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 def...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Цитувати: | A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces. Victor S. Barbosa and Valdir A. Menegatto. SIGMA 16 (2020), 117, 15 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859686736262594560 |
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| author | Barbosa, Victor S. Menegatto, Valdir A. |
| author_facet | Barbosa, Victor S. Menegatto, Valdir A. |
| citation_txt | A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces. Victor S. Barbosa and Valdir A. Menegatto. SIGMA 16 (2020), 117, 15 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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 their many extensions. Necessary and sufficient conditions for the strict positive definiteness of the models are provided when the spaces are metric.
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| first_indexed | 2026-03-15T00:44:36Z |
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| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 117, 15 pages
A Gneiting-Like Method for Constructing Positive
Definite Functions on Metric Spaces
Victor S. BARBOSA † and Valdir A. MENEGATTO ‡
† Centro Tecnológico de Joinville-UFSC,
Rua Dona Francisca, 8300. Bloco U, 89219-600 Joinville SC, Brazil
E-mail: victorrsb@gmail.com
‡ Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo,
Caixa Postal 668, 13560-970, São Carlos - SP, Brazil
E-mail: menegatt@gmail.com
Received June 23, 2020, in final form November 07, 2020; Published online November 19, 2020
https://doi.org/10.3842/SIGMA.2020.117
Abstract. 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. Necessary and
sufficient conditions for the strict positive definiteness of the models are provided when the
spaces are metric.
Key words: positive definite functions; generalized Stieltjes functions; Bernstein functions;
Gneiting’s model; products of metric spaces
2020 Mathematics Subject Classification: 42A82; 43A35
1 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 on a metric space. Write Dρ
X to indicate the diameter-set of (X, ρ),
i.e.,
Dρ
X = {ρ(x, x′) : x, x′ ∈ X}.
This paper is mainly concerned with radial positive definite functions on (X, ρ), that is, conti-
nuous functions f : Dρ
X → R satisfying
n∑
j,k=1
cjckf(ρ(xj , xk)) ≥ 0, (1.1)
for n ≥ 1, reals scalars c1, . . . , cn, and points x1, . . . , xn in X. Functions of this type play
an important role in classical analysis, approximation theory, probability theory, and statistics.
Reference [29] covers what we will need in this paper about radial positive definite functions.
The strict positive definiteness of a radial positive definite function f as above demands that
the inequalities be strict when the xj are distinct and the cj are not all zero. We will write
f ∈ PD(X, ρ) and f ∈ SPD(X, ρ) to indicate that f is positive definite and strictly positive
definite on (X, ρ), respectively.
The two concepts just introduced extend to a product of finitely many quasi-metric spaces.
However, we will formalize the extension only in the setting to be covered in this paper. Unless
victorrsb@gmail.com
menegatt@gmail.com
https://doi.org/10.3842/SIGMA.2020.117
2 V.S. Barbosa and V.A. Menegatto
stated otherwise, throughout the paper, (X, ρ), (Y, σ) and (Z, τ) will denote three quasi-metric
spaces while X×Y ×Z will denote their cartesian product. Here, we will not distinguish among
the spaces X × Y × Z, X × (Y × Z) and (X × Y ) × Z and will not detach any special quasi-
distance in them. A continuous function f : Dρ
X ×Dσ
Y ×Dτ
Z → R is said to be positive definite
on X × Y ×Z (the term radial will be abandoned), and we write f ∈ PD(X × Y ×Z, ρ, σ, τ), if
n∑
j,k=1
cjckf(ρ(xj , xk), σ(yj , yk), τ(zj , zk)) ≥ 0,
for n ≥ 1, reals scalars c1, . . . , cn, and points (x1, y1, z1), . . . , (xn, yn, zn) in X × Y ×Z. A func-
tion f in PD(X×Y ×Z, ρ, σ, τ) is strictly positive definite if the inequalities above are strict when
the (xj , yj , zj) are distinct and the cj are not all zero. Here we write SPD(X × Y × Z, ρ, σ, τ).
Two problems involving the concepts of positive definiteness and strict positive definiteness
are very common in the literature: to characterize PD(X, ρ), SPD(X, ρ), PD(X × Y, ρ, σ), etc,
for fixed choice of the spaces and to determine, explicitly, large families of functions belonging
to them that have some importance in applications.
I.J. Schoenberg characterized in [25] the class PD(Rn, ρ) where ρ is the usual Euclidean
distance. His result states that a continuous function f : [0,∞) → R belongs to PD(Rn, ρ) if
and only if
f(t) =
∫
[0,∞)
Ωn(wt) dµ(w), t ≥ 0,
where µ is a finite and positive measure on [0,∞) while Ωn(x) = Ωn(ρ(x, 0)) is the mean value
of y ∈ Sn−1 7→ eix·y over Sn−1. Here, · denotes the usual inner product in Rn, Sn−1 is the
unit sphere in Rn, if n ≥ 2, while S0 = {−1, 1}. He also characterized the class PD(H, ρ)
where H is an infinite-dimensional Hilbert space and ρ is the distance defined by its norm:
a continuous function f : [0,∞) → R belongs to PD(H, ρ) if and only if t ∈ (0,∞) 7→ f
(
t1/2
)
is completely monotone. Recall that a function f : (0,∞) → R is completely monotone if it
has derivatives of all orders and (−1)nf (n)(t) ≥ 0, for t > 0 and n = 0, 1, . . .. Theorem 7.14
in [30] provides additional information regarding the class PD(H, ρ). In order to obtain the
classes SPD(Rn, ρ), n ≥ 2, and SPD(H, ρ), one needs to eliminate the constant functions from
PD(Rn, ρ) and PD(H, ρ), respectively. Characterizations for some of the classes PD(Lp(A,µ), ρ),
where (A,µ) is a measure space and ρ is given through the p-norm of Lp(A,µ) are presented
in [29, Chapter 2].
