Journal of Fourier Analysis and Applications https://doi.org/10.1007/s00041-018-9631-5

Schoenberg’s Theorem for Positive Definite Functions on Products: A Unifying Framework J. C. Guella1 · V. A. Menegatto1 Received: 9 February 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018

Abstract The main contribution in the present paper is a characterization for positive definiteness and strict positive definiteness of a kernel on the product X × S d , in which X is a nonempty set and S d is the usual d-dimensional unit sphere in Euclidean space, through Fourier-like expansions. The setting presupposes continuity and isotropy on the S d side and no algebraic structure or topology on X . The result may be interpreted as another extension of a classical result of I. J. Schoenberg on positive definite functions on spheres. We take a closer look at our results in the case in which X is a locally compact group, paying special attention to usual Euclidean spaces and high dimensional tori. Keywords Positive definite kernels · Strict positive definiteness · Spheres · Isotropy · Locally compact groups · Torus Mathematics Subject Classification 33C50 · 33C55 · 42A16 · 42A82 · 42C10 · 43A35

1 Introduction For a nonempty set X , a function K mapping X × X into C is called a positive definite kernel on X if n  cμ cν K (u μ , u ν ) ≥ 0, (1.1) μ,ν=1

Communicated by Karlheinz Grochenig.

B

V. A. Menegatto [email protected] J. C. Guella [email protected]

1

Departamento de Matemática, ICMC-USP, Caixa Postal 668, São Carlos, SP 13560-970, Brazil

Journal of Fourier Analysis and Applications

for n ≥ 1, distinct points u 1 , u 2 , . . . , u n in X and complex scalars c1 , c2 , . . . , cn . In other words, the “interpolation matrix" [K (u μ , u ν )]nμ,ν=1 of K at every {u 1 , u 2 , . . . , u n } ⊂ X is positive semi-definite. A positive definite kernel is always Hermitian, that is, K (u, v) = K (v, u), u, v ∈ X , unless it is a real function. In that case, we usually add symmetry to the definition, i.e., we ask that K (u, v) = K (v, u), u, v ∈ X , in order to be able to use only real scalars in (1.1). The kernel is said to be strictly positive definite if the inequalities in (1.1) are strict, unless c1 = c2 = · · · = cn = 0. That provides positive definiteness, in particular, invertibility of the interpolation matrices previously mentioned. Positive definite and strictly positive definite kernels present themselves in many applications in approximation theory, spatial statistics, harmonic analysis, etc. While positive definite kernels can be traced back to many papers from the beginning of the past century, strictly positive definite kernels have found their place more recently. At this point we mention [4] for the basics on positive and strict positive definiteness and [11,19] for some applications. One of the first characterizations for positive definiteness on a set X carrying a group structure is that one provided by Bochner [5] in the case X = R: if f : R → C is continuous, then the kernel (u, v) ∈ R × R → f (u − v) ∈ C is positive definite if, and only if, f is the Fourier transform of a finite positive measure μ on R, i.e.,  f (u) =

R

ei vu dμ(v), u ∈ R.

We refer the reader to [7,23] and references therein for extensions and further discussion on these matters, including a companion result for positive definiteness of radially symmetric kernels and some characterizations for strict positive definiteness. A little later, Schoenberg published his seminal paper [20] characterizing isotropic positive definite kernels on spheres: if f : [−1, 1] → C is continuous and · denotes the usual inner product in Rd+1 , then the kernel (x, y) ∈ S d × S d → f (x · y) is positive definite if, and only if, f (s) =

∞ 

akd Pkd (s), s ∈ [−1, 1],

(1.2)

k=0

where all the coefficients akd are nonnegative, Pkd is the usual Gegenbauer polynomial  of degree k attached to the rational number (d − 1)/2 and k akd Pkd (1) < ∞. After observing that positive definite kernels on S d are positive definite on S m , m ≤ d, Schoenberg extended his representation result to the Hilbert sphere, that is, the unit sphere S ∞ in the real Hilbert space 2 . In that case, we need to use d = ∞ in (1.2) and put Pk∞ (s) = s k . In both cases, f is referred to as the isotropic part of the kernel. The dot notation used above refers to the usual inner product of Rd+1 if d < ∞ and of 2 otherwise.

Journal of Fourier Analysis and Applications

Much later, due to their usefulness in interpolation, the strictly positive definite kernels from the Schoenberg class were characterized as those for which the following additional conditions hold: the set {k : akd > 0} contains infinitely many even and d > 0} infinitely many odd integers in the case d ∈ {2, 3, . . . , ∞} [6,17] and {k : a|k| intersects every full arithmetic progression in Z, i.e, sets of the form N Z + j, N ∈ Z+ , j ∈ {0, 1, . . . , N − 1}, in the case d = 1 [18]. Similar characterizations for positive and strict positive definiteness on compact two-point homogeneous spaces appeared in [1,10], respectively. The paper [9] contains information on positive definite kernels on a general compact abelian group and some pertinent references related to the topic. It describes an abstract characterization for strict positive definiteness on a torsion group and on a product of a finite group and a torus via the concept of ubiquitous set. The case of an m-dimensional torus alone is implicit in a work of Bochner but a detailed proof can be found in [21]. If f is a continuous and 2π -periodic function on (each component of) Rm and Tm := {u ∈ Rm : −π ≤ u j < π ; j = 1, 2, . . . , m}, the kernel (u, v) ∈ Tm × Tm → f (v − u) is positive definite if, and only if, f (u) =



fˆ(n)ei(n·u) , u ∈ Rm ,

n∈Zm

in which all the Fourier coefficients fˆ(n) are nonnegative and 

fˆ(n) := lim

n∈Zm

N →∞



fˆ(n) < ∞.

|n|≤N

According to [13], a kernel as above is strictly positive definite if, and only if, the set {n ∈ Zm : fˆ(n) > 0} intersects all the translations (cosets) of each subgroup (a1 Z, a2 Z, . . . , am Z), a1 , a2 , . . . , am ∈ Z+ \{0} of Zm . Recently, positive and strict positive definiteness on a product of spaces began to attract more attention due to their applicability in some specific problems from statistics, mainly those involving random fields on spaces across time. The paper [2] investigated positive definite kernels on a product of the form G × S d , in which G is an arbitrary locally compact group, keeping both, the group structure of G and the isotropy of S d 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 that paper can be described as follows: if f : G × [−1, 1] → C is continuous, the kernel ((u, x), (v, y)) ∈ (G × S d )2 → f (u −1 ∗ v, x · y) is positive definite if, and only if, f has the form f (u, s) =

