Relations between Schoenberg Coefficients on Real and Complex Spheres of Different Dimensions

Positive definite functions on spheres have received an increasing interest in many branches of mathematics and statistics. In particular, the Schoenberg sequences in the spectral representation of positive definite functions have been studied by several mathematicians in the last years. This paper provides a set of relations between Schoenberg sequences defined over real as well as complex spheres of different dimensions. We illustrate our findings describing an application to strict positive definiteness.

Fourier analysis on spheres is related to the so called Schoenberg sequences (see Daley and Porcu, 2013, also called sequences of Schoenberg coefficients) that are related to the dimension where any positive definite function on real or complex spheres is defined. There has been a recent interest on Schoenberg sequences, especially after the list of research problems in Gneiting (2013) and in Porcu et al. (2018a). Recursive relations between Schoenberg coefficients on d-dimensional spheres have been first proposed by Gneiting (2013). Fiedler (2014)

has then
solved an open problem in Gneiting (2013), related to other types of recursions involving Schoenberg coefficients. Ziegel (2014) has used Schoenberg sequences to find the convolution roots of positive definite functions on spheres. Truebner and Ziegel (2017) illustrated the differentiability properties of positive definite functions on spheres through their Schoenberg sequences. Recently, Arafat et al. (2017) have solved Gneiting's research problem number 3 making extensive use of Schoenberg sequences. Projections from Hilbert into finite dimensional spheres have been considered by Møller et al. (2017). Finally, Schoenberg sequences have been shown to be central to the study of geometric properties of Gaussian fields on spheres (Lang and Schwab, 2013) or spheres cross time (Lang and Schwab, 2013). This last, on the basis of the fundamental result in Berg and Porcu (2017).
Literature on complex spheres has been sparse. After the tour de force in Menegatto (2014) there has been a recent interest on complex spheres as reported from Berg et al. (2018) and in Massa et al. (2017).
This paper completes the picture about Schoenberg sequences on spheres of R d and C q , respectively. Specifically, we show recursive relations that have been lacking from the previously mentioned literature. Section 2 deals with real valued d-dimensional spheres.
Section 3 is instead related to complex spheres. Some implications in terms of strict positive definiteness are provided in Section 4. The paper ends with a discussion.

Background and Notation
For a positive integer d, let S d = {x ∈ R d+1 , x = 1} denote the d-dimensional unit sphere embedded in R d+1 , with · being the Euclidean norm. We define the great circle distance as the continuous mapping defined through for all n ≥ 1, distinct points x 1 , x 2 , . . . , x n on S d and real numbers c 1 , . . . , c n , is called positive definite. Further, if the inequality is strict, unless the vector (c 1 , . . . , c n ) ⊤ is the zero vector, then it is called strictly positive definite (see Bingham, 1973, with the references therein). If, in addition, for some mapping ψ : [0, π] → R, then C is called a geodesically isotropic covariance by Porcu et al. (2018b). With no loss of generality, we assume through the paper that the function ψ is continuous along with the normalization ψ(0) = 1. Gneiting (2013) calls Ψ d the class of continuous functions ψ : [0, π] → R with ψ(0) = 1 such that the function C in Equation (1) is positive definite. The inclusions Ψ d ⊃ Ψ d+1 , d ≥ 1, are known to be strict. Following Schoenberg (1942), for every continuous function ψ : [0, π] → R with ψ(0) = 1, and every integer d ≥ 2, define where, for any λ > 0, C λ n denotes the n-th Gegenbauer polynomial of order λ (Abramowitz and Stegun, 1964), and Moreover, we define Note that in the cases d = 1 (the circle) and d = 2 (the unit sphere of R 3 ), Gegenbauer polynomials simplify to Chebyshev and Legendre polynomials (Abramowitz and Stegun, 1964), respectively.
The coefficient sequences {b n,d } ∞ n=0 play a crucial role in the spectral representations for positive definite functions on spheres, which are the equivalent of Bochner and Schoenberg's theorems in Euclidean spaces (see Daley and Porcu, 2013, with the references therein) and are provided by Schoenberg (1942), who shows that a mapping ψ : [0, π] → R belongs to the class Ψ d if and only if it can be uniquely written as where c λ n denotes the normalized λ-Gegenbauer polynomial of degree n, namely, and {b n,d } ∞ n=0 is a probability mass sequence. The series (5) is known to be uniformly convergent. We follow Daley and Porcu (2013) when calling the sequence {b n,d } ∞ n=0 in (5) the d-Schoenberg sequence of coefficients, to emphasize the dependence on the index d in the class Ψ d . Accordingly, we say that (ψ, {b n,d }) is a uniquely determined d-Schoenberg pair if ψ belongs to the class Ψ d and admits the expansion (5) with d-Schoenberg sequence {b n,d } ∞ n=0 . The following recursive relations among the coefficients b n,d and b n,d+2 attached to a d-Schoenberg pair (ψ, {b n,d+2 }) (Gneiting, 2013 Møller et al. (2017). Yet, there are some relations that have not been discovered and these will be illustrated throughout.