Schoenberg also provided characterizations for the classes PD
(
Sd, ρ
)
, d ≥ 1, where ρ is now
the geodesic distance on Sd. His result also included a characterization for the class PD(S∞, ρ),
where S∞ is the unit sphere in the real Hilbert space `2 while ρ is its geodesic distance [26].
R. Gangolli [9] extended Schoenberg results to PD(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 PD(H, ρ), if and only if f has a series representation in the form
f(t) =
∞∑
k=0
aHk P
H
k (cos t), t ∈ [0, π],
where aHk ≥ 0 for all k and
∑∞
k=0 a
H
k P
H
k (1) <∞. Here, PHk is the monomial xk if H = S∞ and
a Jacobi polynomial of degree k that depends on the space H being used, otherwise. The classes
SPD
(
Sd, ρ
)
, SPD(S∞, ρ), and SPD(H, ρ) were described in [3, 8] through additional conditions
on the sets
{
k : aHk > 0
}
.
The same was done for the classes PD(X × Y, ρ, σ) and SPD(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
A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces 3
with their respective Riemannian distances ρ and σ, the characterization for PD(X × Y, ρ, σ)
appeared in [4, 13]: a continuous function f : [0, π]2 → R belongs to SPD(X × Y, ρ, σ) if and
only if f has a series representation in the form
f(t, u) =
∞∑
k,l=0
aX,Yk,l P
X
k (cos t)P Yl (cosu), t, u ∈ [0, π],
with aX,Yk,l ≥ 0 for all k and l and the series being convergent at (t, u) = (0, 0). As for SPD(X ×
Y, ρ, σ), a description can be found in [4, 11, 14, 15] and depends on additional assumptions on
the sets
{
k − l : aX,Yk,l > 0
}
. The cases in which (X, ρ) is the usual metric space Rn and Y is
either a compact two-point homogeneous space or S∞ were considered recently: PD(X×Y, ρ, σ)
was described in [6, 7, 12, 27] while a description for SPD(X×Y, ρ, σ) can be inferred from [12].
As for the explicit determination of large families in either PD(X, ρ) or SPD(X, ρ), the
most efficient techniques make use of completely monotone functions and conditionally negative
definite functions on (X, ρ). A continuous function f : Dσ
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 cj satisfy
∑n
j=1 cj = 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 PD(X, ρ). Further, f ◦ g
belongs to SPD(X, ρ) if and only if f is nonconstant and g(t) > g(0), for t ∈ Dρ
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 PD(X × Y, ρ, σ). Further, f ◦ g belongs to SPD(X × Y, ρ, σ) if and only if f is nonconstant
and g(t, u) > g(0, 0), for (t, u) ∈ Dρ
X × Dσ
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 Fr given by
Fr(t, u) =
1
h(u)r
f
(
g(t)
h(u)
)
, (t, u) ∈ Dρ
X ×D
σ
Y , (1.2)
belongs to PD(X × Y, ρ, σ), as long as f is a bounded generalized Stieltjes function of order
λ > 0 [31] and r ≥ λ. Further, in the case in which (X, ρ) and (Y, σ) are metric spaces
and X has at least two points, Fr belongs to SPD(X × Y, ρ, σ) if and only if f is nonconstant,
g(t) > g(0) for t ∈ Dρ
X \ {0}, and h(u) > h(0) for u ∈ Dσ
Y \ {0}. 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 PD(X×Y ×Z, ρ, σ, τ) and SPD(X×Y ×
Z, ρ, σ, τ) that can be generalized to finitely many quasi-metric spaces. From a practical point
of view, we envisage 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 resemble some of the models discussed in [1, 2] involving positive definiteness for the product
of three metric spaces but focusing on the strict positive definiteness of the models. 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. They will be used in the subsequent material and
are not frequently dealt with in the literature, except in some very particular cases. We will
4 V.S. Barbosa and V.A. Menegatto
provide two simple techniques to construct functions in CND(X×Y, ρ, σ) and a third one specific
for case in which X is the usual metric space Rn. In Section 3, we begin describing the main
contributions of the paper. We propose a model to construct strictly positive definite functions
in a product of three metric spaces 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 metric spaces based on generalized Stieltjes functions
of order λ > 0. Section 5 contains adaptations of the results proved in Section 4 in order to
produce models based on generalized complete Bernstein functions of order λ > 0. In Section 6,
we address two examples that can serve as applications of the main results proved in the paper.
