions on the sets {k : a H k > 0}. The same was done for the classes P D(X × Y, ρ, σ) and SP D(X × Y, ρ, σ) for some choices of (X, ρ) and (Y, σ). For the case where X and Y are compact two-point homogeneous spaces with their respective Riemannian distances ρ and σ, the characterization for P D(X × Y, ρ, σ) appeared in [4,13]: a continuous function f : [0, π] 2 → R belongs to SP D(X × Y, ρ, σ) if, and only if, f has a series representation in the form a X,Y k,l P X k (cos t)P Y l (cos u), t, u ∈ [0, π], with a X,Y k,l ≥ 0 for all k and l and the series being convergent at (t, u) = (0, 0). As for SP D(X × Y, ρ, σ), a description can be found in [4,12,14,15] and depends on additional assumptions on the sets {k − l : a X,Y k,l > 0}. The cases in which (X, ρ) is the usual metric space R n and Y is either a compact two-point homogeneous space or S ∞ were considered recently: P D(X × Y, ρ, σ) was described in [6,7,11,27] while a description for 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 conditionally negative definite functions on (X, ρ). A continuous function f : D σ X → R is conditionally negative definite on (X, ρ), and we write f ∈ CND(X, ρ), if the quadratic forms in (1.1) are nonpositive when the coefficients c j satisfy n j=1 c j = 0. Clearly, this notion can be extended to a cartesian product of quasi-metric spaces so that the symbol CND(X × Y, ρ, σ) also makes sense.
The following construction providing an efficient technique follows from Theorem 3.5 in [20] along with Lemma 2.5 in [23]: if f is a bounded and completely monotone function and g is a nonnegative valued function in CND(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 ρ X \ {0}. 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 function in CND(X × Y, ρ, σ), then f • g belongs to P D(X × Y, ρ, σ). Further, f • g belongs to SP D(X × Y, ρ, σ) if, and only if, f is nonconstant and g(t, u) > g(0, 0), for (t, u) ∈ D ρ X × D σ Y with t + u > 0. If we drop the boundedness of f , then the results above still hold as long as we assume g is positive valued.
Motivated by a celebrated result of Gneiting in [10], an interesting procedure to construct positive definite functions on a cartesian product of quasi-metric spaces was described in [21]. If f is a bounded and completely monotone function, g is a nonnegative valued function in CND(X, ρ) and h is a positive valued function in CND(Y, σ), then the function F r given by belongs to P D(X ×Y, ρ, σ), as long as f is a bounded generalized Stieltjes function of order λ > 0 and 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, With some adaptations on the assumptions and specifying r accordingly, similar results can be expanded to the case where f is an unbounded complete monotone function.
In this paper, the target is to establish extensions of the criterion described in the previous 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 in random fields evolving temporally over either a torus or a cylinder. On the other hand, we also intend to prove mathematical results that either encompass or resemble some of the models discussed in [1,2] involving positive definiteness for the product of three metric spaces. The outline of the paper is as follows: in Section 2, we tackle the construction of conditionally negative definite functions on a product of quasi-metric spaces, a notion required 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 construct functions in CND(X × Y, ρ, σ) and another one in the specific case in which X is the usual metric space R n . In Section 3, we discuss a model to construct positive definite and strictly positive definite functions in a product of three quasi-metric space given by products of compositions of completely monotone functions and nonnegative valued conditionally negative definite functions. In Section 4, we focus on extensions of the model (1.2) to three quasi-metric spaces based on generalized Stieltjes functions of order λ > 0 while Section 5 contains adaptations of the results in Section 4 to produce models based on generalized complete Bernstein functions of order λ > 0. In Section 6, we address some examples that can serve as applications of the main results proved in the paper.
2 Functions in the class CN D(X × Y, ρ, σ) Results that deliver large classes of functions in CND(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 the specific case where X is the usual metric space R n . Two of them depend upon Bernstein functions (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 (−1) n−1 f (n) (t) ≥ 0, for t > 0 and n = 1, 2, . . .. A Bernstein function f has an integral representation in the form where a, b ≥ 0 and µ is a positive measure on (0, ∞) satisfying A Bernstein function f can be continuously extended to 0 by setting f (0) = lim w→0 + f (w). It is well known that if f is a Bernstein function and g is a nonnegative positive valued function in CND(X, ρ), then f • g belongs to CND(X, ρ). Theorem 2.1 provides a generalization of this fact.