∞  k=0

bkd (u)Pkd (s), (u, s) ∈ G × [−1, 1],

Journal of Fourier Analysis and Applications

in which Pkd is as before, {bkd } is a sequence of continuous functions on G defining  positive definite kernels (u, v) ∈ G 2 → bkd (u −1 ∗ v), and k bkd (e)Pkd (1) < ∞, with uniform convergence of the series for (u, t) ∈ G × [−1, 1]. No strict positive definiteness issues were addressed in [2]. It is worth mentioning that the results in [2] have been generalized by replacing S d with a compact Gelfand pair (see [3]). An adaptation of the result above leads to positive and strict positive definiteness of continuous kernels of the form ((u, x), (v, y)) ∈ (S m × S d )2 → f (u · v, x · y), in which · stands now for the inner products in both Rd+1 and Rm+1 , cases that were extensively studied in [12,14–16]. In this paper, we propose a unified study for positive definiteness and strict positive definiteness on a product of the form X × S d , d ∈ {1, 2, . . . , ∞}, fixing no algebraic or topological structures on X , but keeping the isotropy of S d . In Sect. 2, we characterize positive definiteness and indicate how to recover some known cases from our characterization. In Sect. 3, we present several equivalences for strict positive definiteness in the setting adopted, the last two being explicit characterizations for that concept. Section 4 is devoted to an analysis of our results in the case in which X is the group (Rm , +), with the deduction of some criteria for strict positive definiteness, with or without radiality in Rm . Finally, in Sect. 5, we point how to translate our results to the case in which X is the m-dimensional torus.

2 Positive Definiteness on X × Sd In this section, we provide a characterization for the positive definiteness of a kernel ((u, x), (v, y)) ∈ (X × S d )2 → f (u, v, x · y), under the assumption that f (u, v, ·) is continuous on [−1, 1], for each fixed u, v ∈ X . Recall that the positive definiteness of a complex-valued kernel as above corresponds to the positive semi-definiteness of the matrices 

f (u μ , u ν , xμ · xν )

n μ,ν=1

,

whenever n ≥ 1 and (u 1 , x1 ), (u 2 , x2 ), . . . , (u n , xn ) are distinct points in X × S d , that is, to  n ct f (u μ , u ν , xμ · xν ) μ,ν=1 c ≥ 0, c ∈ Cn . The matrices mentioned above will be referred to as the interpolation matrices of the kernel at the set {(u 1 , x1 ), (u 2 , x2 ), . . . , (u n , xn )}. Strict positive definiteness corresponds to strict inequalities in the inequalities above for nonzero c. The basic properties of a positive definite kernel as above are collected in the lemma below. The proof is standard and is omitted. Lemma 2.1 For a positive definite kernel ((u, x), (v, y)) ∈ (X ×S d )2 → f (u, v, x·y), we have f (u, u, 1) ≥ 0, u ∈ X ,

Journal of Fourier Analysis and Applications

f (u, v, s) = f (v, u, s), u, v ∈ X , s ∈ [−1, 1], and | f (u, v, s)| ≤ [ f (u, u, 1) f (v, v, 1)]1/2 1 ≤ [ f (u, u, 1) + f (v, v, 1)], u, v ∈ X , s ∈ [−1, 1]. 2 Further, for each u ∈ X fixed, the function f (u, u, ·) is the isotropic part of a positive definite kernel on S d . The following lemma will play a key role below to obtain information on positive definite kernels on X × S d . Lemma 2.2 Let ((u, x), (v, y)) ∈ (X × S d )2 → f (u, v, x · y) be positive definite and assume that each function f (u, v, ·) is continuous on [−1, 1]. For distinct points u 1 , u 2 , . . . , u p in X and complex numbers c1 , c2 , . . . , c p , define g(s) =

p 

cμ cν f (u μ , u ν , s), s ∈ [−1, 1].

μ,ν=1

The following assertions hold: (i) The function g is real and continuous. (ii) The function g is the isotropic part of a positive definite kernel on S d . (iii) If the cν are not all zero and the kernel is strictly positive definite, then g is the isotropic part of a strictly positive definite kernel on S d . Proof The continuity of g on [−1, 1] is obvious. Since, g(s) =

p 

cμ cν f (u μ , u ν , s) =

μ,ν=1

p 

cμ cν f (u ν , u μ , s) = g(s), s ∈ [−1, 1],

μ,ν=1

g is clearly a real function. In order to verify (ii) holds, observe that if x1 , x2 , . . . , xq are distinct points on S d and b = (b1 , b2 , . . . , bq ) belongs to Rq , then p q    q bt g(xα · xβ ) α,β=1 b = bα cμ cν bβ f (u μ , u ν , xα · xβ )

=

α,β=1 μ,ν=1 p q  

(bα cμ )(bβ cν ) f (u μ , u ν , xα · xβ ).

α,β=1 μ,ν=1

But, the resulting quadratic form above matches that in the definition of positive definiteness when we employ the points (u 1 , x1 ), (u 1 , x2 ), . . . , (u 1 , xq ), (u 2 , x1 ), (u 2 , x2 ), . . . , (u 2 , xq ), . . . , (u p , x1 ), (u p , x2 ), . . . , (u p , xq ) on X × S d and the complex numbers b1 c1 , b2 c1 , . . . , bq c1 , b1 c2 , b2 c2 , . . . , bq c2 , . . . , b1 c p , b2 c p , . . . , bq c p . The

Journal of Fourier Analysis and Applications

positive definiteness of the kernel ((u, x), (v, y)) ∈ (X × S d )2 → f (u, v, x · y) guarantees that  q bt g(xα · xβ ) α,β=1 b ≥ 0. Finally, assume that at least one cμ is nonzero. If at least one bα is nonzero, then one of the numbers bα cμ is likewise nonzero. Hence, if the kernel is strictly positive definite, then the previous inequality is actually strict.

The main theorem in this section is as follows. Theorem 2.3 For a kernel ((u, x), (v, y)) ∈ (X × S d )2 → f (u, v, x · y), the following assertions are equivalent: (i) The kernel is positive definite and each function f (u, v, ·) is continuous on [−1, 1]. (ii) Each function f (u, v, ·) has a series representation in the form f (u, v, s) =

∞ 

akd (u, v)Pkd (s), s ∈ [−1, 1],

k=0

where akd (u, v) ∈ C, k ∈ Z+ , u, v ∈ X , each kernel (u, v) ∈ X × X → akd (u, v)  d d is positive definite and ∞ k=0 ak (u, u)Pk (1) < ∞, u ∈ X . Moreover, if f satisfies either (i) or (ii) and d < ∞, then the following representation formula holds for the akd :  akd (u, v) = C(k, d)

1 −1

f (u, v, s)Pkd (s)(1 − s 2 )(d−2)/2 ds, u, v ∈ X ,

in which  C(k, d) =

1 −1



Pkd (s)

2

2 (d−2)/2

(1 − s )

−1 ds

.