2 Functions in the class CND(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 Rn. 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−1f (n)(t) ≥ 0, for t > 0 and n = 1, 2, . . .. A Bernstein
function f has an integral representation in the form
f(w) = a+ bw +
∫
(0,∞)
(1− e−sw) dµ(s), w ≥ 0,
where a, b ≥ 0 and µ is a positive measure on (0,∞) satisfying∫
(0,∞)
(1 ∧ s) dµ(s) <∞.
A Bernstein function f can be continuously extended to 0 by setting f(0) = limw→0+ f(w).
It follows from [5, Proposition 2.9] 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
φ(t, u) = f(g(t) + h(u)), (t, u) ∈ Dρ
X ×D
σ
Y ,
belongs to CND(X × Y, ρ, σ).
Proof. Assume g and h are as in the statement of the lemma. Let n be a positive integer,
c1, . . . , cn real numbers satisfying
∑n
j=1 cj = 0, and (x1, y1), . . . , (xn, yn) points in X×Y . Direct
calculation shows that
n∑
j,k=1
cjckf(g(ρ(xj , xk)) + h(σ(yj , yk))) = b
n∑
j,k=1
cjck [g(ρ(xj , xk)) + h(σ(yj , yk))]
−
∫
[0,∞)
n∑
j,k=1
cjcke
−sg(ρ(xj ,xk))−sh(σ(yj ,yk)) dµ(s).
Since the function x ∈ (0,∞) 7→ e−x is bounded and completely monotone and the matrix
[−sg(ρ(xj , xk)) − sh(σ(yj , yk))]
n
j,k=1 is almost positive semi-definite, then Lemma 2.5 in [23]
implies that
n∑
j,k=1
cjcke
−sg(ρ(xj ,xk))−sh(σ(yj ,yk)) ≥ 0.
A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces 5
Thus,
n∑
j,k=1
cjckf(g(ρ(xj , xk)) + h(σ(yj , yk))) ≤ 0,
and the proof is complete. �
Here are some examples of functions in CND(X × Y, ρ, σ) provided by Theorem 2.1 with g
and h as there
φ(t, u) = f(t) + g(u), φ(t, u) = f(t) + g(u) +
√
1 + 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 to present is based on positive-valued Bernstein functions and
holds when one of the spaces is the usual Rn.
Theorem 2.2. Assume Rn is endowed with its usual Euclidean distance ρ. If (Y, σ) is a quasi-
metric space, f is a positive-valued Bernstein function and h is a positive-valued function in
CND(Y, σ), then
(t, u) ∈ [0,∞)×Dσ
Y 7→ −
1
h(u)n/2
e−f(t
2/h(u))
belongs to CND(Rn × Y, ρ, σ). Further, the formula
(t, u) ∈ [0,∞)×Dσ
Y 7→
1
h(0)n/2
− 1
h(u)n/2
e−f(t
2/h(u)), w > 0,
defines a bounded positive-valued function in CND(Rn × Y, ρ, σ).
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. So, if f is a Bernstein function, then
the Bernstein–Widder theorem [32, p. 161] leads to the representation
e−wf(t
2/h(u)) =
∫
[0,∞)
e−st
2/h(u) dµwf (s), (t, u) ∈ [0,∞)×Dσ
Y , w > 0,
for some finite and positive measure µwf on [0,∞). Since Theorem 3.2(i) in [22] shows that the
functions
(t, u) ∈ [0,∞)×Dσ
Y 7→
1
h(u)n/2
e−st
2/h(u), s > 0,
belong to PD(Rn × Y, ρ, σ), we may infer that so do
(t, u) ∈ [0,∞)×Dσ
Y 7→
1
h(u)n/2
e−wf(t
2/h(u)), w > 0.
The theorem follows after we take w = 1. �
Under the setting in Theorem 2.2, the formula
(t, u) ∈ [0,∞)×Dσ
Y 7→
1
h(0)n/2
− 1
h(u)n/2
e−wf(t
2/h(u)), w > 0,
defines bounded functions in CND(Rn × Y, ρ, σ).
6 V.S. Barbosa and V.A. Menegatto
Finally, 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 Sλ, if it can be represented in the form
f(w) = Cf +
Df
wλ
+
∫
(0,∞)
1
(w + s)λ
dµf (s), w > 0, (2.1)
where Cf = limw→∞ f(w), Df ≥ 0, and µf is a positive measure on (0,∞) such that∫
(0,∞)
1
(1 + s)λ
dµf (s) <∞.
It is not hard to see that a generalized Stieltjes function f of order λ is bounded if and only if
Df = 0 and
∫
(0,∞)
1
sλ
dµf (s) <∞.
The set of all bounded functions from Sλ will be written as Sbλ. Examples and additional
properties of functions in both Sλ and Sbλ can be found in [18, 19, 21, 24, 28] and references
quoted in there. It is known that every function in Sλ is completely monotone.
Theorem 2.3. Let f be a function in Sbλ, g a nonnegative valued function in CND(X, ρ), and
h a function in CND(Y, σ). If the function Fr in (1.2) is bounded from above by M > 0, then
M − Fr belongs to CND(X × Y, ρ, σ).