Theorem 2.1. Let f be a Bernstein function. If g is a nonnegative valued function in CND(X, ρ) and h is a nonnegative valued function in CND(Y, σ), then the function φ given by Proof. Assume g and h are as in the statement of the lemma. Let n be a positive integer, c 1 , . . . , c n real numbers satisfying n j=1 c j = 0, and (x 1 , y 1 ), . . . , (x n , y n ) points in X × Y . Direct calculation shows that n j,k=1 Since the function x ∈ (0, ∞) → e −x is bounded and completely monotone and the matrix [−sg(ρ(x j , x k )) − sh(σ(y j , y k ))] n j,k=1 is almost positive semi-definite, then Lemma 2.5 in [23] implies that n j,k=1 and the proof is complete.
Here are some examples of functions in CND(X × Y, ρ, σ) provided by Theorem 2.1. The functions g and h need to be as in the statement of the theorem: The second method we want present is based on positive valued Bernstein functions and holds when one of the spaces is the usual R n .
Proof. Theorem 3.7 in [24] shows that a function f : (0, ∞) → (0, ∞) is a Bernstein function if, and only if, e −wf is completely monotone for all w > 0. Hence, if f and h are as in the statement of the theorem, then the Bernstein-Widder's Theorem leads to the representation for some finite and positive measure µ u f on [0, ∞). Consequently, Since Theorem 3.2-(i) in [22] shows that belongs to P D(R n × Y, ρ, σ), we may infer that so does By Theorem 2.2 in [5], it follows that At last, we will provide a method to construct functions in CND(X × Y, ρ, σ) via generalized 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 It is not hard to see that a generalized Stieltjes function f of order λ is bounded if, and only if, The set of all bounded functions from S λ will be written as S b λ . Examples and additional properties of functions in both S λ and S b λ can be found in [18,19,21,24,28] and references quoted in there. It is known that every function in S λ is completely monotone.
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).

Products in
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 and conditionally negative definite functions. This methodology, and also the others to come in Sections 4 and 5, presupposes the existence of conditionally negative definite functions on a product of quasi-metric spaces as discussed in Section 2.
The Schur Product Theorem implies that if f 1 and f 2 are completely monotone functions and g and h are positive valued functions in CND(X, ρ) and CND(Y × Z, σ, τ ), respectively, then the function F given by And, if f 1 and f 2 are bounded, we can even assume g and h are nonnegative valued. Theorem 3.1 provides a setting in which the strict positive definiteness of the model can be granted.
we can pick two distinct points (x 1 , y 1 , z 1 ) and (x 2 , y 2 , z 2 ) in X ×Y ×Z with x 1 = x 2 , σ(y 1 , y 2 ) = u, and τ (z 1 , z 2 ) = 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, first we invoke the Bernstein-Widder Theorem to write where µ 1 and µ 2 are (not necessarily finite) positive measures on [0, ∞). Recalling the proof of Theorem 2.1, we know already that the functions belong to P D(X × Y × Z, ρ, σ, τ ). Hence, so do the functions If f 2 is nonconstant, F will belong to SP D(X × Y × Z, ρ, σ, τ ) if we can show that the functions in (3.5) belong to SP D(X × Y × Z, ρ, σ, τ ). However, if f 1 is nonconstant, it is promptly seen that F will belong to SP D(X × Y × Z, ρ, σ, τ ) as long as can show that the functions 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 are positive definite whenever s, s ′ > 0 and (x 1 , y 1 , z 1 ), . . . , (x n , y n , z n ) are distinct points in X × Y × Z. If n = 1, there is nothing to be proved. If n ≥ 2, according to Lemma 2.5 in [23], the aforementioned positive definiteness will hold if, and only if, If x j = x k , then ρ(x j , x k ) > 0 and the assumption on g implies that g(ρ(x j , x k )) > g(0). If y j = y k , then σ(y j , y k ) > 0 and the assumption on h implies that h(σ(y j , y k ), τ (z j , z k )) > h(0, 0). The same can be inferred if 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 are separated from t. Since separability is usually not present in models that come from applications, the results in the next sections may be interpreted as an attempt to provide models with no such inconvenience.

Models based on generalized Stieltjes functions
Here, we will extend and analyze the model (1.2) for three quasi-metric spaces. Since there 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.