Proof If (i) holds, the last assertion in Lemma 2.1 implies that each function f (u, u, ·) is continuous and, in addition, the isotropic part of a positive definite kernel on S d . Schoenberg’s characterization for the continuous and isotropic positive definite kernels on S d provides a series representation for each f (u, u, ·) according to (ii). As a matter of fact, we have that akd (u, u) ≥ 0, u ∈ X . If u, v ∈ X and u = v, Lemma 2.2 implies that all three functions f (u, u, ·) + f (v, v, ·) + f (u, v, ·) + f (v, u, ·), f (u, u, ·) + f (v, v, ·) − f (u, v, ·) − f (v, u, ·) and f (u, u, ·) + f (v, v, ·) − i f (u, v, ·) + i f (v, u, ·)

Journal of Fourier Analysis and Applications

are continuous and, also, the isotropic parts of positive definite kernels on S d . Given that f (u, v, ·) is a linear combination of them (with complex coefficients), it is promptly seen, via Schoenberg’s Theorem once again, that f (u, v, ·) has a series representation as in the statement of the theorem. Next, we verify the positive definiteness of the kernels (u, v) ∈ X × X → akd (u, v), k ∈ Z+ . If y1 , y2 , . . . , y p are distinct points on X and c = (c1 , c2 , . . . , c p ) ∈ C p , Lemma 2.2 reveals that p 

g(s) =

cμ cν f (yμ , yν , s), s ∈ [−1, 1],

μ,ν=1

is real, continuous and the isotropic part of a positive definite kernel on S d . Since g(s) =

∞   p  ct akd (yμ , yν )

μ,ν=1

k=0

c Pkd (s), s ∈ [−1, 1],

Schoenberg’s result along with the usual orthogonality relations for the Pkd in the case d < ∞ [8, P. 10] and standard properties of power series in the case d = ∞, imply that  p c ≥ 0. ct akd (yμ , yν ) μ,ν=1

This shows that (i) implies (ii). Conversely, assume that each f (u, v, ·) has a series representation satisfying the properties given in (ii). Since each kernel (u, v) ∈ X × X → akd (u, v) is positive definite, we may infer that ∞ 

|akd (u, v)Pkd (1)| =

k=0

≤

∞ 

|akd (u, v)|Pkd (1)

k=0 ∞  

1 2

akd (u, u) + akd (v, v) Pkd (1) < ∞, u, v ∈ X .

k=0

In particular, each f (u, v, ·) is a continuous function on [−1, 1]. To conclude the k on [−1, 1] by the formula proof, for u, v ∈ X and k ≥ 0, let us define functions gu,v k (s) = akd (u, v)Pkd (s), s ∈ [−1, 1]. gu,v

For distinct points (u 1 , x1 ), (u 2 , x2 ), . . . , (u n , xn ) in X × S d , 

guk μ ,u ν (xμ · xν )

n μ,ν=1

 n = akd (u μ , u ν )

μ,ν=1

 n • Pkd (xμ · xν )

μ,ν=1

,

in which • denotes the Schur product of matrices. Since both matrices on the righthand side of the equality are positive semi-definite, so is the matrix on the left-hand

Journal of Fourier Analysis and Applications

side, due to the Schur product Theorem. In particular, each kernel ((u, x), (v, y)) ∈ k (x · y) is positive definite. Since positive definiteness is closed (X × S d )2 → gu,v under pointwise limits, the formula ((u, x), (v, y)) ∈ (X × S ) → d 2

∞ 

k gu,v (x · y)

k=0

defines a positive definite kernel. In other words, the kernel ((u, x), (v, y)) ∈ (X × S d )2 → f (u, v, x · y) is positive definite. The last statement of the theorem follows from the orthogonality properties of the Gegenbauer polynomials.

We now illustrate Theorem 2.3 for some relevant choices of the set X . Example 2.4 Let X be endowed with a topology and let f : X × X × [−1, 1] → C be a continuous function such that ((u, x), (v, y)) ∈ (X × S d )2 → f (u, v, x · y) is positive definite. If d < ∞, the integral representation for akd provided by Theorem 2.3 implies that the coefficient functions akd are continuous. Indeed, for each pair (u, v) ∈ X 2 , the set {(u, v, t) : t ∈ [−1, 1]} is compact in X × X × [−1, 1] and then, we can conclude that for any given  > 0, there are open sets U and V in X such that | f (x, y, t) − f (u, v, t)| < , whenever u, x ∈ U and v, y ∈ V . As for the continuity of ak∞ , it is a little bit trickier. First of all, we need to observe that for u, v ∈ X fixed, lim akd (u, v)Pkd (1) = ak∞ (u, v),

d→∞

(2.1)

a property implied by results proved in [20]. On the other hand, proceeding as in Corollary 3 in [11], we can deduce the recurrence relation akd+2 (u, v)Pkd+2 (1) =

(k + d − 1)(k + d) d ak (u, v)Pkd (1) d(2k + d − 1) (k + 1)(k + 2) d d a (u, v)Pk+2 − (1), u, v ∈ X , d(2k + d + 3) k+2

which yields an inequality of the form

C

∞

ak (u, v) − akd (u, v)Pkd (1) ≤ [ f (u, u, 1) + f (v, v, 1)], u, v ∈ X , d where C is a positive constant depending on k. Hence, we have local uniform convergence in (2.1) and the continuity of ak∞ follows from that of the akd , d < ∞. Example 2.5 Let X be a group G with operation ∗ and g : G × [−1, 1] → C a continuous function. Define f (u, v, s) := g(u −1 ∗ v, s), u, v ∈ G, s ∈ [−1, 1].

Journal of Fourier Analysis and Applications

By Theorem 2.3, the kernel ((u, x), (v, y)) ∈ (G × S d )2 → g(u −1 ∗v, x · y) is positive definite if, and only if, g(u −1 ∗ v, s) =

∞ 

akd (u, v)Pkd (s), u, v ∈ G, s ∈ [−1, 1],

k=0

where akd (u, v) ∈ C, k ∈ Z+ , u, v ∈ G, each kernel (u, v) ∈ G × G → akd (u, v)  d d is positive definite, and ∞ k=0 ak (u, u)Pk (1) < ∞, u ∈ G. In the case d < ∞, the integral representation for the coefficient functions akd (u, v) takes the form  akd (u, v) = C(k, d)

1 −1

g(u −1 ∗ v, s)Pkd (s)(1 − s 2 )(d−2)/2 ds, u, v ∈ G.