Proof. This follows from Theorem 2.4(i) in [21] where it is proved that Fr belongs to
PD(X × Y, ρ, σ). �
3 Products in PD(X × Y × Z, ρ, σ, τ )
In this section, we will present models that may belong to either PD(X × Y × Z, ρ, σ, τ)
or SPD(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 Sec-
tions 4 and 5, presupposes the existence of conditionally negative definite functions on a product
of quasi-metric spaces, reason why Section 2 was included here.
The Schur product theorem [16, p. 479] implies that if f1 and f2 are completely monotone
functions and g and h are positive-valued functions in CND(X, ρ) and CND(Y × Z, σ, τ), re-
spectively, then the function F given by
F (t, u, v) = f1(g(t))f2(h(u, v)), (t, u, v) ∈ Dρ
X ×D
σ
Y ×Dτ
Z , (3.1)
belongs to PD(X × Y ×Z, ρ, σ, τ). And, if f1 and f2 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.
Theorem 3.1. Assume (X, ρ), (Y, σ) and (Z, τ) are metric spaces. Let f1 and f2 be noncon-
stant completely monotone functions and g and h positive-valued functions in CND(X, ρ) and
CND(Y ×Z, σ, τ), respectively. The following assertions concerning the function F given by (3.1)
are equivalent:
(i) F belongs to SPD(X × Y × Z, ρ, σ, τ).
(ii) g(t) > g(0), for t ∈ Dρ
X \ {0}, and h(u, v) > h(0, 0), for (u, v) ∈ Dσ
Y ×Dτ
Z , with u+ v > 0.
A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces 7
Proof. If g(t) = g(0) for some t ∈ Dρ
X \ {0}, we can pick two distinct points (x1, y1, z1) and
(x2, y2, z2) in X × Y ×Z with ρ(x1, x2) = t, y1 = y2, and z1 = z2 in order to obtain the singular
matrix
[F (ρ(xj , xk), σ(yj , yk), τ(zj , zk))]
2
j,k=1 = [f1(g(0))f2(h(0, 0))]2j,k=1 .
If h(u, v) = h(0, 0) for some (u, v) ∈ Dσ
Y ×Dτ
Z with u + v > 0, we can pick two distinct points
(x1, y1, z1) and (x2, y2, z2) in X×Y ×Z with x1 = x2, σ(y1, y2) = u, and τ(z1, z2) = v in order to
obtain the very same singular matrix. In either case, F cannot belong to SPD(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
f1(g(t))f2(h(u, v)) =
∫
[0,∞)
[∫
[0,∞)
e−g(t)s−h(u,v)s
′
dµ1(s)
]
dµ2(s
′),
where µ1 and µ2 are (not necessarily finite) positive measures on [0,∞). Recalling the proof
of Theorem 2.1, we know already that the functions
(t, u, v) ∈ Dρ
X ×D
σ
Y ×Dτ
Z 7→ e−g(t)s−h(u,v)s
′
, s, s′ > 0,
belong to PD(X × Y × Z, ρ, σ, τ). Hence, so do the functions
(t, u, v) ∈ Dρ
X ×D
σ
YD
τ
Z 7→
∫
[0,∞)
e−g(t)se−h(u,v)s
′
dµ1(s), s′ > 0. (3.2)
If f2 is nonconstant, F will belong to SPD(X × Y ×Z, ρ, σ, τ) if we can show that the functions
in (3.2) belong to SPD(X × Y × Z, ρ, σ, τ). However, if f1 is nonconstant, it is promptly seen
that F will belong to SPD(X × Y × Z, ρ, σ, τ) as long as can show that the functions
(t, u, v) ∈ Dρ
X ×D
σ
Y ×Dτ
Z 7→ e−g(t)s−h(u,v)s
′
, s, s′ > 0,
belong to SPD(X × Y × Z, ρ, σ). So, in order to complete the proof, we will show that, under
the assumptions in (ii), the matrices[
e−g(ρ(xj ,xk))s−h(σ(yj ,yk),τ(zj ,zk))s
′]n
j,k=1
are positive definite whenever s, s′ > 0 and (x1, y1, z1), . . . , (xn, yn, zn) 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
g(0)s+ h(0)s′ < g(ρ(xj , xk))s+ h(σ(yj , yk), τ(zj , zk))s
′, j 6= k. (3.3)
If xj 6= xk, then ρ(xj , xk) > 0 and the assumption on g implies that g(ρ(xj , xk)) > g(0).
If yj 6= yk, then σ(yj , yk) > 0 and the assumption on h implies that h(σ(yj , yk), τ(zj , zk)) >
h(0, 0). The same can be inferred if zj 6= zk. Thus, in any case, (3.3) holds. �
The model given by (3.1) has a considerable drawback: the variables u and v 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.
8 V.S. Barbosa and V.A. Menegatto
4 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 this, 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 be a function in Sλ, g a positive-valued function in CND(X, ρ) and h a posi-
tive-valued function in CND(Y × Z, σ, τ). For r ≥ λ, set
Gr(t, u, v) =
1
h(u, v)r
f
(
g(t)
h(u, v)
)
, (t, u, v) ∈ Dρ
X ×D
σ
Y ×Dτ
Z . (4.1)
The following assertions hold:
(i) Gr belongs to PD(X × Y × Z, ρ, σ, τ).
