Strictly Positive Definite Kernels on a Product of Spheres II
We present, among other things, a necessary and sufficient condition for the strict positive definiteness of an isotropic and positive definite kernel on the cartesian product of a circle and a higher dimensional sphere. The result complements similar results previously obtained for strict positive...
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irk-123456789-1480042019-02-17T01:25:56Z Strictly Positive Definite Kernels on a Product of Spheres II Guella, J.C. Menegatto, V.A. Peron, A.P. We present, among other things, a necessary and sufficient condition for the strict positive definiteness of an isotropic and positive definite kernel on the cartesian product of a circle and a higher dimensional sphere. The result complements similar results previously obtained for strict positive definiteness on a product of circles [Positivity, to appear, arXiv:1505.01169] and on a product of high dimensional spheres [J. Math. Anal. Appl. 435 (2016), 286-301, arXiv:1505.03695]. 2016 Article Strictly Positive Definite Kernels on a Product of Spheres II / J.C. Guella, V.A. Menegatto, A.P. Peron // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 20 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33C50; 33C55; 42A16; 42A82; 42C10; 43A35 DOI:10.3842/SIGMA.2016.103 http://dspace.nbuv.gov.ua/handle/123456789/148004 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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We present, among other things, a necessary and sufficient condition for the strict positive definiteness of an isotropic and positive definite kernel on the cartesian product of a circle and a higher dimensional sphere. The result complements similar results previously obtained for strict positive definiteness on a product of circles [Positivity, to appear, arXiv:1505.01169] and on a product of high dimensional spheres [J. Math. Anal. Appl. 435 (2016), 286-301, arXiv:1505.03695]. |
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Guella, J.C. Menegatto, V.A. Peron, A.P. |
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Guella, J.C. Menegatto, V.A. Peron, A.P. Strictly Positive Definite Kernels on a Product of Spheres II Symmetry, Integrability and Geometry: Methods and Applications |
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Guella, J.C. Menegatto, V.A. Peron, A.P. |
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Guella, J.C. |
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Strictly Positive Definite Kernels on a Product of Spheres II |
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Strictly Positive Definite Kernels on a Product of Spheres II |
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Strictly Positive Definite Kernels on a Product of Spheres II |
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Strictly Positive Definite Kernels on a Product of Spheres II |
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Strictly Positive Definite Kernels on a Product of Spheres II |
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strictly positive definite kernels on a product of spheres ii |
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Інститут математики НАН України |
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Strictly Positive Definite Kernels on a Product of Spheres II / J.C. Guella, V.A. Menegatto, A.P. Peron // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 20 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT guellajc strictlypositivedefinitekernelsonaproductofspheresii AT menegattova strictlypositivedefinitekernelsonaproductofspheresii AT peronap strictlypositivedefinitekernelsonaproductofspheresii |
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2025-07-11T03:25:57Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 103, 15 pages
Strictly Positive Definite Kernels
on a Product of Spheres II
Jean C. GUELLA, Valdir A. MENEGATTO and Ana P. PERON
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo,
Caixa Postal 668, 13560-970, São Carlos - SP, Brazil
E-mail: jeanguella@gmail.com, menegatt@icmc.usp.br, apperon@icmc.usp.br
Received June 01, 2016, in final form October 24, 2016; Published online October 28, 2016
http://dx.doi.org/10.3842/SIGMA.2016.103
Abstract. We present, among other things, a necessary and sufficient condition for the
strict positive definiteness of an isotropic and positive definite kernel on the cartesian
product of a circle and a higher dimensional sphere. The result complements similar re-
sults previously obtained for strict positive definiteness on a product of circles [Positivity,
to appear, arXiv:1505.01169] and on a product of high dimensional spheres [J. Math. Anal.
Appl. 435 (2016), 286–301, arXiv:1505.03695].
Key words: positive definite kernels; strictly positive definiteness; isotropy; covariance func-
tions; sphere; circle
2010 Mathematics Subject Classification: 33C50; 33C55; 42A16; 42A82; 42C10; 43A35
1 Introduction
The theory of positive definite and strictly positive definite kernels on manifolds and groups
can not be separated from the seminal paper of I.J. Schoenberg [17] published in the first
half of the past century. The major contribution in that paper refers to kernels of the form
(x, y) ∈ Sm × Sm → f(x · y), in which · is the usual inner product of Rm+1 and the “generating
function” f is real and continuous in [−1, 1]. The kernel is positive definite if, and only if, the
generating function f has a series representation in the form
f(t) =
∞∑
k=0
amk P
m
k (t), t ∈ [−1, 1],
where all the coefficients amk are nonnegative, Pmk denotes the usual Gegenbauer polynomial of
degree k attached to the rational number (m− 1)/2 and
∑
k a
m
k P
m
k (1) <∞. The normalization
for the Gegenbauer polynomials used here is
Pmn (1) =
(
n+m− 2
n
)
, n = 0, 1, . . . .
Recall that the positive definiteness of a general real kernel (x, y) ∈ X2 → K(x, y) on a nonempty
set X, demands that K(x, y) = K(y, x), x, y ∈ X, and the inequality
n∑
µ,ν=1
cµcνK(xµ, xν) ≥ 0,
whenever n is a positive integer, x1, x2, . . . , xn are n distinct points on X and c1, c2, . . . , cn are
real scalars. For complex kernels we do not need the symmetry assumption and we have to use
mailto:jeanguella@gmail.com
menegatt@icmc.usp.br
apperon@icmc.usp.br
http://dx.doi.org/10.3842/SIGMA.2016.103
2 J.C. Guella, V.A. Menegatto and A.P. Peron
complex scalars instead. Isotropy on Sm refers to the fact that the variables x and y of Sm are
tied to each other via the inner product of Rm+1.