Obviously, the previous relation implies that akd (u, v) = bkd (u −1 ∗ v), u, v ∈ G, in which  bkd (w) = C(k, d)

1 −1

g(w, s)Pkd (s)(1 − s 2 )(d−2)/2 ds, w ∈ G.

The positive definiteness of (u, v) ∈ G 2 → bkd (u −1 ∗ v) follows from that of (u, v) ∈ G × G → akd (u, v). As for the positive definiteness of (u, v) ∈ G × G → ak∞ (u, v), it follows from the positive definiteness of (u, v) ∈ G × G → akd (u, v), d < ∞, and the limit formula in Example 2.4. If X is a locally compact group, then Examples 2.4 and 2.5 combined is all that is needed in order to recover the main results proved in [2]. Example 2.6 Here we let X be a d -dimensional compact two-point homogeneous space Hd and consider a continuous function g : [−1, 1]2 → C. We define

f (u, v, s) := g(cos(|uv|/2), s), u, v ∈ Hd , s ∈ [−1, 1],

in which |uv| is the usual geodesic distance between u and v in Hd , here normalized so that the diameter of Hd is 2π . Theorem 2.3 is applicable to the kernel ((u, x), (v, y)) ∈ (Hd × S d )2 → f (u, v, x · y). Since for each s fixed, f (u, v, s) depends upon u and v isotropically, additional work via group representations leads to a series representation for the kernel with coefficients given by akd (u, v) =

∞ 



(d −2)/2,β

d ,d bk,l Pl



(cos(|uv|/2)), u, v ∈ Hd ,

l=0

d ,d in which bk,l ≥ 0 for all k and l, β is a rational number attached to the space

(d −2)/2,β

Hd and Pl

is the usual Jacobi polynomial of degree l associated to the pair

Journal of Fourier Analysis and Applications



((d − 2)/2, β) [1]. In the case in which Hd is a sphere S d , the representation above reduces itself to akd (u, v) =

∞ 







d ,d d bk,l Pl (u · v), u, v ∈ S d ,

l=0

in which · is the usual inner product of Rd +1 . The deduction of the formula above in the spherical case is presented in [2] while the general case follows in a similar fashion. In the spherical setting, the series representation for the positive definite kernel ((u, x), (v, y)) ∈ S d × S d )2 → f (u, v, x · y) described above matches the one previously deduced in [15]. For u, v ∈ X fixed, the convergence of the series representation for f (u, v, ·) in Theorem 2.3 is uniform for s ∈ [−1, 1]. We close this section providing a complement to Theorem 2.3, if improved convergence is wanted for the series representation of the kernel. Theorem 2.7 Let ((u, x), (v, y)) ∈ (X × S d )2 → f (u, v, x · y) be a kernel satisfying the conditions in Theorem 2.3(ii). Assume that there exists a sequence {bk } so that akd (u, u) ≤ bk , u ∈ X , k = 0, 1, . . . ,  d and ∞ k=0 bk Pk (1) < ∞. Then the series representation provided by Theorem 2.3 is uniformly convergent for (u, v, s) ∈ X 2 × [−1, 1]. Proof Let  be given and select k0 so that

∞

d k=k0 +1 bk Pk (1)

< . Now observe that

k0 ∞

 

d d ak (u, v)Pk (s) ≤ |akd (u, v)||Pkd (s)|

f (u, v, s) −

k=0

≤

k=k0 +1 ∞ 

|akd (u, v)|Pkd (1)

k=k0 +1

while the positive definiteness of (u, v) ∈ X 2 → akd (u, v) implies that ∞ 

|akd (u, v)|Pkd (1) ≤

k=k0 +1

≤

∞  1   d ak (u, u) + akd (v, v) Pkd (1) 2 k=k0 +1 ∞ 

bk Pkd (1) < .

k=k0 +1

The result follows.



Journal of Fourier Analysis and Applications

3 Strict Positive Definiteness In this section, we will present necessary and sufficient conditions on a positive definite kernel representable as in Theorem 2.3 in order that it be strictly positive definite, making use of the well-established results that characterize continuous and isotropic strictly positive definite kernels on S d . The arguments needed in the proofs of the technical results in this section are adapted from ideas originally developed in [14]. For a subset {u 1 , u 2 , . . . , u p } of X and a subset {x1 , x2 , . . . , xq } of S d which contains no pairs of antipodal points, the enhanced subset of X × S d generated by them is the ordered set {(u 1 , x1 ), (u 2 , x1 ), . . . , (u p , x1 ), (u 1 , x2 ), (u 2 , x2 ), . . . , (u p , x2 ), . . . , (u 1 , xq ), (u 2 , xq ), . . . , (u p , xq ), (u 1 , −x1 ), (u 2 , −x1 ), . . . , (u p , −x1 ), (u 1 , −x2 ), (u 2 , −x2 ), . . . , (u p , −x2 ), . . . (u 1 , −xq ), (u 2 , −xq ), . . . , (u p , −xq )}. The positive numbers p and q in the definition above are independent of each other. We are fixing an order among the elements of the set above so that the writing of the upcoming results can be made easier. It is not hard to see that if B = {(u 1 , x1 ), (u 2 , x2 ), . . . , (u n , xn )} is a subset of X × S d , then there exists an enhanced subset A of X × S d that contains B. Indeed, if {u 1 , u 2 , . . . , u p } encompasses the distinct elements among the u μ , the subset {x1 , x2 , . . . , xq } of S d contains no pairs of antipodal points of S d and satisfies xβ ∈ {±x1 , ±x2 , . . . , ±xq }, β = 1, 2, . . . , n, then the enhanced subset of X × S d generated by {u 1 , u 2 , . . . , u p } and {x1 , x2 , . . . , xq } provides an example. If ((u, x), (v, y)) ∈ (X × S d )2 → f (u, v, x · y) is positive definite and B and A are subsets of X × S d as in the previous paragraph, the interpolation matrix of the kernel at B is a principal sub-matrix of the interpolation matrix of the kernel at A. In this case, the later will be referred to as the augmented interpolation matrix of the kernel at A and will be denoted by I ( f , A). In particular, if I ( f , A) is positive definite, so is the interpolation matrix of the kernel at B. These comments contain the arguments that justify the following lemma. Lemma 3.1 Let ((u, x), (v, y)) ∈ (X × S d )2 → f (u, v, x · y) be a positive definite kernel. It is strictly positive definite if, and only if, for each enhanced subset A of X × S d , the augmented interpolation matrix I ( f , A) is positive definite. For a kernel ((u, x), (v, y)) ∈ (X × S d )2 → f (u, v, x · y) satisfying Theorem 2.3(ii), an augmented interpolation matrix I ( f , A) has an induced decomposition I ( f , A) =