(ii) If Gr belongs to SPD(X×Y ×Z, ρ, σ, τ), then g(t) > g(0), for t ∈ Dρ
X \{0}, and h(u, v) >
h(0, 0), for (u,w) ∈ Dσ
Y ×Dτ
Z with u+ v > 0.
Proof. Inserting the integral representation (2.1) for f in (4.1) leads to the formula
Gr(t, u, v) =
Cf
h(u, v)r
+
Df
g(t)λh(u, v)r−λ
+
1
h(u, v)r−λ
∫
(0,∞)
1
[g(t) + sh(u, v)]λ
dµf (s).
In order to prove (i), it suffices to show that each of the three summands above belongs
to PD(X × Y × Z, ρ, σ, τ). Once the functions
w ∈ (0,∞) 7→ 1
wα
, α = λ, r, r − λ,
are known to be completely monotone, some of the basic results quoted at the introduction
of the paper show that t ∈ Dρ
X 7→ g(t)−λ belongs to PD(X, ρ), while (u,w) ∈ Dσ
Y × Dτ
Z 7→
h(u,w)α, α = r, r− λ, belongs to PD(Y ×Z, σ, τ). Hence, it is easily seen that all the functions
(t, u, v) ∈ Dρ
X × Dσ
Y × Dτ
Z 7→ g(t)−λ and (t, u, v) ∈ Dρ
X × Dσ
Y × Dτ
Z 7→ h(u, v)α, α = r, r − λ,
belong to PD(X ×Y ×Z, ρ, σ, τ). The fact that PD(X ×Y ×Z, ρ, σ, τ) is closed under products
is all that is needed in order to see that (t, u, v) ∈ Dρ
X × Dσ
Y × Dτ
Z 7→ g(t)−λh(u, v)r−λ also
belongs to PD(X × Y × Z, ρ, σ, τ). It remains to show that the third summand belongs to
PD(X × Y × Z, ρ, σ, τ). Since w ∈ (0,∞) 7→ e−w is completely monotone, the same reasoning
reveals that (t, u, v) ∈ Dρ
X×Dσ
Y ×Dτ
Z 7→ exp(−wg(t)−wh(u, v)) belongs to PD(X×Y ×Z, ρ, σ, τ)
for w > 0. 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−swe−twwλ−1 dw, s, t > 0, (4.2)
now imply that all the functions
(t, u, v) ∈ Dρ
X ×D
σ
Y ×Dτ
Z 7→
1
[g(t) + sh(u, v)]λ
, s > 0,
belong to PD(X × Y × Z, ρ, σ, τ). But, since PD(X × Y × Z, ρ, σ, τ) is closed under products,
we see that the remaining third summand
(t, u, v) ∈ Dρ
X ×D
σ
Y ×Dτ
Z 7→
1
h(u, v)r−λ
∫
(0,∞)
1
[g(t) + sh(u, v)]λ
dµf (s), s > 0,
A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces 9
also belongs to PD(X × Y × Z, ρ, σ, τ), completing the proof of (i). If g(t) = g(0), for some
t ∈ Dρ
X \ {0}, by picking two distinct points (x1, y1, z1) and (x2, y2, z2) in X × Y × Z such that
ρ(x1, x2) = t, y1 = y2, and z1 = z2, we obtain the singular matrix
[Gr(ρ(xj , xk), σ(yj , yk), τ(zj , zk))]
2
j,k=1 =
[
1
h(0, 0)r
f
(
g(0)
h(0, 0)
)]2
j,k=1
.
If h(u, v) = h(0, 0), for (u, v) ∈ Dσ
Y × Dτ
Z with u + v > 0, we can take two distinct points
(x1, y1, z1) and (x2, y2, z2) in X × Y × Z such that x1 = x2, σ(y1, y2) = u, and τ(z1, z2) = v
in order to obtain the very same singular matrix. In either case, we may infer that Gr cannot
belong to SPD(X × Y × Z, ρ, σ, τ). In any case, Gr cannot belong to SPD(X × Y × Z, ρ, σ, τ)
and (ii) follows. �
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 definiteness of the
model in Theorem 4.1 in some specific cases.
Theorem 4.2. Let f be a function in Sλ, g a positive-valued function in CND(X, ρ) and
h a positive-valued function in CND(Y × Z, σ, τ). The following assertion holds for the func-
tion Gr in (4.1):
(i) If (X, ρ) is nontrivial and Gr belongs to SPD(X×Y ×Z, ρ, σ, τ), then either Df > 0 or µf
is not the zero measure.
Further, in the case in which r = λ and Df > 0, the following additional conclusion holds:
(ii) If either (Y, σ) or (Z, τ) is nontrivial and Gλ belongs to SPD(X × Y × Z, ρ, σ, τ), then
either Cf > 0 or µf is not the zero measure.