Some fifty years later, the very same positive definite functions were found useful for solving
scattered data interpolation problems on spheres. But that demanded strictly positive definite
functions, and thus a characterization of these functions was needed at the start. We recall that
the strict positive definiteness of a general positive definite kernel as in the above definition
requires that the previous inequalities be strict whenever at least one cµ is nonzero. In other
words, the interpolation matrices [K(xµ, xν)]nµ,ν=1 of K at each set {x1, x2, . . . , xn} need to
be positive definite. The strict positive definiteness on spheres was an issue for some time
until Schoenberg’s result was complemented by a result of Debao Chen et al. in 2003 [6] and
by Menegatto et al. [14]. A real, continuous, isotropic and positive definite kernel (x, y) ∈
Sm×Sm → f(x · y) is strictly positive definite if, and only if, the following additional condition
holds for the coefficients in Schoenberg’s series representation for the generating function f :
– (m = 1) the set {k ∈ Z : a1|k| > 0} intersects each full arithmetic progression in Z;
– (m ≥ 2) the set {k : amk > 0} contains infinitely many even and infinitely many odd
integers.
As a matter of fact, positive definite and strictly positive definite functions and kernels on
spheres play a fundamental role in several other applications. For instance, two recent papers
authored by Beatson and Zu Castell [3, 4] provide new families of strictly positive definite
functions on spheres via the so-called half-step operators, a spherical analogue of Matheron’s
montée and descente operators on Rm+1. Additional applications are mentioned in [5, 10].
The spheres belong to a much larger class of metric spaces, that is, they are compact two-point
homogeneous spaces. Positive definiteness on these spaces in the same sense we explained above
was investigated by Gangolli [8] while strict positive definiteness was completely characterized
in [1].
In 2011, the paper [7] considered strictly positive definite functions on compact abelian
groups taking into account a paper on strict positive definiteness on S1 previously written by
X. Sun [19]. Among other things, the paper included abstract characterizations for strict positive
definiteness on a torsion group and on a product of a finite group and a torus.
In the past two years, the attention shifted all the way to positive definiteness on a product
of spaces, the main motivation coming from problems involving random fields on spaces across
time. The first important reference we would like to mention along this line is [5], where the
authors investigated positive definite kernels on a product of the form G×Sm, in which G is an
arbitrary locally compact group, keeping both the group structure of G and the isotropy of Sm
in the setting. Let us denote by e the neutral element of G, ∗ the operation of the group G
and by u−1 the inverse of u ∈ G with respect to ∗. The main contribution in [5] states that
a continuous kernel of the form ((u, x), (v, y)) ∈ (G×Sm)2 → f(u−1 ∗ v, x · y) is positive definite
if, and only if, the function f has a representation in the form
f(u, t) =
∞∑
n=0
fmn (u)Pmn (t), (u, t) ∈ G× [−1, 1],
in which {fmn } is a sequence of continuous functions on G for which
∑
fmn (e)Pmn (1) <∞, with
uniform convergence of the series for (u, t) ∈ G× [−1, 1]. As a matter of fact, the functions fmn
are positive definite on G in the sense that the kernel (u, v) ∈ G2 → fmn (u−1 ∗ v) is positive
definite as previously defined. The paper [5] did not considered any strict positive definiteness
issues. It is worth to mention that [5] may be linked to [9] where the reader can find a possible
reason for considering positive definiteness on a product of a locally compact abelian group with
a classical space.
Strictly Positive Definite Kernels on a Product of Spheres II 3
Simultaneously, positive definiteness and strict positive definiteness on a product of spheres
was investigated in [11, 12, 13]. A characterization for the isotropic and positive definite kernels
on Sm×SM was deduced in [12] and that agreed with the characterization mentioned in [5] and
also with [18, Chapter 4]. By the way, a real, continuous and isotropic kernel ((x, z), (y, w)) ∈
(Sm×SM )2 → f(x · y, z ·w) is positive definite if, and only if, the function f has a double series
representation in the form
f(t, s) =
∞∑
k,l=0
am,Mk,l Pmk (t)PMl (s), t, s ∈ [−1, 1], (1.1)
in which am,Mk,l ≥ 0, k, l ∈ Z+ and
∞∑
k,l=0
am,Mk,l Pmk (1)PMl (1) < ∞. As before, we will call f the
generating function of the kernel.
One of the main theorems in [11] reveals that, in the case in which m,M ≥ 2, a positive
definite kernel as above is strictly positive definite if, and only if, the set {(k, l) : am,Mk,l > 0}
contains sequences from each one of the sets 2Z+ × 2Z+, 2Z+ × (2Z+ + 1), (2Z+ + 1) × 2Z+,
and (2Z+ + 1) × (2Z+ + 1), all of them having both component sequences unbounded. The
very same paper contains some other intriguing results related to strict positive definiteness,
including a notion of strict positive definiteness that holds in product spaces only. In the case
m = M = 1, the condition becomes this one [13]: the set {(k, l) : a1,1|k|,|l| > 0} intersects all the
translations of each subgroup of Z2 having the form {(pa, qb) : q, p ∈ Z}, a, b > 0. Even with
the completion of these papers, it became clear very soon that a similar characterization for the
remaining case, that is, strict positive definiteness on S1 × Sm, m ≥ 2, would demand much
more work, perhaps a mix of the techniques used in [11, 13].
This is the point where we explain what the contributions in this paper are. In the next
section, we present two abstract results that describe how to obtain continuous, isotropic and
strictly positive kernels on S1 × Sm via the characterization for strict positive definiteness of
continuous, isotropic and positive definite kernels on either S1 or Sm separately. In Section 3,
we present necessary and sufficient conditions in order that a real, continuous, isotropic and
positive definite kernel on S1 × Sm, m ≥ 2, be strictly positive definite, thus filling in the
missing gap in the previous papers on the subject. In Section 4, we indicate how to obtain
a similar characterization after we replace the sphere Sm with an arbitrary compact two-point
homogeneous space.
2 Abstract sufficient conditions
In this section, we describe two abstract sufficient conditions for strict positive definiteness on
S1×Sm, both derived from strict positive definiteness on single spheres. The results show that
transferring strict positive definiteness from the factors S1 and Sm to strict positive definiteness
of the product S1 × Sm is not so obvious as it seems.
Here and in the next sections, we will assume all the generating functions of the kernels are
real-valued continuous functions and that the dimension m in Sm is at least 2. But the reader
is advised that the results hold true for complex kernels after some obvious modifications.