∞ 

I ( f , A, k)

k=0

in which I ( f , A, k) is the interpolation matrix of the kernel ((u, x), (v, y)) ∈ (X × S d )2 → akd (u, v)Pkd (x · y)

Journal of Fourier Analysis and Applications

at A. The order we have fixed for the elements in the enhanced subset A of X × S d determines a special block representation for each matrix I ( f , A, k), that is, I ( f , A, k) = [Mρσ (k)]2ρ,σ =1 , where each block Mρσ (k) has its own block structure αβ Mρσ (k) = [Mρσ (k)]α,β=1 , ρ, σ = 1, 2, q

defined by  p αβ Mρσ (k) = akd (u μ , u ν )

μ,ν=1

(−1)k(σ +ρ) Pkd (xα · xβ ), α, β = 1, 2, . . . , q.

The sign (−1)k(σ +ρ) derives from the parity of Pkd with respect to k. The facts described above and the previous results lead to the following characterization for strict positive definiteness based on augmented interpolation matrices. One item in the theorem depends upon the block sub-matrices  p M11 (k) = akd (u μ , u ν )

μ,ν=1

q Pkd (xα

· xβ )

α,β=1

.

Lemma 3.2 Let ((u, x), (v, y)) ∈ (X × S d )2 → f (u, v, x · y) be a kernel as in Theorem 2.3(ii). The following assertions are equivalent; (i) The kernel is strictly positive definite; (ii) If A is an enhanced subset of X × S d generated by {u 1 , u 2 , . . . , u p } ⊂ X and the subset {x1 , x2 , . . . , xq } of S d containing no pairs of antipodal points, then the only solution (c1 , c2 ) ∈ (C pq )2 to the system [c1 + (−1)k c2 ]t M11 (k)[c1 + (−1)k c2 ] = 0, k ∈ Z+ , is the trivial solution c1 = c2 = 0. (iii) If A is an enhanced subset of X × S d generated by {u 1 , u 2 , . . . , u p } ⊂ X and the subset {x1 , x2 , . . . , xq } of S d containing no pairs of antipodal points, then q q the only vector (c11 , c12 , . . . , c1 , c21 , c22 , . . . , c2 ) of (C p )2q satisfying q  t  p  c1α + (−1)k c2α akd (u μ , u ν ) α,β=1

μ,ν=1

  β k β Pkd (xα · xβ ) = 0, c1 + (−1) c2

for all k ∈ Z+ , is the zero vector. Proof In order to prove (i) and (ii) are equivalent, it suffices to observe that M11 (k) = M22 (k) = (−1)k M12 (k) = (−1)k M21 (k) and to apply Lemma 3.1. The equivalence between (ii) and (iii) follows after we introduce components in the vectors c1 and c2 appearing in (ii).



Journal of Fourier Analysis and Applications

In the next result, we break the main equation in Lemma 3.2(iii) according to the parity of the indices k involved. Proposition 3.3 Let ((u, x), (v, y)) ∈ (X × S d )2 → f (u, v, x · y) be a kernel as in Theorem 2.3(ii). The following assertions are equivalent: (i) The kernel is strictly positive definite; (ii) If p is a positive integer, u 1 , u 2 , . . . , u p are distinct points in X and {x1 , x2 , . . . , xq } is a subset of S d containing no pairs of antipodal points, then the only solution q q (d11 , d12 , . . . , d1 , d21 , d22 , . . . , d2 ) in (C p )2q to the system ⎧ q   p  ⎪ ⎪ β α t d ⎪ ⎪ (d Pkd (xα · xβ ) = 0, k ∈ 2Z+ + 1 a ) (u , u ) d μ ν 1 k ⎪ ⎨ μ,ν=1 1 α,β=1 q   p  ⎪ ⎪ β α t d ⎪ (d Pkd (xα · xβ ) = 0, k ∈ 2Z+ a ) (u , u ) d ⎪ 2 k μ ν ⎪ ⎩ μ,ν=1 2 α,β=1

is the trivial one. Proof For distinct points u 1 , u 2 , . . . , u p in X and a subset {x1 , x2 , . . . , xq } of S d containing no pairs of antipodal points, assume that the system in (ii) has a nontrivial q q solution (d11 , d12 , . . . , d1 , d21 , . . . , d2 ). Defining c1α = −d1α + d2α and c2α = d1α + d2α , q q α = 1, 2, . . . , q, it is promptly seen that (c11 , c12 , . . . , c1 , c21 , c22 , . . . , c2 ) belongs to p 2q (C ) \{0} and that  

q    p β β (c1α + (−1)k c2α )t ak (u μ , u ν ) μ,ν=1 c1 + (−1)k c2 Pkd (xα · xβ ) = 0,

α,β=1

(3.1) for all k ∈ Z+ , that is, Lemma 3.2(iii) does not hold. In particular, the kernel in the statement of the theorem is not strictly positive definite. This shows that (i) implies (ii). The converse follows along the same lines. If (i) does not hold, Lemma 3.2 guarantees the existence of distinct points u 1 , u 2 , . . . , u p in X , a subset {x1 , x2 , . . . , xq } of S d q q containing no pairs of antipodal points and a vector (c11 , c12 , . . . , c1 , c21 , c22 , . . . , c2 ) in (C p )2q \{0} so that (3.1) holds for all k ∈ Z. But the system is equivalent to that one q q q q described in (ii). The vector (c11 −c21 , c12 −c22 , . . . , c1 −c2 , c11 +c21 , c12 +c22 , . . . , c1 +c2 )

of (C p )2q then provides a non-trivial solution to the system in (ii). We are now ready to prove the main results in this section. Before we do so, recall the characterizations of strict positive definiteness on spheres proved in [6,17,18]. Theorem 3.4 Let f be the isotropic part of a continuous and isotropic positive definite kernel on S d and consider its Fourier-Gegenbauer expansion (1.2). The kernel is strictly positive definite if, and only if, the conditions below hold: (i) (d ∈ {2, 3, . . . , ∞}) The set {k : akd > 0} contains infinitely many even and infinitely many odd integers. d > 0} intersects every full arithmetic progression in Z. (ii) (d = 1) The set {k : a|k|