Proof. If (X, ρ) is nontrivial, Df = 0 and µf is the zero measure, then we can take two distinct
points (x1, y1, z1) and (x2, y2, z2) in X × Y × Z with y1 = y2 and z1 = z2 in order to obtain the
singular matrix
[Gr(ρ(xj , xk), σ(yj , yk), τ(zi, zj))]
2
j,k=1 =
[
Cf
h(0, 0)r
]2
j,k=1
.
Similarly, if either (Y, σ) or (Z, τ) is nontrivial, r = λ, Cf = 0 < Df and µf is the zero measure,
then we can take two distinct points (x1, y1, z1) and (x2, y2, z2) in X × Y × Z with x1 = x2
in order to obtain the singular matrix
[Gλ(ρ(xj , xk), σ(yj , yk), τ(zj , zk))]
2
j,k=1 =
[
Df
g(0)λ
]2
j,k=1
,
In either case, Gr cannot belong to SPD(X × Y × Z, ρ, σ, τ). �
Theorem 4.3 achieves a necessary and sufficient condition for the strict positive definiteness
of Gr in the case in which r > λ and Df > 0 in the representation for f .
Theorem 4.3. Assume (X, ρ), (Y, σ) and (Z, τ) are metric spaces. Let f be a function in Sλ,
g a positive-valued function in CND(X, ρ) and h a positive-valued function in CND(Y ×Z, σ, τ).
If Df > 0 and r > λ, then the following assertions for Gr in (4.1) are equivalent:
(i) Gr belongs to SPD(X × Y × Z, ρ, σ, τ).
(ii) g(t) > g(0), for t ∈ Dρ
X \ {0}, and h(u, v) > h(0, 0), for (u, v) ∈ Dσ
Y ×Dτ
Z with u+ v > 0.
10 V.S. Barbosa and V.A. Menegatto
Proof. In view of Theorem 4.1(ii), only the implication (ii)⇒ (i) needs to be proved. Assume
Df > 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
(t, u, v) ∈ Dρ
X ×D
σ
Y ×Dτ
Z 7→
Df
g(t)λh(u, v)r−λ
belongs to SPD(X × Y × Z, ρ, σ, τ). As the other two summands appearing in the equation
defining Gr(t, u, v) belong to PD(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 Df = 0, and r ≥ λ.
Theorem 4.4. Assume (X, ρ), (Y, σ) and (Z, τ) are metric spaces. Let f be a function in Sλ,
g a positive-valued function in CND(X, ρ) and h a positive-valued function in CND(Y ×Z, σ, τ).
If (X, ρ) is nontrivial, Df = 0, and r ≥ λ, then the following assertions for Gr as (4.1) are
equivalent:
(i) Gr belongs to SPD(X × Y × Z, ρ, σ, τ).
(ii) f is nonconstant, g(t) > g(0), for t ∈ Dρ
X\{0}, and h(u,w) > h(0, 0), for (u,w) ∈ Dσ
Y ×Dτ
Z
with u+ w > 0.
Proof. If Gr belongs to SPD(X × Y × Z, ρ, σ, τ), Theorem 4.2(i) shows that µf is nonzero.
In particular, f is nonconstant. On the other hand, Theorem 4.1(ii) reveals that the other
two conditions in (ii) also hold. Thus, (i) implies (ii). Conversely, if f is nonconstant, the
assumption Df = 0 implies that the measure µf is nonzero. That being said, (i) will follow if
we can prove that
(t, u, v) ∈ Dρ
X ×D
σ
Y ×Dτ
Z 7→
∫
(0,∞)
1
[g(t) + sh(u, v)]λ
dµf (s)
belongs to SPD(X × Y × Z, ρ, σ, τ) under the other two assumptions in (ii). Indeed, since
(t, u, v) ∈ Dρ
X ×Dσ
Y ×Dτ
Z 7→ h(u, v)λ−r belongs to PD(X × Y × Z, ρ, σ, τ) and h(0, 0) > 0, the
Oppenheim–Schur inequality [16, p. 509] will lead to (i). Since µf is nonzero, it suffices to show
that
(t, u, v) ∈ Dρ
X ×D
σ
Y ×Dτ
Z 7→
1
[g(t) + sh(u, v)]λ
belongs to SPD(X × Y × Z, ρ, σ, τ), for s > 0. By (4.2), what needs to be proved is that the
functions
(t, u, v) ∈ Dρ
X ×D
σ
Y ×Dτ
Z 7→ e−g(t)w−h(u,v)sw, w, s > 0,
belong to SPD(X ×Y ×Z, ρ, σ, τ). But that follows by the same argument 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.
Theorem 4.5. Assume (X, ρ), (Y, σ), and (Z, τ) are metric spaces. Let f be a function in Sλ,
g a positive-valued function in CND(X, ρ), and h a positive-valued function in CND(Y ×Z, σ, τ).
If Df > 0, r = λ, and µf is nonzero, then the following assertions for Gr as (4.1) are equivalent:
(i) Gλ belongs to SPD(X × Y × Z, ρ, σ, τ).
(ii) g(t) > g(0), for t ∈ Dρ
X \ {0}, and h(u, v) > h(0, 0), for (u, v) ∈ Dσ
Y ×Dτ
Z with u+w > 0.