The results to be presented here will depend upon a technical lemma that provides an al-
ternative formulation for the strict positive definiteness of a continuous, isotropic and positive
definite kernel on S1×Sm, to be described below. Let (x1, w1), (x2, w2), . . . , (xn, wn) be distinct
points on S1 × Sm and represent the components in S1 in polar form:
xµ = (cos θµ, sin θµ), θµ ∈ [0, 2π), µ = 1, 2, . . . , n.
4 J.C. Guella, V.A. Menegatto and A.P. Peron
Write A to denote the interpolation matrix of ((x, z), (y, w)) ∈ (S1 × Sm)2 → f(x · y, z · w)
at {(x1, z1), (x2, z2), . . . , (xn, zn)} and consider the quadratic form ctAc, where ct indicates the
transpose of c. If the kernel is positive definite, then the addition theorem for spherical har-
monics [15, p. 18] and the representation (1.1) imply that
ctAc =
∞∑
k,l=0
C(k, l,m)a1,mk,l
d(l,m)∑
j=1
∣∣∣∣∣∣
n∑
µ=1
cµe
ikθµY m
l,j (zµ)
∣∣∣∣∣∣
2
,
in which c = (c1, c2, . . . , cn), {Y m
l,j : j = 1, 2, . . . , d(l,m)} is a basis for the space of all spherical
harmonics of degree l in m + 1 variables and C(k, l,m) is a positive constant that depends
upon k, l and m. The deduction of the equality above requires the formulas P 1
0 ≡ 1 and
P 1
k (cos θ) =
2
k
cos kθ, t ∈ [0, 2π), k = 1, 2, . . . .
The equality ctAc = 0 is equivalent to
n∑
µ=1
cµe
ikθµY m
l,j (zµ) = 0, j = 1, 2, . . . , d(l,m), (k, l) ∈
{
(k, l) : a1,mk,l > 0
}
.
If this last piece of information holds, we can invoke the addition theorem once again in order
to see that
n∑
µ=1
cµe
ikθµPml (zµ · z) = 0, (k, l) ∈
{
(k, l) : a1,mk,l > 0
}
, z ∈ Sm.
Finally, if the condition above holds, we can multiply the equality in it by e−iθ and split the
equation once again via the addition theorem to obtain
d(l,m)∑
j=1
n∑
µ=1
cµe
ik(θµ−θ)Y m
l,j (zµ)
Y m
l,j (z) = 0, (k, l) ∈
{
(k, l) : a1,mk,l > 0
}
, z ∈ Sm.
Since {Y m
l,j : j = 1, 2, . . . , d(l,m)} is linearly independent, we reach
n∑
µ=1
cµY
m
l,j (zµ)eik(θµ−θ) = 0, j = 1, 2, . . . , d(l,m), (k, l) ∈
{
(k, l) : a1,mk,l > 0
}
,
for θ ∈ [0, 2π). Multiplying the real part of the equality in the previous formula by cos kθ and
integrating in [0, 2π] with respect to θ, we obtain
n∑
µ=1
cµY
m
l,j (zµ) cos kθµ = 0, j = 1, 2, . . . , d(l,m), (k, l) ∈
{
(k, l) : a1,mk,l > 0
}
.
Similarly, it is easily seen that
n∑
µ=1
cµY
m
l,j (zµ) sin kθµ = 0, j = 1, 2, . . . , d(l,m), (k, l) ∈
{
(k, l) : a1,mk,l > 0
}
.
The above computations justify the following lemma.
Lemma 2.1. Let f be the generating function of an isotropic and positive definite kernel
on S1 × Sm and consider its series representation (1.1). The following statements are equivalent:
Strictly Positive Definite Kernels on a Product of Spheres II 5
(i) the kernel ((x, z), (y, w)) ∈ (S1 × Sm)2 → f(x · y, z · w) is strictly positive definite;
(ii) if n is a positive integer and (x1, z1), (x2, z2), . . . , (xn, zn) are distinct points on S1 × Sm,
then the only solution c = (c1, c2, . . . , cn) of the system
n∑
µ=1
cµe
ikθµPml (zµ · z) = 0, (k, l) ∈
{
(k, l) : a1,mk,l > 0
}
, z ∈ Sm,
is c = 0.
A particular case of the lemma is pertinent (see Theorem 2 in [6]).
Lemma 2.2. Let g be the generating function of an isotropic and positive definite kernel on Sm
and consider its series representation according to Schoenberg. The following statements are
equivalent:
(i) the kernel (z, w) ∈ (Sm)2 → g(z · w) is strictly positive definite;
(ii) if n is a positive integer and z1, z2, . . . , zn are distinct points on Sm, then the only solution
c = (c1, c2, . . . , cn) of the system
n∑
µ=1
cµP
m
k (zµ · z) = 0, k ∈
{
k : amk > 0
}
, z ∈ Sm,
is c = 0.
Another consequence is this one (see Theorem 5.1 in [16]).
Lemma 2.3. Let g be the generating function of an isotropic and positive definite kernel on S1
and consider its series representation according to Schoenberg. The following statements are
equivalent:
(i) the kernel (x, y) ∈ (S1)2 → g(x · y) is strictly positive definite;
(ii) if n is a positive integer and x1, x2, . . . , xn are distinct points on S1 given in polar form
xµ = (cos θµ, sin θµ), µ = 1, 2, . . . , n, then the only solution c = (c1, c2, . . . , cn) of the
system
n∑
µ=1
cµe
ikθµ = 0, k ∈
{
k : a1k > 0
}
,
is c = 0.
If f generates an isotropic positive definite kernel on S1 × Sm, we will adopt the following
notation attached to its double series representation (1.1):
Jf :=
{
(k, l) : a1,mk,l > 0
}
.
If I is a subset of Z+, we will write I ∈ SPD(Sm) to indicate that there exists a continuous
and positive definite kernel (z, w) ∈ (Sm)2 → g(z ·w) for which the set {l : aml > 0} attached to
the series representation for g in Schoenberg’s result is precisely I. This definition is well posed
once strict positive definiteness does not depend upon the actual values of the numbers aml in
the series representation for the generating function but only on the set {l : aml > 0} itself.
The first important contribution in this section is as follows.