Inserting the integral representation for f in (4.7) leads to the formula In order to prove (i), it suffices to show that each one of the three summands above belong to P D(X × Y × Z, ρ, σ, τ ). Once the functions are completely monotone, some of the basic results quoted at the introduction of the paper It remains to show that the third summand belongs to P D(X × Y × Z, ρ, σ, τ ). Since w ∈ (0, ∞) → e −w is completely monotone, the same reasoning reveals that (t, The fact that integration with respect to an independent parameter does not affect positive definiteness and the elementary identity Γ(λ) (s + t) λ = ∞ 0 e −sw e −tw w λ−1 dw, s, t > 0. (4.8) now implies that all the functions belong to P D(X × Y × Z, ρ, σ, τ ). But, since P D(X × Y × Z, ρ, σ, τ ) is closed under products, we now see that also do. The proof of (ii) needs to be done by contradiction. If g(t) = g(0), for some t ∈ D ρ X \ {0}, by picking two distinct points (x 1 , y 1 , z 1 ) and (x 2 , y 2 , z 2 ) in X × Y × Z such that ρ(x 1 , x 2 ) = t, y 1 = y 2 , and z 1 = z 2 , we obtain the singular matrix If h(u, v) = h(0, 0), for (u, v) ∈ D σ Y × D τ Z with u + v > 0, we can take two distinct points (x 1 , y 1 , z 1 ) and (x 2 , y 2 , z 2 ) in X × Y × Z such that x 1 = x 2 , σ(y 1 , y 2 ) = u, and τ (z 1 , z 2 ) = 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 two points. Theorem 4.2 provides additional necessary conditions for the strict positive definite of the model in Theorem 4.1 in some specific cases. Further, in the case in which r = λ and D f > 0, the following additional assumption holds: Proof. If (X, ρ) is nontrivial, D f = 0 and µ f is the zero measure, then we can take two distinct points (x 1 , y 1 , z 1 ) and (x 2 , y 2 , z 2 ) in X × Y × Z with y 1 = y 2 and z 1 = z 2 in order to obtain the singular matrix Similarly, if either (Y, σ) or (Z, τ ) is nontrivial, r = λ, C f = 0 < D f and µ f is the zero measure, then we can take two distinct points (x 1 , y 1 , z 1 ) and (x 2 , y 2 , z 2 ) in X × Y × Z with x 1 = x 2 in order to obtain the singular matrix In either case, we may infer that G r cannot belong to SP D(X × Y × Z, ρ, σ, τ ).   Z, σ, τ ). If D f > 0 and r > λ, then the following assertions for G r as in (4.7) are equivalent: Proof. In view of Theorem 4.1-(ii), only the implication (ii) ⇒ (i) needs to be proved. Assume D f > 0, r > λ, and also the two assumptions on g and h quoted in (ii). Theorem 3.1 coupled with arguments justified in the proof of Theorem 4.1 reveal that belongs to SP D(X ×Y ×Z, ρ, σ, τ ). As the other two summands appearing in the equation defining 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 in the case in which D f = 0, and r ≥ λ.  If (X, ρ) is nontrivial, D f = 0, and r ≥ λ, then the following assertions for G r as in (4.7) are equivalent: 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. On the 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 that the measure µ f is nonzero. That being said, (i) will follow if we can prove that and h(0, 0) > 0, Oppenheim's inequality ([16, P.480]) will lead to (i). Since µ f is nonzero, it suffices to show that belongs to SP D(X × Y × Z, ρ, σ, τ ), for s > 0. Reporting to (4.8), what needs to be proved is that the functions belong to SP D(X × Y × Z, ρ, σ, τ ). But that follows by the same trick employed at the end of the proof of Theorem 3.1.
The proof of Theorem 4.4 justifies the following complement of Theorem 4.3. . If D f > 0, r = λ, and µ f is nonzero, then the following assertions for G r as in (4.7) are equivalent: It remains to consider the case in which D f > 0, r = λ and µ f = 0. However, Theorem 4.2-(ii) shows that, essentially, what needs to be analyzed is the case where C f D f > 0, r = λ and µ f = 0 and also imposing the non triviality of some of the spaces involved. In this case G r takes the form 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 the model with r, g and h as before. The obvious adjustments and the details on that will be left to the readers.