Journal of Fourier Analysis and Applications

Our next theorem provides a characterization of strictly positive definite kernels on X × S d when d = 1. Theorem 3.5 Let d ∈ {2, 3, . . . , ∞} and let ((u, x), (v, y)) ∈ (X ×S d )2 → f (u, v, x· y) be a kernel as in Theorem 2.3(ii). The following assertions are equivalent: (i) The kernel is strictly positive definite; (ii) If p is a positive integer, u 1 , u 2 , . . . , u p are distinct points in X and c is a nonzero vector in C p , then  p {k ∈ Z+ : ct akd (u μ , u ν )

μ,ν=1

c > 0}

contains infinitely many even and infinitely many odd integers. Proof If the kernel in the statement of the theorem is strictly positive definite, then for distinct points u 1 , u 2 , . . . , u p in X , and a nonzero vector c in C p , Lemma 2.2(iii) reveals that s ∈ [−1, 1] →

p 

cμ cν f (u μ , u ν , s)

μ,ν=1

is the isotropic part of a strictly positive definite kernel on S d . On the other hand, it p is an easy matter to verify that the numbers ct akd (u μ , u ν ) μ,ν=1 c are the FourierGegenbauer coefficients in the expansion of the same function. Hence, Theorem 3.4 implies that (ii) holds. Thus, (i) implies (ii). Conversely, assume that (ii) holds but the kernel is not strictly positive definite. We can find distinct points u 1 , u 2 , . . . , u p in X , a subset {y1 , y2 , . . . , yq } of S d containing no pairs of antipodal points and a q q nonzero vector (d11 , d12 , . . . , d1 , d21 , d22 , . . . , d2 ) in (C p )2q so that the two equations q in Proposition 3.3(ii) hold. We will proceed assuming that (d21 , d22 , . . . , d2 ) is nonzero and that q   p  (d2α )t akd (u μ , u ν )

β (d ) μ,ν=1 2

α,β=1

Pkd (yα · yβ ) = 0, k ∈ 2Z+ ,

(3.2)

and will reach a contradiction. The other possibility can be handled similarly. Without loss of generality, we can assume that d21 = 0. Since the set   p k ∈ 2Z+ : (d21 )t akd (u μ , u ν )

μ,ν=1

d21 > 0



is infinite, we can select an infinite subset Q from it and θ ∈ {1, 2, . . . , q} so that  p (d2θ )t akd (u μ , u ν )

μ,ν=1

for α = 1, 2, . . . , q and k ∈ Q.

 p d2θ ≥ (d2α )t akd (u μ , u ν )

dα, μ,ν=1 2

Journal of Fourier Analysis and Applications

Clearly,  p (d2θ )t akd (u μ , u ν )

dθ μ,ν=1 2

> 0, k ∈ Q.

Now for k ∈ Q, rewrite (3.2) as p   (d2α )t akd (u μ , u ν ) μ,ν=1 d2α P d (yα · yα ) k 0 = 1+ p  d θ θ t Pkd (1) α=θ (d2 ) ak (u μ , u ν ) μ,ν=1 d2  p β  (d2α )t akd (u μ , u ν ) μ,ν=1 (d2 ) P d (yα · yβ ) k + , k ∈ Q.   θ )t a d (u , u ) p θ Pkd (1) (d d μ ν α=β 2 k μ,ν=1 2 Since each kernel (u, v) ∈ X × X → akd (u, v) is positive definite, we have that  p (d2α )t akd (u μ , u ν ) μ,ν=1 d2α 0≤ ≤ 1, k ∈ Q, p  (d2θ )t akd (u μ , u ν ) μ,ν=1 d2θ while the Cauchy-Schwarz inequality implies that

α t d  β

(d ) a (u μ , u ν ) p d k

2 μ,ν=1 2 ≤ 1, α = β, k ∈ Q.   p (d2θ )t akd (u μ , u ν ) μ,ν=1 d2θ Since yα · yβ ∈ (−1, 1), α = β, Lemma B1.1 in [8] implies the limit formula lim

Q k→∞

Pkd (yα · yβ ) Pkd (1)

= 0, α = β.

Obviously, the same formula holds for d = ∞. Hence, taking k ∈ Q large enough so that Pkd (yα · yβ ) Pkd (1)

≤

1 , 2

we can conclude that 0 ≥ 1 + 0 − 1/2 = 1/2, a contradiction.



A characterization similar to Theorem 3.5 for the d = 1 case remains open. The easy part is reported below without proof. Theorem 3.6 Let ((u, x), (v, y)) ∈ (X × S 1 )2 → f (u, v, x · y) be a kernel as in Theorem 2.3(ii). If it is strictly positive definite, then for any positive integer p (≤ the

Journal of Fourier Analysis and Applications

cardinality of X ), distinct points u 1 , u 2 , . . . , u p in X and a nonzero vector c in C p , the set   p t 1 k ∈ Z : c a|k| (u μ , u ν ) c>0 μ,ν=1

intersects every full arithmetic progression in Z. The next result depends upon some new notation. For a positive definite kernel ((u, x), (v, y)) ∈ (X × S d )2 → f (u, v, x · y) as in Theorem 2.3(ii), we introduce the truncated functions  d aγo (u, v) = a2k+1 (u, v), u, v ∈ X 2k+1≥γ

and 

aγe (u, v) =

d a2k (u, v), u, v ∈ X .

2k≥γ

Since Pkd (1) ≥ 1, d ≥ 2, k ≥ 0, and each kernel (u, v) ∈ X × X → akd (u, v) is positive definite, we have that



k2 k2



 d

d

d

≤

a (u, v) (u, v)

a

Pk (1) 2k 2k

k=k1

k=k1 ≤

k2 

d d [a2k (u, u) + a2k (v, v)]Pkd (1), u, v ∈ X , k2 ≥ k1 .

k=k1

 d d e Since ∞ k=0 ak (u, u)Pk (1) < ∞, u ∈ X , we may infer that each function aγ is well defined. A similar argument leads to the same conclusion for the functions aγo . Our final characterization for strict positive definiteness is based upon the two functions introduced above. Theorem 3.7 Let d ∈ {2, 3, . . . , ∞} and let ((u, x), (v, y)) ∈ (X ×S d )2 → f (u, v, x· y) be a kernel as in Theorem 2.3(ii). The following assertions are equivalent: (i) The kernel is strictly positive definite; (ii) For each γ ≥ 0, the kernels (u, v) ∈ X 2 → aγo (u, v) and (u, v) ∈ X 2 → aγe (u, v) are strictly positive definite. Proof If u 1 , u 2 , . . . , u p are distinct points in X and c ∈ C p , we have that  p ct aγo (u μ , u ν )

μ,ν=1

c=

 2k+1≥γ

 p ct ak (u μ , u ν ) μ,ν=1 c.