6 J.C. Guella, V.A. Menegatto and A.P. Peron
Theorem 2.4. Let f be the generating function of an isotropic and positive definite kernel
on S1 × Sm and consider its series representation (1.1). If{
k : {l : (k, l) ∈ Jf} ∈ SPD
(
Sm
)}
∈ SPD
(
S1
)
,
then the kernel ((x, z), (y, w)) ∈ (S1×Sm)2 → f(x·y, z ·w) is strictly positive definite on S1×Sm.
Proof. We will show that, under the assumption{
k : {l : (k, l) ∈ Jf} ∈ SPD
(
Sm
)}
∈ SPD
(
S1
)
,
the alternative condition in Lemma 2.1 holds. In particular, the notation employed in that
lemma will be adopted here. Let (x1, z1), (x2, z2), . . . , (xn, zn) be distinct points in S1×Sm and
suppose that
n∑
µ=1
cµe
ikθµPml (zµ · z) = 0, (k, l) ∈ Jf , z ∈ Sm.
Let M be a maximal subset of {1, 2, . . . , n} that indexes the distinct elements among the zj .
Writing Mj := {µ : zµ = zj}, j ∈M , the previous equality becomes
∑
j∈M
∑
µ∈Mj
cµe
ikθµ
Pml (zj · z) = 0, (k, l) ∈ Jf , z ∈ Sm.
In particular,
∑
j∈M
∑
µ∈Mj
cµe
ikθµ
Pml (zj · z) = 0, l ∈ {l : (k, l) ∈ Jf}, z ∈ Sm,
whenever k ∈ {k : {l : (k, l) ∈ Jf} ∈ SPD(Sm)}. Since the zj in the expression above are all
distinct, Lemma 2.2 yields that∑
µ∈Mj
cµe
ikθµ = 0, k ∈
{
k : {l : (k, l) ∈ Jf} ∈ SPD
(
Sm
)}
,
for every j ∈M . An application of Lemma 2.3 for each j plus the help of our original assumption
leads to cµ = 0, µ ∈Mj , j ∈M . But this corresponds to c = 0. �
In a similar manner, but with slightly easier arguments, the following cousin theorem can be
proved.
Theorem 2.5. Let f be the generating function of an isotropic and positive definite kernel
on S1 × Sm and consider its series representation (1.1). If{
l : {k : (k, l) ∈ Jf} ∈ SPD
(
S1
)}
∈ SPD
(
Sm
)
,
then the kernel ((x, z), (y, w)) ∈ (S1×Sm)2 → f(x·y, z ·w) is strictly positive definite on S1×Sm.
We close this section presenting realizations for the previous theorems. They follow from the
characterizations for strict positive definiteness on singles spheres described in the Introduction.
Corollary 2.6. Let f be the generating function for an isotropic and positive definite kernel
on S1 × Sm and consider its series representation (1.1). Either condition below is sufficient for
the kernel ((x, z), (y, w)) ∈ (S1 × Sm)2 → f(x · y, z · w) to be strictly positive definite:
Strictly Positive Definite Kernels on a Product of Spheres II 7
(i) Jf contains sequences {(kµ, 2lµν) : µ, ν ∈ Z+} and {(k′µ, 2l′µν + 1): µ, ν ∈ Z+} so that
{±kµ : µ ∈ Z+} ∩ (nZ + j) 6= ∅, j = 0, 1, . . . , n− 1, n ≥ 1,
{±k′µ : µ ∈ Z+} ∩ (nZ + j) 6= ∅, j = 0, 1, . . . , n− 1, n ≥ 1,
and
lim
ν→∞
lµν = lim
ν→∞
l′µν =∞, µ ∈ Z+.
(ii) Jf contains sequences {(kµν , 2lµ) : µ, ν ∈ Z+} and {(k′µν , 2l′µ + 1): µ, ν ∈ Z+} so that
{±kµν : ν ∈ Z+} ∩ (nZ + j) 6= ∅, j = 0, 1, . . . , n− 1, n ≥ 1, µ ∈ Z+,
{±k′µν : ν ∈ Z+} ∩ (nZ + j) 6= ∅, j = 0, 1, . . . , n− 1, n ≥ 1, µ ∈ Z+,
and
lim
µ→∞
lµ = lim
µ→∞
l′µ =∞.
Finally, we would like to point that the previous theorems can be reproduced in other settings,
with or without the presence of isotropy (for example Sm × Sm and the product of Sm and
a torus). Details will not be included here.
3 Characterizations for strict positive definiteness on S1 × Sm
In order to obtain a characterization for the strict positive definiteness of an isotropic positive
definite kernel on S1 × Sm, we need to look at the concept of strict positive definiteness in an
enhanced form. We begin this section explaining what we mean by that and introducing the
additional concepts needed.
It is an obvious matter to see that we can write the generating function f of a positive definite
kernel ((x, z), (y, w)) ∈ (S1 × Sm)2 → f(x · y, z · w) in the form
f(t, s) =
∞∑
l=0
fl(t)P
m
l (s), t, s ∈ [−1, 1], (3.1)
in which
fl(t) :=
∞∑
k=0
a1,mk,l P
1
k (t), t ∈ [−1, 1], l = 0, 1, . . . .
Since Pml (1) ≥ 1, m ≥ 2, l = 0, 1, . . ., the series
∞∑
k=0
a1,mk,l P
1
k (1) < ∞ converges. In particular,
each fl is the generating function of a continuous, isotropic and positive definite kernel on S1.
In the statement of the next lemma, we will employ the following additional notation related
to another one we have previously introduced:
Jkf := {l : (k, l) ∈ Jf}.
In particular,
∪kJkf = {l : fl 6= 0}.
Among other things, the lemma suggests how a characterization for the strict positive definite-
ness of an isotropic and positive definite kernel on S1 × Sm should look like.
8 J.C. Guella, V.A. Menegatto and A.P. Peron
Lemma 3.1. Let f be the generating function of an isotropic and positive definite kernel
on S1 × Sm and consider the alternative series representation (3.1) for f . If p is a positive
integer, x1, x2, . . . , xp are distinct points on S1 and c is a real vector in Rp, then the continuous
function g given by
g(s) =
∑
l∈∪kJkf
{
ct[fl(xi · xj)]pi,j=1c
}
Pml (s), s ∈ [−1, 1],
generates an isotropic and positive definite kernel on Sm. If c is nonzero and the kernel
((x, z), (y, w)) ∈ (S1 × Sm)2 → f(x · y, z · w) is strictly positive definite, then (z, w) ∈ (Sm)2 →
g(z · w) is strictly positive definite as well.