Models based on generalized complete Bernstein functions
In this section, we will point how to extend the results proved in Section 4 to models defined by functions coming from the class B λ of generalized complete Bernstein functions of order λ > 0, that is, functions f having a representation in the form (5.9) where A f , B f ≥ 0 and ν f is a positive measure on (0, ∞) for which The class B 1 is more common in the literature. Functions in it may receive different names depending where they are used: operator monotone functions, Löwner functions, Pick functions, Nevanlinna functions, etc. Many examples of functions in B λ can be found scattered in [24].
As we shall see below, the proofs of the results to be enunciated in this section are very similar to those of the theorems proved in Section 3. For that reason, most of the details will be omitted.
We begin with a version of Theorem 4.1 for models generated by functions in B λ .
(ii) If I r belongs to SP D(X × Y, ρ, σ), then g(t) > g(0), for t ∈ D ρ X \ {0}, and h(u, v) > h(0, 0), Proof. It suffices to use the formula that derives from the integral representation for f and to mimic the proof of Theorem 4.1.  (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 > 0 or ν f is not the zero measure.
In the case in which r = λ and D f > 0, 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 > 0 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. . If B f > 0 and r > λ, then the following assertions 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 ρ X \ {0}, and h(u, v) > h(0, 0), for (u, v) ∈ D σ Y × D τ Z , u + v > 0. 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 CND(X, ρ), and h a positive valued function in CND(Y × Z, σ, τ ). If B f > 0, r = λ, and ν f is nonzero, then the following assertions for I r as in (5.10) are equivalent: Remark 5.6. All the theorems proved so far in this section can be re-stated and proved for the model with r, g and h as before and with some small adjustments. Once again, we leave the proofs to the interested readers.

Some concrete realizations
This section contains some illustrations of the theorems proved in Section 4. All of them can 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 geodesic distance ρ d and let Y = [0, π/2] and Z = R n both endowed with their usual Euclidean distances σ and τ respectively. The function g given by the formula belongs to CND(S d , ρ d ) while results proved in [17] points that, if s ∈ (0, 2], then the function h given by belongs to CND(Y × Z, σ, τ ). It is also easily seen that g(t) > g(0) for all t ∈ (0, π] and defines a function G r in SP D(X, Y, Z, ρ d , σ, τ ), whenever f comes from S λ . A similar conclusion holds for the model 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 the same. As a matter of fact, we can let (Z, τ ) be a quasi-metric space which is isometrically embedded in an infinite dimensional Hilbert space.
Example 6.2. Here we consider X = R endowed with its Euclidean norm ρ. On the other hand, 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 CND(Y, σ d ) and CND(Z, τ d ′ ), then the mapping h : [0, π] 2 → R given by h(u, v) = c+u+v defines a positive valued function that belongs to CND(Y × Z, σ d , τ d ′ ), whenever c is a positive constant. In addition, h(u, v) > c = h(0, 0), whenever u + v > 0. On the other hand, g : [0, ∞) → R given by g(t) = t s , t ≥ 0, belongs to CND(X, ρ), as long as s ∈ (0, 2]. Hence, c + g is a positive valued function that belongs to CND(X, ρ) for which g(t) > c = g(0) for t > 0. With this in mind, it is now clear that under the setting of either Theorem 4.3 or Theorem 4.4, the model 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 interested reader can implement quite more complicated examples along the same lines by using the characterization of functions in CND(S d , σ d ) obtained in [20] and the many concrete examples of functions in CND(R, ρ) listed in [17].
defines a function G r in SP D(X, Y, Z, ρ, σ, τ ), whenever f comes from S λ . Two important particular 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 2 (t 2 ), t ≥ 0 and h(v) = f 3 (v 2 ), v ≥ 0, in which f 2 and f 3 are positive valued Bernstein functions. Under the setting of either Theorem 4.3 or Theorem 4.4, we may infer that G r (t, u, v) = 1 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 2 (t) = t, t ≥ 0, and using Oppenheim's inequality, we obtain a model compatible with the one analyzed in Proposition 1 in [2].
-Let X = R n be endowed with its usual distance and Z = S d ′ endowed with its geodesic distance τ d ′ . Here, under the setting of either Theorem 4.3 or Theorem 4.4, we may infer that G r given by belongs to SP D(X, Y, Z, ρ, σ, τ d ′ ). This model is compatible with the model described in Theorem 3.2 in [1]. We observe that neither [1] nor [2] discussed the strict positive definiteness of the models they analyzed.