Results
We start with a very simple result, that we report formally for the convenience of the reader.
pair, then the d ′ -Schoenberg sequence of coefficients of ψ is uniquely determined as follows.
We are not aware of any closed form expression for the integrals appearing in (10) and (9), and therefore of the relationships between the sequences {b n,d } ∞ n=0 and {b n,d ′ } ∞ n=0 attached to a d ′ -Schoenberg pair (ψ, b n,d ′ ), apart from the specific case where d ′ = d+2. Indeed, Gneiting (2013) provides a closed form expression for {b n,d+2 } ∞ n=0 as a function of {b n,d } ∞ n=0 that is given by (6)-(8). An explicit expression for the inverse function is provided throughout.
Theorem 2. If (ψ, {b n,3 }) is a 3-Schoenberg pair, then the 1-Schoenberg sequence of coefficients of ψ is given by Proof. From Identity (7), if (ψ, {b n,3 }) is a 3-Schoenberg pair, we have that Hence, for every nonnegative integer j, and for any positive integer n, Summing up both sides of (13) from 0 to m, we obtain We now use the fact that the right-hand side in Equation (13) Since ψ belongs to Ψ 1 , the series ∞ n=0 b n,1 converges to 1 and, therefore, the sequence {b n,1 } ∞ n=0 converges to zero. We can thus take the limit for m → ∞ in Equation (15) and this will provide (12). In particular, we now take n = 2 to deduce that b 2,1 = 2 ∞ j=1 b 2j,3 /{1+2j} which combined with (6) yields (11).
We are now able to provide an extension of Theorem 2 for d > 3. For a positive integer m and x > 0, (x) m will denote the standard rising factorial (Pochhammer symbol).
The proof is completed.

Schoenberg Sequences on Complex Spheres
In analogy with the results obtained in Section 2, we consider similar results on complex spheres.

Background and Notation
For a positive integer q, denote by Ω 2q the unit sphere in C q . A mapping C : for all n ≥ 1, distinct points z 1 , . . . , z n of Ω 2q and complex numbers c 1 , . . . , c n . Let "·" denote the usual inner product in C q . If q ≥ 2 and for some function ϕ : B[0, 1] → C. This nomenclature is not universal but it is quite adequate in our setting. Observe that in the case q = 1, if z, w ∈ Ω 2 , then z·z ∈ Ω 2 . Hence, the previous definition becomes an extreme case once the domain of ϕ needs to be Ω 2 itself.
Keeping the analogy with the previous section, for q ≥ 2, we call Υ 2q the class of continuous functions ϕ, with ϕ(1) = 1 such that C in (20) is positive definite. For a neater illustration of the applications in Section 4, we call Υ + 2q the class of functions ϕ belonging to Υ 2q such that C in (20) is strictly positive definite. Both classes Υ 2q and Υ + 2q are nested, that is, if q ≤ q ′ , then Υ 2q ′ ⊂ Υ 2q and Υ + 2q ′ ⊂ Υ + 2q . To present the characterization of the class Υ 2q described in Menegatto and Peron (2001), we denote by R q−2 m,n the disk polynomial of bi-degree (m, n) with respect to the nonnegative integer q − 2. The set {R q−2 m,n : m, n = 0, 1, . . .} is a complete orthogonal system in L 2 (B[0, 1], ν q−2 ), with with i being the complex number such that i 2 = −1. In particular, with δ denoting the Kronecker delta, and where Expressions and main properties of disk polynomials can be found in Wünsche (2005) and in references quoted there. We recall the following recursion satisfied for every z in B[0, 1], m ≥ 1 and n ≥ 0 (Menegatto, 2014): For every continuous function ϕ : B[0, 1] → C and every triplet (m, n, q) of nonnegative integers, we can define The functions belonging to the class Υ 2q are uniquely characterized through the expansion (Menegatto and Peron, 2001) where a q−2 m,n ≥ 0, m, n ∈ Z + and ∞ m,n=0 a q−2 m,n = 1. Following Section 2, we finally define a 2q-Schoenberg pair (ϕ, {a q−2 m,n }) any function belonging to the class Υ 2q with expansion defined according to (26). In this case, the double sequence {a q−2 m,n } ∞ m,n=0 will be called the 2q-Schoenberg sequence of coefficients of ϕ.

Results
Since the classes Υ 2q are nested, here we prove a recursive relation among the coefficients a q−1 m,n and a q−2 m,n attached to a 2(q + 1) Schoenberg pair (ϕ, {a q−1 m,n }) that resembles (8).

This yields (27).
Here is the main result of the section.
4 Applications Involving the Classes Ψ + d and Υ + 2q In this section, we present applications of the previous results involving the classes Ψ + d and Υ + 2q .
A similar result holds for real spheres with a similar proof. In particular, if ψ belongs to Ψ d ∩ Ψ + d ′ , then ψ belongs to Ψ + d . However, this result was proved earlier in Corollary 1 in Gneiting (2013) via a slightly different argument.
Theorem 6 allows the following obvious consequences. If ϕ is a function in Υ 2q , we write ϕ r to indicate the restriction of ϕ to [−1, 1].

Discussion
This paper has completed the picture about the classes Ψ d , Υ 2q and Υ + 2q in terms of their Schoenberg sequences. Yet, there are many challenges that involve Schoenberg sequences, for instance in product spaces. Berg and Porcu (2017) consider the analogue of Schoenberg pairs introduced in this paper, but on the product space S d × G, for G a locally compact group.
Generalizations of the results in Berg and Porcu (2017) have been provided by Guella et al. (2017). It would be very interesting to inspect whether the results provided in this paper can be generalized to these cases. Another important challenge would be to inspect the Schoenberg pairs related to matrix valued kernels (see open problem 2 in Porcu et al., 2018a).