Proof. Write c = (c1, c2, . . . , cp). If z1, z2, . . . , zq are distinct points on Sm and d1, d2, . . . , dq
are real numbers, then
q∑
µ,ν=1
dµdνg(zµ · zν) =
q∑
µ,ν=1
p∑
i,j=1
(dµci)(dνcj)f(xi · xj , zµ · zν).
But, the last expression above corresponds to a quadratic form involving f and the pq distinct
points (xi, zµ), i = 1, 2, . . . , p, µ = 1, 2, . . . , q, of S1 × Sm. In particular,
q∑
µ,ν=1
dµdνg(zµ · zν) ≥ 0.
If the real numbers dµ are not all zero and c 6= 0, then at least one of the scalars dµci is likewise
nonzero. Further, if ((x, z), (y, w)) ∈ (S1 × Sm)2 → f(x · y, z · w) is strictly positive definite,
then the quadratic form above is, in fact, positive. In particular, (z, w) ∈ (Sm)2 → g(z · w) is
strictly positive definite. �
Another useful technical result is as follows.
Lemma 3.2. Let f be the generating function of an isotropic and positive definite kernel
on S1 × Sm and consider the alternative series representation (3.1) for f . If (x1, z1), (x2, z2),
. . . , (xn, zn) are distinct points in S1 × Sm and c1, c2, . . . , cn are real scalars, then the following
assertions are equivalent:
(i)
n∑
i,j=1
cicjf(xi · xj , zi · zj) = 0;
(ii)
n∑
i,j=1
cicjfl(xi · xj)Pml (zi · zj) = 0, l ∈ ∪kJkf .
Proof. One direction is immediate while the other follows simply from the observation that
(t, s) ∈ [−1, 1]2 → fl(t)P
m
l (s) is the generating function of a positive definite kernel
on S1 × Sm. �
Next, we formalize the definition of enhancement we use in this section.
Definition 3.3. Let p and q be positive integers, {x1, x2, . . . , xp} ⊂ S1 and {z1, z2, . . . , zq} an
antipodal free subset of Sm, that is, a set containing no pairs of antipodal points. An enhanced
subset of S1 × Sm generated by them is the set{
(x1, z1), (x2, z1), . . . , (xp, z1), (x1, z2), (x2, z2), . . . , (xp, z2), . . . ,
(x1, zq), (x2, zq), . . . , (xp, zq), (x1,−z1), (x2,−z1), . . . , (xp,−z1),
(x1,−z2), (x2,−z2), . . . , (xp,−z2), . . . , (x1,−zq), (x2,−zq), . . . , (xp,−zq)
}
.
Strictly Positive Definite Kernels on a Product of Spheres II 9
The positive numbers p and q in the previous definition may have no connection at all. The
order in which the elements in an enhanced subset of S1×Sm are displayed is not relevant, but
the writing of the upcoming results will take into account the order adopted above and inherited
from those in the subsets of S1 and Sm involved. An enhanced set as above contains 2pq distinct
points.
The following lemma is concerned with the existence of enhanced sets.
Lemma 3.4. If B′ = {(x′1, z′1), (x′2, z′2), . . . , (x′n, z′n)} is a subset of S1 × Sm, then there exists
an enhanced subset B of S1 × Sm that contains B′.
Proof. It suffices to consider the enhanced subset of S1 × Sm generated by the set {x1, x2,
. . . , xp} that encompasses the distinct elements among the x′i and an antipodal free subset
{z1, z2, . . . , zq} of Sm satisfying z′i ∈ {z1, z2, . . . , zq}, i = 1, 2, . . . , n. �
If f is the generating function of an isotropic and positive definite kernel on S1 × Sm and B
is an enhanced set as previously described, we will write E(f,B) to denote the interpolation
matrix of ((x, z), (y, w)) ∈ (S1 × Sm)2 → f(x · y, z · w) at B, keeping the order for the points
of B. If A is a subset of S1 × Sm and B is an enhanced subset of S1 × Sm containing A, then
the interpolation matrix of ((x, z), (y, w)) ∈ (S1 × Sm)2 → f(x · y, z · w) at A is a principal
sub-matrix of E(f,B). In particular, if E(f,B) is positive definite, so is A. These comments
justify the following lemma.
Lemma 3.5. Let f be the generating function of an isotropic and positive definite kernel
on S1 × Sm and consider the alternative series representation (3.1) for f . The following as-
sertions are equivalent:
(i) the kernel ((x, z), (y, w)) ∈ (S1 × Sm)2 → f(x · y, z · w) is strictly positive definite;
(ii) If B is an enhanced subset of S1 × Sm, then the matrix E(f,B) is positive definite.
Due to the decomposition for the generating function f of an isotropic positive definite kernel
as described in (3.1), we can write
E(f,B) =
∞∑
l=0
E(f,B, l)
in which E(f,B, l) is the interpolation matrix of the kernel (t, s) ∈ S1×Sm → fl(t)P
m
l (s) at B.
The order in which the elements of an enhanced subset of S1 × Sm appears forces the matrix
E(f,B, l) to have a very distinctive block representation. Precisely,
E(f,B, l) =
(
M11 M12
M21 M22
)
,
where each block Mρσ = Mρσ(f,B, l) has its own block structure
Mρσ = [Mµν
ρσ ]qµ,ν=1, ρ, σ = 1, 2,
defined by
Mµν
ρσ = [fl(xi · xj)]pi,j=1(−1)l(σ+ρ)Pml (zµ · zν), µ, ν = 1, 2, . . . , q.
Implicitly used in the writing of the block decomposition above is the fact that Gegenbauer
polynomials of even degree are even functions while those of odd degree are odd functions.
In particular, since M22 = M11 and M12 = M21 = (−1)lM11, the matrix E(f,B, l) depends
upon M11 only.