A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces 11
It remains to consider the case in which Df > 0, r = λ and µf = 0. However, Theorem 4.2(ii)
shows that, essentially, what needs to be analyzed is the case where CfDf > 0, r = λ and µf = 0
and also imposing the non-triviality of some of the spaces involved. In this case Gr takes the
form
Gλ(t, u, v) =
Cf
h(u, v)λ
+
Df
g(t)λ
, (t, u, v) ∈ Dρ
X ×D
σ
Y ×Dτ
Z ,
with CfDf > 0. So far, the strict positive definiteness of Gr in this case remains an open
question.
Remark 4.6. All the theorems proved so far can be re-stated and demonstrated for the model
Hr(t, u, v) =
1
g(t)r
f
(
h(u, v)
g(t)
)
, (t, u, v) ∈ Dρ
X ×D
σ
Y ×Dτ
Z , f ∈ Sλ,
with r, g and h as before. The obvious adjustments and the details on that will be left to the
reader.
5 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 Bλ, here called the class of generalized complete Bernstein
functions of order λ > 0, that is, functions f having a representation in the form
f(w) = Af +Bfw
λ +
∫
(0,∞)
(
w
w + s
)λ
dνf (s), x > 0,
where Af , Bf ≥ 0 and νf is a positive measure on (0,∞) for which∫
(0,∞)
1
(1 + s)λ
dνf (s) <∞.
The class B1 is more common in the literature. Functions in it may receive different names
depending where they are used: operator monotone functions, Löwner (Loewner) functions,
Pick functions, Nevanlinna functions, etc. Many examples of functions in Bλ 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 Bλ.
Theorem 5.1. Let f be a function in Bλ, g a positive-valued function in CND(X, ρ), and
h a positive-valued function in CND(Y × Z, σ, τ). For r ≥ λ, set
Ir(t, u, v) =
1
g(t)r
f
(
g(t)
h(u, v)
)
, (t, u, v) ∈ Dρ
X ×D
σ
Y ×Dτ
Z . (5.1)
The following assertions hold:
(i) Ir belongs to PD(X × Y × Z, ρ, σ, τ).
(ii) If Ir belongs to SPD(X×Y, ρ, σ), then g(t) > g(0), for t ∈ Dρ
X \{0}, and h(u, v) > h(0, 0),
for (u, v) ∈ Dσ
Y ×Dτ
Z with u+ v > 0.
12 V.S. Barbosa and V.A. Menegatto
Proof. It suffices to use the formula
Ir(t, u, v) =
Af
g(t)r
+
Bf
h(u, v)λg(t)r−λ
+
1
g(t)r−λ
∫
(0,∞)
1
[g(t) + sh(u, v)]λ
dνf (s)
that derives from the integral representation for f and to mimic the proof of Theorem 4.1. �
Theorem 4.2 takes the following form.
Theorem 5.2. Let f be a function in Bλ, g a positive-valued function in CND(X, ρ), and
h a positive-valued function in CND(Y × Z, σ, τ). The following assertion holds for the func-
tion Ir in (5.1):
(i) If either (Y, σ) or (Z, τ) is nontrivial and Ir belongs to SPD(X×Y ×Z, ρ, σ, τ), then either
Bf > 0 or νf is not the zero measure.
In the case in which r = λ and Df > 0, the following additional assumption holds:
(ii) If (X, ρ) is nontrivial and Iλ belongs to SPD(X ×Y ×Z, ρ, σ, τ), then either Af > 0 or νf
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 be a function in Bλ,
g a positive-valued function in CND(X, ρ), and h a positive-valued function in CND(Y ×Z, σ, τ).
If Bf > 0 and r > λ, then the following assertions for Ir in (5.1) are equivalent:
(i) Ir belongs to SPD(X × Y × Z, ρ, σ, τ).
(ii) g(t) > g(0), for t ∈ Dρ
X \ {0}, and h(u, v) > h(0, 0), for (u, v) ∈ Dσ
Y ×Dτ
Z with u+ v > 0.
Theorem 5.4. Assume (X, ρ), (Y, σ), and (Z, τ) are metric spaces. Let f be a function in Bλ,
g a positive-valued function in CND(X, ρ), and h a positive-valued function in CND(Y ×Z, σ, τ).
If either (Y, σ) or (Z, τ) is nontrivial, Bf = 0, and r ≥ λ, then the following assertions for Ir
in (5.1) are equivalent:
(i) Ir belongs to SPD(X × Y × Z, ρ, σ, τ).
(ii) f is nonconstant, g(t) > g(0), for t ∈ Dρ
X\{0}, and h(u, v) > h(0, 0), for (u, v) ∈ Dσ
Y ×Dτ
Z ,
u+ v > 0.
Theorem 5.5. Assume (X, ρ), (Y, σ), and (Z, τ) are metric spaces. Let f be a function in Bλ,
g a positive-valued function in CND(X, ρ), and h a positive-valued function in CND(Y ×Z, σ, τ).
If Bf > 0, r = λ, and νf is nonzero, then the following assertions for Ir in (5.1) are equivalent:
(i) Iλ belongs to SPD(X × Y × Z, ρ, σ, τ).