Journal of Fourier Analysis and Applications

Hence, if the kernel is strictly positive definite and c = 0, then Theorem 3.5 implies that  p ct aγo (u μ , u ν ) c > 0. μ,ν=1

In particular, (u, v) ∈ X 2 → aγo (u, v) is strictly positive definite. A similar argument leads to the strict positive definiteness of (u, v) ∈ X 2 → aγe (u, v). Conversely, if both kernels in (ii) are strictly positive definite, it is promptly seen that Condition (ii) in Theorem 3.5 holds. Therefore, the kernel is strictly positive definite.

Altering a little bit the definition of the functions used in the Theorem 3.7, one can prove the following result. Theorem 3.8 Let ((u, x), (v, y)) ∈ (X × S 1 )2 → f (u, v, x · y) be a kernel as in Theorem 2.3(ii). If the kernel is strictly positive definite, then for each full arithmetic progression P of Z, the kernel (u, v) ∈ X 2 →



1 a|k| (u, v), u, v ∈ X ,

k∈P

is strictly positive definite. A converse of Theorem 3.8 is clearly dependent on a converse of Theorem 3.6, which is still open. We conclude this section with a useful sufficient condition for strict positive definiteness. Theorem 3.9 Let ((u, x), (v, y)) ∈ (X × S d )2 → f (u, v, x · y) be a kernel as in Theorem 2.3(ii). In order that it be strictly positive definite it is sufficient that: (i) (d ∈ {2, 3, . . . , ∞}) The kernels (u, v) ∈ X 2 → akd (u, v) be strictly positive definite for infinitely many even k and infinitely many odd k. 1 (u, v) is strictly positive definite} (ii) (d = 1) The set {k : (u, v) ∈ X × X → a|k| intersects every full arithmetic progression in Z. Proof Assertion (i) follows from Theorem 3.7. As for (ii), assume that 1 K := {k : (u, v) ∈ X × X → a|k| (u, v) is strictly positive definite}

intersects each full arithmetic progression in Z and let A be an enhanced subset of X × S 1 . We will show that the augmented interpolation matrix I ( f , A) is positive definite. If A is generated by {u 1 , u 2 , . . . , u p } ⊂ X and {x1 , x2 , . . . , xq } ⊂ S 1 , the quadratic form associated to I ( f , A) is q ∞   k=0 α,β=1

p

cαt [ak1 (u μ , u ν )]μ,ν=1 cβ Pk1 (xα · xβ ), cα ∈ C p , α = 1, 2, . . . , q.

Journal of Fourier Analysis and Applications

If it is zero, then q  α,β=1

p

cαt [ak1 (u μ , u ν )]μ,ν=1 cβ Pk1 (xα · xβ ) = 0, k ∈ Z+ .

(3.3)

Writing xα = (cos θα , sin θα ), α = 1, 2, . . . , p, for distinct angles in [0, 2π ) and invoking the definition of Pk1 (they are Tchebyshev polynomials), Equation (3.3) implies that q 

p

α,β=1

[cos(|k|θα )cα ]t [ak1 (u μ , u ν )]μ,ν=1 [cos(|k|θβ )cβ ] +

q  α,β=1

p

[sin(|k|θα )cα ]t [ak1 (u μ , u ν )]μ,ν=1 [sin(|k|θβ )cβ ] = 0, k ∈ K .

The definition of K leads to cos |k|θα cα = sin |k|θα cα = 0, α = 1, 2, . . . , q, k ∈ K , and, consequently, q 

ei|k|θα cα = 0, k ∈ K .

α=1 p

Writing cα = (cα1 , cα2 , . . . , cα ), α = 1, 2, . . . , q, it is promptly seen that q  α=1

cαμ ei|k|θα = 0, k ∈ K ; μ = 1, 2, . . . , p.

The symmetric nature of K leads to q  α=1

(Re

cαμ )ei|k|θα

=

q  α=1

(Im cαμ )ei|k|θα = 0, k ∈ K ; μ = 1, 2, . . . , p.

Due to our assumption on K and results from the Appendix section in [1], we conclude μ that all the cα are zero.



4 An Application to the Case in Which X is the Group (Rm , +) In this section, we discuss two implications of the results in Sect. 3 for the case in which X is the group (Rm , +). Expanding a little bit what we have mentioned about positive definiteness on R at the introduction, a result of Bochner [5] states that if

Journal of Fourier Analysis and Applications

f : Rm → C is continuous, then the kernel (u, v) ∈ (Rm )2 → f (v − u) is positive definite if, and only if, f is the Fourier transform of a finite positive Borel measure on Rm . If we move to strict positive definiteness, we have the following specialization [23, P. 74]. Theorem 4.1 Let g : Rm → C be continuous and integrable. The kernel (u, v) ∈ (Rm )2 → g(v − u) is strictly positive definite if, and only if, g is bounded and its Fourier transform is nonnegative and not identically zero. Taking into account what we have proved so far, we can demonstrate the following results in the setting Rm × S d . We will use notation from Example 2.5 with (G, ∗) = (Rm , +). Theorem 4.2 Let d ∈ {2, 3, . . . , ∞} and let f : Rm × [−1, 1] → C be continuous. Assume that the kernel ((u, x), (v, y)) ∈ (Rm × S d )2 → f (v − u, x · y) is positive definite and consider the series representation for f according to Example 2.5. If each bkd is integrable in Rm , then the following assertions are equivalent: (i) The kernel is strictly positive definite; (ii) The set {k ∈ Z+ : bkd ≡ 0} contains infinitely many even and infinitely many odd integers. Proof If (i) holds, Theorem 3.7 reveals that, for each γ ≥ 0, the kernels 

(u, v) ∈ (Rm )2 → aγo (u, v) :=

d b2k+1 (v − u)

2k+1≥γ

and (u, v) ∈ (Rm )2 → aγe (u, v) :=



d b2k+1 (v − u)

2k≥γ

are strictly positive definite. In particular, for each γ ≥ 0, there are indices k1 , k2 ≥ γ d d are not identically zero. Hence, (ii) must hold. and b2k so that the functions b2k 1 +1 2 Next, assume (ii) holds. Since Theorem 2.3 guarantees that each (u, v) ∈ (Rm )2 → bkd (v − u) is positive definite, it follows that |bkd (w)| ≤ bkd (0), w ∈ Rm . In particular, each bkd is a bounded continuous function (continuity comes from Example 2.5). Hence, if we assume each bkd is integrable in Rm , an application of Corollary 6.12 in [23] leaves us with two possibilities: for each k, either the Fourier transform of bkd vanishes or (u, v) ∈ (Rm )2 → bkd (v−u) is strictly positive definite. Since a continuous integrable positive definite function can be recovered from its Fourier transform by integration, the previous conclusion corresponds to the following statement: for each k, either bkd vanishes or (u, v) ∈ (Rm )2 → bkd (v − u) is strictly positive definite. It is now clear that if (ii) holds, then Condition (ii) in Theorem 3.7 holds as well and, consequently, the kernel ((u, x), (v, y)) ∈ (Rm × S d )2 → f (v − u, x · y) is strictly positive definite.