10 J.C. Guella, V.A. Menegatto and A.P. Peron
Keeping all the notation introduced so far, Lemmas 3.2 and 3.5 and the comments above lead
to the following characterization for the strict positive definiteness of the kernel ((x, z), (y, w)) ∈
(S1 × Sm)2 → f(x · y, z · w) via
M11 =
[
[fl(xi · xj)]pi,j=1P
m
l (zµ · zν)
]q
µ,ν=1
.
Lemma 3.6. Let f be the generating function of an isotropic and positive definite kernel
on S1 × Sm and consider the alternative series representation (3.1) for f . The following as-
sertions are equivalent:
(i) the kernel ((x, z), (y, w)) ∈ (S1 × Sm)2 → f(x · y, z · w) is strictly positive definite;
(ii) if B is an enhanced subset of S1×Sm generated by a subset {x1, x2, . . . , xp} of S1 and the
antipodal free subset {z1, z2, . . . , zq} of Sm, then the only solution (c1, c2) ∈ (Rpq)2 of the
system[
c1 + (−1)lc2
]t
M11
[
c1 + (−1)lc2
]
= 0, l ∈ ∪kJkf ,
is the trivial one, that is, c1 = c2 = 0.
Introducing components for the vectors ci in the previous lemma, we obtain the following
reformulation.
Lemma 3.7. Let f be as in the previous lemma. The following assertions are equivalent:
(i) the kernel ((x, z), (y, w)) ∈ (S1 × Sm)2 → f(x · y, z · w) is strictly positive definite;
(ii) if B is an enhanced subset of S1 × Sm generated by a subset {x1, x2, . . . , xp} of S1 and
the antipodal free subset {z1, z2, . . . , zq} of Sm, then the only solution (c11, c
2
1, . . . , c
q
1, c
1
2, c
2
2,
. . . , cq2) ∈ (Rp)2q of the system
q∑
µ,ν=1
{
(cµ1 + (−1)lcµ2 )t[fl(xi · xj)]pi,j=1(c
ν
1 + (−1)lcν2)
}
Pml (zµ · zν) = 0, l ∈ ∪kJkf ,
is the trivial one.
Next, we break up the system in the previous lemma, according to the parity of the elements
in ∪kJkf .
Proposition 3.8. Let f be as in the previous lemma. The following assertions are equivalent:
(i) the kernel ((x, z), (y, w)) ∈ (S1 × Sm)2 → f(x · y, z · w) is strictly positive definite;
(ii) if p and q are positive integers, x1, x2, . . . , xp are distinct points on S1 and {z1, z2, . . . ,
zq} is an antipodal free subset of Sm, then the only solution (d11, d
2
1, . . . , d
q
1, d
1
2, d
2
2, . . . , d
q
2)
in (Rp)2q of the system
q∑
µ,ν=1
{
(dµ1 )t[fl(xi · xj)]pi,j=1d
ν
1
}
Pml (zµ · zν) = 0, l ∈ (2Z+ + 1) ∩
(
∪k Jkf
)
,
q∑
µ,ν=1
{
(dµ2 )t[fl(xi · xj)]pi,j=1d
ν
2
}
Pml (zµ · zν) = 0, l ∈ (2Z+) ∩
(
∪k Jkf
)
,
is the trivial one.
Strictly Positive Definite Kernels on a Product of Spheres II 11
Proof. Assume that for some distinct points x1, x2, . . . , xp in S1 and some antipodal free subset
{z1, z2, . . . , zq} of Sm, the system in (ii) has a nontrivial solution (d11, d
2
1, . . . , d
q
1, d
1
2, d
2
2, . . . , d
q
2).
If dµ1 6= 0 for some µ, we define cµ1 = −cµ2 = 2−1dµ1 , µ = 1, 2, . . . , q. Otherwise, we define
cµ1 = cµ2 = 2−1dµ2 , µ = 1, 2, . . . , q. In both cases, the vector (c11, c
2
1, . . . , c
q
1, c
1
2, c
2
2, . . . , c
q
2) is
nonzero and, in addition,
q∑
µ,ν=1
{(
cµ1 + (−1)lcµ2
)t
[fl(xi · xj)]pi,j=1
(
cν1 + (−1)lcν2
)}
Pml (zµ · zν) = 0,
for l ∈ [(2Z+ + 1) ∩ (∪kJkf )] ∪ [(2Z+) ∩ (∪kJkf )] = ∪kJkF . In other words, Condition (ii) in
Lemma 3.7 does not hold for the enhanced set B generated by the subset {x1, x2, . . . , xp}
of S1 and the antipodal free subset {z1, z2, . . . , zq} of Sm. Thus, ((x, z), (y, w)) ∈ (S1 ×
Sm)2 → f(x · y, z · w) is not strictly positive definite. Conversely, if (i) does not hold, the
previous lemma assures the existence of a subset {x1, x2, . . . , xp} of S1, an antipodal free subset
{z1, z2, . . . , zq} of Sm, an enhanced subset A of S1×Sm generated by them and a nonzero vector
(c11, c
2
1, . . . , c
q
1, c
1
2, c
2
2, . . . , c
q
2) ∈ (Rp)2q so that
q∑
µ,ν=1
{(
cµ1 + (−1)lcµ2
)t
[fl(xi · xj)]pi,j=1
(
cν1 + (−1)lcν2
)}
Pml (zµ · zν) = 0, l ∈ ∪kJkf .
However, this last piece of information corresponds to
q∑
µ,ν=1
{(
cµ1 − c
µ
2
)t
[fl(xi · xj)]pi,j=1
(
cν1 − cν2
)}
Pml (zµ · zν) = 0, l ∈ (2Z+ + 1) ∩
(
∪k Jkf
)
,
and
q∑
µ,ν=1
{(
cµ1 + cµ2
)t
[fl(xi · xj)]pi,j=1
(
cν1 + cν2
)}
Pml (zµ · zν) = 0, l ∈ (2Z+) ∩
(
∪k Jkf
)
.
On the other hand, it is easily verifiable that the vector(
c11 − c12, c21 − c22, . . . , c
q
1 − c
q
2, c
1
1 + c12, c
2
1 + c22, . . . , c
q
1 + cq2
)
∈
(
Rp
)2q
is nonzero. Thus, (ii) does not hold due to Lemma 3.7. �
We are about ready to prove the crucial result in this section.