(ii) g(t) > g(0), for t ∈ Dρ
X \ {0}, and h(u, v) > h(0, 0), for (u, v) ∈ Dσ
Y ×Dτ
Z with u+ v > 0.
Remark 5.6. All the theorems proved so far in this section can be re-stated and proved for the
model
Jr(t, u, v) =
1
h(u, v)r
f
(
h(u, v)
g(t)
)
, (t, u, v) ∈ Dρ
X ×D
σ
Y ×Dτ
Z , f ∈ Bλ,
with r, g and h as before and with some small adjustments. Once again, we leave the proofs to
the interested reader.
A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces 13
6 Two 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 reader.
Example 6.1. Let X be the unit sphere Sd in Rd+1 endowed with is usual geodesic distance ρd
and let Y = [0, π/2] and Z = Rn both endowed with their usual Euclidean distances σ and τ
respectively. The function g given by the formula
g(t) = 3− cos t, t ∈ [0, π],
belongs to CND
(
Sd, ρd
)
while results proved in [17] show that, if s ∈ (0, 2], then the function h
given by
h(u, v) = 1 + sinu+ vs, (u, v) ∈ [0, π/2]× [0,∞),
belongs to CND(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/2] × [0,∞) with u + v > 0. Under the setting of either
Theorem 4.3 or Theorem 4.4, the model
Gr(t, u, v) =
1
[1 + sinu+ vs]r
f
(
3− cos t
1 + sinu+ vs
)
, (t, u, v) ∈ [0, π]× [0, π/2]× [0,∞),
defines a function Gr in SPD(X,Y, Z, ρd, σ, τ), whenever f comes from Sλ. A similar conclusion
holds for the model
Hr(t, u, v) =
1
[3− cos t]r
f
(
1 + sinu+ vs
3− cos t
)
, (t, u, v) ∈ [0, π]× [0, π/2]× [0,∞),
under the setting in Remark 4.6. These examples can be expanded, by letting Z be a Hilbert
space and τ the distance induced by its norm, keeping all the rest the same. In 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 = Sd and Z = Sd
′
, both endowed with their geodesic distances σd and τd′ . Since
t ∈ [0, π] 7→ t belongs to both CND(Y, σd) and CND(Z, τd′), 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, σd, τd′),
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) = ts, 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
Gr(t, u, v) =
1
[c+ u+ v]r
f
(
c+ ts
c+ u+ v
)
, (t, u, v) ∈ [0,∞)× [0, π]× [0, π],
defines a function Gr in SPD(X,Y, Z, ρ, σd, τd′), as long as f comes from Sλ. The interested
reader can implement considerably more complicated examples along the same lines by using the
characterization of functions in CND
(
Sd, σd
)
obtained in [20] and the many concrete examples
of functions in CND(R, ρ) listed in [17].
The examples point that for the right choice of the quasi-metric spaces, the models discussed
in the paper may lead to flexible, interpretable and even computationally feasible classes of cross-
covariance functions for multivariate random fields adopted in statistics. Hopefully, that will be
confirmed in the near future.
14 V.S. Barbosa and V.A. Menegatto
Acknowledgements
The authors express their gratitude to the anonymous referees for their comments and remarks
which led to an improved version of the paper.
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1 Introduction
2 Functions in the class CND(X x Y,rho,sigma)
3 Products in PD(X x Y x Z, rho,sigma,tau)
4 Models based on generalized Stieltjes functions
5 Models based on generalized complete Bernstein functions
6 Two concrete realizations
References
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| id | nasplib_isofts_kiev_ua-123456789-211003 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T00:44:36Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Barbosa, Victor S. Menegatto, Valdir A. 2025-12-22T09:23:24Z 2020 A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces. Victor S. Barbosa and Valdir A. Menegatto. SIGMA 16 (2020), 117, 15 pages 1815-0659 2020 Mathematics Subject Classification: 42A82; 43A35 arXiv:2006.12217 https://nasplib.isofts.kiev.ua/handle/123456789/211003 https://doi.org/10.3842/SIGMA.2020.117 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 their many extensions. Necessary and sufficient conditions for the strict positive definiteness of the models are provided when the spaces are metric. The authors express their gratitude to the anonymous referees for their comments and remarks, which led to an improved version of the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces Article published earlier |
| spellingShingle | A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces Barbosa, Victor S. Menegatto, Valdir A. |
| title | A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces |
| title_full | A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces |
| title_fullStr | A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces |
| title_full_unstemmed | A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces |
| title_short | A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces |
| title_sort | gneiting-like method for constructing positive definite functions on metric spaces |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211003 |
| work_keys_str_mv | AT barbosavictors agneitinglikemethodforconstructingpositivedefinitefunctionsonmetricspaces AT menegattovaldira agneitinglikemethodforconstructingpositivedefinitefunctionsonmetricspaces AT barbosavictors gneitinglikemethodforconstructingpositivedefinitefunctionsonmetricspaces AT menegattovaldira gneitinglikemethodforconstructingpositivedefinitefunctionsonmetricspaces |