Journal of Fourier Analysis and Applications

We now examine the case where d = 1 in Theorem 4.2, i.e., strictly positive definite kernels on Rm × S 1 . Theorem 4.3 Let f : Rm × [−1, 1] → C be continuous and assume that the kernel ((u, x), (v, y)) ∈ (Rm × S 1 )2 → f (v − u, x · y) is positive definite. If each bk1 is integrable in Rm , then the following assertions are equivalent: (i) The kernel is strictly positive definite; (ii) The set 1 {k ∈ Z+ : b|k| ≡ 0}

intersects each full arithmetic progression in Z. Proof Theorem 3.8 along with the very same procedure employed in the second half of the proof of Theorem 4.2 settles that (i) implies (ii). In order to prove the converse, we follow the steps in the proof of Theorem 4.2 until the point we reach that for each k, either bk1 vanishes or (u, v) ∈ (Rm )2 → bk1 (v − u) is strictly positive definite. It is now clear that the result follows from Theorem 3.9.

We now look at the case in which the function f is radial. Recall that a function g : Rm → C is said to be radial if g(u) = g(v) whenever u, v ∈ Rm and u = v ( here  ·  denotes the usual norm in Rm ). Similarly, we will say that a continuous function f : Rm × [−1, 1] → C is radial if f (u, t) = f (v, t), whenever u, v ∈ Rm , u = v and t ∈ [−1, 1]. We now specialize Theorem 4.3 to the case where f is radial. Note that for any radial function g : Rm → C, there exists a function gr : [0, ∞) → C so that g(u) = gr (u), u ∈ Rm . Theorem 4.4 Let m ≥ 2, d ∈ {2, 3, . . . , ∞}, and f : Rm ×[−1, 1] → C a continuous radial function for which the kernel ((u, x), (v, y)) ∈ (Rm × S d )2 → f (v − u, x · y) is positive definite. The following assertions are equivalent: (i) The kernel ((u, x), (v, y)) ∈ (Rm × S d )2 → f (v − u, x · y) is strictly positive definite. (ii) The set {k ∈ Z+ : (bkd )r is nonconstant} contains infinitely many even and infinitely many odd integers. Proof It suffices to observe that for m ≥ 2, the main result in [22] reveals that a continuous and positive definite kernel (u, v) ∈ Rm × Rm → g(v − u) is strictly positive definite if, and only if, g is not constant.

A similar version holds when d = 1, but is omitted.

Journal of Fourier Analysis and Applications

5 The Case in Which X is a Torus In this section, we consider the case in which X is the m-dimensional torus Tm as defined in the introduction. The Fourier analysis on the torus closest to the material described in the introduction and here in this section is developed in [21]. We will keep the notation in Example 2.5, replacing (G, ∗) with (Tm , +). If g : Rm × [−1, 1] → C is a continuous function which is 2π -periodic in (each component of) Rm , Examples 2.4 and 2.5 imply that the kernel ((u, x), (v, y)) ∈ (Tm × S d )2 → g(v − u, x · y) is positive definite if, and only if, g(u, s) =

∞ 

bkd (u)Pkd (s), u ∈ Tm , s ∈ [−1, 1],

k=0

where each function bkd : Tm → C is continuous, each kernel (u, v) ∈ Tm × Tm →  d d bkd (v − u) is positive definite, and ∞ k=0 bk (e)Pk (1) < ∞. The characterization of the continuous positive definite kernels on Tm (see [21]) leads to bkd (w) =



d cn,k ei(n·w) , w ∈ Tm ,

n∈Zm

 d < ∞, with nonnegative coefficients cd depending upon n, k and where n∈Zm cn,k n,k d. We collect these findings in the theorem below. Theorem 5.1 Let g : Rm × [−1, 1] → C be a continuous function which is 2π periodic in Rm . The kernel ((u, x), (v, y)) ∈ (Tm × S d )2 → g(v − u, x · y) is positive definite if, and only if, g(w, t) =

∞  

d cn,k ei(n·w) Pkd (t), (w, t) ∈ Tm × [−1, 1],

k=0 n∈Zm d ≥ 0, (n, k) ∈ Zm × Z , and ∞  d d in which cn,k + k=0 n∈Zm cn,k Pk (1) < ∞. In the case d < ∞, the coefficients in the expansion are computable through the formula

 d cn,k



= c(k, d, m) Tm

1 −1

ei(n·u) g(u, t)Pkd (t)(1 − t 2 )(d−2)/2 dtdu,

for some positive constant c(k, d, m) depending upon k, d and m. The theorem above can also be extracted from Corollary 3.5 in [2], a fact that the interested reader can easily verify. Using the results in Sect. 3 and the characterization of strictly positive definite kernels on Tm presented in Sect. 1, we also obtain the following criterion for strict positive definiteness on Tm × S d .

Journal of Fourier Analysis and Applications

Theorem 5.2 Let d ∈ {2, 3, . . . , ∞} and let g : Rm × [−1, 1] → C be a continuous function which is 2π -periodic in Rm . Assume that the kernel ((u, x), (v, y)) ∈ (Tm × S d )2 → g(v − u, x · y) is positive definite and consider its representation as provided by Theorem 5.1. The following assertions are equivalent: (i) The kernel is strictly positive definite. (ii) The sets   d {(n, k) : ck,n > 0 and k ≥ γ } ∩ 2Z+ × (a1 Z + j1 , a2 Z + j2 , . . . , am Z + jm ) and d {(k, n) : cn,k > 0 and k ≥ γ }   ∩ (2Z+ + 1) × (a1 Z + j1 , a2 Z + j2 , . . . , am Z + jm )

are non empty whenever γ ≥ 0, a1 , a2 , . . . , am ∈ Z+ \{0} and ji ∈ {0, 1, . . . , ai }, i = 1, 2, . . . , m. Acknowledgements The authors want to thank two independent referees for useful comments and suggestions. The second author acknowledges partial support by FAPESP, Grant 2016/09906-0.

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