Theorem 3.9. Let f be the generating function of an isotropic and positive definite kernel
on S1 × Sm and consider the alternative series representation (3.1) for f . The following asser-
tions are equivalent:
(i) the kernel ((x, z), (y, w)) ∈ (S1 × Sm)2 → f(x · y, z · w) is strictly positive definite;
(ii) if p is a positive integer, x1, x2, . . . , xp are distinct points in S1 and c ∈ Rp \ {0}, then the
set {
l ∈ ∪kJkf : ct[fl(xi · xj)]pi,j=1c > 0
}
contains infinitely many even and infinitely many odd integers.
12 J.C. Guella, V.A. Menegatto and A.P. Peron
Proof. Lemma 3.1 justifies one implication. As for the other, assume the condition in the
statement of the theorem holds but ((x, z), (y, w)) ∈ (S1 × Sm)2 → f(x · y, z · w) is not strictly
positive definite. Hence, we can find a positive integer p, distinct points x1, x2, . . . , xp in S1, an
antipodal free subset {y1, y2, . . . , yn} of Sm and a nonzero vector (d11, d
2
1, . . . , d
q
1, d
1
2, d
2
2, . . . , d
q
2)
in (Rp)2q so that the two equations in Proposition 3.8(ii) hold. We will proceed assuming that
(d12, d
2
2, . . . , d
q
2) is a nonzero vector and that
q∑
µ,ν=1
{
(dµ2 )t[fl(xi · xj)]pi,j=1d
ν
2
}
Pml (zµ · zν) = 0, l ∈ 2Z+ ∩
(
∪k Jkf
)
and will reach a contradiction. The other possibility can be handled similarly but the details
will be omitted. Without loss of generality, we can assume that the vector d12 is nonzero. Since
the set{
l ∈ ∪kJkf :
(
d12
)t
[fl(xi · xj)]pi,j=1d
1
2 > 0
}
∩ 2Z
is infinite by assumption, we can select an infinite subset Q of it and a number θ = θ(l) in
{1, 2, . . . , q} so that(
dθ2
)t
[fl(xi · xj)]pi,j=1d
θ
2 ≥
(
dµ2
)t
[fl(xi · xj)]pi,j=1d
µ
2 , µ = 1, 2, . . . , q, l ∈ Q.
In particular,(
dθ2
)t
[fl(xi · xj)]pi,j=1d
θ
2 > 0, l ∈ Q.
Returning to the initial equality we can write
0 = 1 +
q∑
µ=1
µ6=θ
(dµ2 )t[fl(xi · xj)]pi,j=1d
µ
2
(dθ2)
t[fl(xi · xj)]pi,j=1d
θ
2
Pml (zµ · zµ)
Pml (1)
+
∑
µ6=ν
(dµ2 )t[fl(xi · xj)]pi,j=1d
ν
2
(dθ2)
t[fl(xi · xj)]pi,j=1d
θ
2
Pml (zµ · zν)
Pml (1)
, l ∈ Q.
Since each fl is the continuous and isotropic part of a positive definite kernel on S1, we have
that (dµ2 )t[fl(xi · xj)]pi,j=1d
µ
2 ≥ 0, µ = 1, 2, . . . , q. In particular,
q∑
µ=1
µ 6=θ
(dµ2 )t[fl(xi · xj)]pi,j=1d
µ
2
(dθ2)
t[fl(xi · xj)]pi,j=1d
θ
2
∈ [0, q − 1],
while the Cauchy–Schwarz inequality implies that
0 ≤
∣∣∣∣∣(d
µ
2 )t[fl(xi · xj)]pi,j=1d
ν
2
(dθ2)
t[fl(xi · xj)]pi,j=1d
θ
2
∣∣∣∣∣ ≤ 1, µ 6= ν.
Since zµ · zν ∈ (−1, 1), µ 6= ν, a well-known property of the Gegenbauer polynomials provides
the limit formula [20, p. 196]
lim
l→∞
l∈Q
Pml (zµ · zν)
Pml (1)
= 0, µ 6= ν.
Consequently, we may apply the definition of limit conveniently (l large enough), to conclude
that 0 ≥ 1 + 0− 1/2 = 1/2, a contradiction. �
Strictly Positive Definite Kernels on a Product of Spheres II 13
The next theorem demands the truncated sum functions (γ ≥ 0)
foγ =
∑
2l+1≥γ
f2l+1 and f eγ =
∑
2l≥γ
f2l
attached to the generating function of an isotropic and positive definite kernel. Since Pml (1) ≥ 1,
m ≥ 2, l ≥ 0, it follows that
|fl(t)| ≤
∞∑
k=0
a1,mk,l P
1
k (1) ≤
∞∑
k=0
a1,mk,l P
1
k (1)Pml (1), l ≥ γ.
In particular, since
∞∑
l=0
∞∑
k=0
a1,mk,l P
1
k (1)Pml (1) <∞, the functions foγ and f eγ are continuous.
Theorem 3.10. Let f be the generating function of an isotropic and positive definite kernel
on S1 × Sm and consider the alternative series representation (3.1) for f . The following asser-
tions are equivalent:
(i) the kernel ((x, z), (y, w)) ∈ (S1 × Sm)2 → f(x · y, z · w) is strictly positive definite;
(ii) for each γ ≥ 0, the functions foγ and f eγ are the generating functions of isotropic and
strictly positive definite kernels on S1.
Proof. If (i) holds, we can apply Lemma 3.1 in order to see that
ctfoγ (xi · xj)c =
∑
2l+1≥γ
ctf2l+1(xi · xj)c > 0,
whenever c ∈ Rp \ {0}, γ ≥ 0 and x1, x2, . . . , xp are distinct points in S1. Obviously, a similar
property is valid for f eγ . Thus, (ii) follows. Conversely, if (ii) holds, then Condition (ii) in the
previous theorem holds as well, due to the fact that (ii) is valid for all γ ≥ 0. Thus, Theorem 3.9
guarantees the strict positive definiteness of ((x, z), (y, w)) ∈ (S1 × Sm)2 → f(x · y, z · w). �
The coefficient in the series expansion of foγ attached to the polynomial P 1
k is∑
γ≤2l+1∈Jkf
a1,mk,2l+1.
It is positive if, and only if, Jkf contains an odd integer greater than or equal to γ. A similar
remark applies to the coefficients in the series of f eγ . Taking into account the characterization for
strict positive definiteness on S1 quoted at the introduction, we have the following consequence
of the previous theorem and our final characterization for strict positive definiteness on S1×Sm.
Theorem 3.11. Let f be the generating function of an isotropic and positive definite kernel
on S1 × Sm. The following assertions are equivalent:
(i) the kernel ((x, z), (y, w)) ∈ (S1 × Sm)2 → f(x · y, z · w) is strictly positive definite;
(ii) for each γ ≥ 0, the sets{
k ∈ Z : J
|k|
f ∩ {γ, γ + 1, . . .} ∩ (2Z + 1) 6= ∅
}
and {
k ∈ Z : J
|k|
f ∩ {γ, γ + 1, . . .} ∩ 2Z+ 6= ∅
}
intersect every arithmetic progression in Z.
14 J.C. Guella, V.A. Menegatto and A.P. Peron
4 Replacing Sm with a compact two-point homogeneous space
The results obtained so far in the paper can be adapted to hold for kernels on a product of the
form S1 ×Md, in which Md is a compact two-point homogeneous space. The case S1 × S1 was
covered in [13] while the case Sd ×Md, d ≥ 3, was covered in [2, Theorem 4.5]. The results
sketched in this section complement these two cases.
Let us write |zw| to denote the usual normalized surface distance between z and w in Md.
As described in [2], an isotropic kernel ((x, z), (y, w)) ∈ (S1 ×Md)2 → f(x · y, cos(|zw|/2)) is
positive definite if, and only if, the generating function f has a double series representation in
the form
f(t, s) =
∞∑
k,l=0
adk,lP
1
k (t)P d,βl (s), t, s ∈ [−1, 1]2,
in which adk,l ≥ 0, k, l ∈ Z+, P d,βl is the Jacobi polynomial associated to the pair ((d− 2)/2, β),
β is a number from the list −1/2, 0, 1, 3, depending on the respective category Md belongs to,
that is, the real projective spaces Pd(R), d = 2, 3, . . ., the complex projective spaces Pd(C),
d = 4, 6, . . ., the quaternionic projective spaces Pd(H), d = 8, 12, . . ., and the Cayley projective
plane Pd(Cay), d = 16, and
∞∑
k,l=0
ak,lP
1
k (1)P d,βl (1) <∞.
In this setting, the procedure adopted in Section 3 can be considerably simplified. An alter-
native series representation for the generating function of the kernel can be likewise defined and
the fact that a point in Md possesses infinitely many antipodal points permits the deduction of
a version of Theorem 3.9 without considering any enhancements and augmentations. Precisely,
we have the following result.
Theorem 4.1. Let f be the generating function of an isotropic and positive definite kernel
on S1 ×Md and consider the alternative series representation for f . The following assertions
are equivalent:
(i) the kernel ((x, z), (y, w)) ∈ (S1 ×Md)2 → f(x · y, z · w) is strictly positive definite;
(ii) if p is a positive integer, x1, x2, . . . , xp are distinct points in S1 and c ∈ Rp \ {0}, then the
set {
l ∈ ∪kJkf : ct[fl(xi · xj)]pi,j=1c > 0
}
contains infinitely many integers.
Taking into account the characterization for strict positive definiteness obtained in [1], the
final characterization in these remaining cases is this one.
Theorem 4.2. Let f be the generating function of an isotropic and positive definite kernel
on S1 ×Md. Assume Md is not a sphere. The following assertions are equivalent:
(i) the kernel ((x, z), (y, w)) ∈ (S1 ×Md)2 → f(x · y, z · w) is strictly positive definite;
(ii) for each γ ≥ 0, the set{
k ∈ Z : J
|k|
f ∩ {γ, γ + 1, . . .} 6= ∅
}
intersects every arithmetic progression in Z.
Strictly Positive Definite Kernels on a Product of Spheres II 15
Acknowledgements
The authors are grateful to the referees for their valuable comments and suggestions. Second
author acknowledges partial financial support from FAPESP, grant 2014/00277-5. Likewise, the
third author acknowledges support from the same foundation, under grants 2014/25796-5 and
2016/03015-7.
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http://dx.doi.org/10.7153/mia-19-54
http://arxiv.org/abs/1505.00591
http://arxiv.org/abs/1605.07071
http://arxiv.org/abs/1510.08658
http://dx.doi.org/10.3842/SIGMA.2016.043
http://arxiv.org/abs/1601.07743
http://dx.doi.org/10.1007/s00365-016-9323-9
http://arxiv.org/abs/1505.05682
http://dx.doi.org/10.1090/S0002-9939-03-06730-3
http://dx.doi.org/10.1090/S0002-9939-2010-10533-6
http://arxiv.org/abs/1002.3017
http://dx.doi.org/10.1198/016214502760047113
http://dx.doi.org/10.3150/12-BEJSP06
http://arxiv.org/abs/1111.7077
http://dx.doi.org/10.1016/j.jmaa.2015.10.026
http://arxiv.org/abs/1505.03695
http://dx.doi.org/10.1215/17358787-3649260
http://arxiv.org/abs/1503.08174
http://dx.doi.org/10.1007/s11117-016-0425-1
http://arxiv.org/abs/1505.01169
http://dx.doi.org/10.1016/j.camwa.2006.04.006
http://dx.doi.org/10.1007/978-1-4612-0581-4
http://dx.doi.org/10.1090/S0025-5718-96-00780-6
http://dx.doi.org/10.1215/S0012-7094-42-00908-6
http://dx.doi.org/10.1201/b10811
http://dx.doi.org/10.1201/b10811
http://dx.doi.org/10.1090/S0025-5718-04-01668-0
1 Introduction
2 Abstract sufficient conditions
3 Characterizations for strict positive definiteness on S1 Sm
4 Replacing Sm with a compact two-point homogeneous space
References
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