Positive Definite Functions on Complex Spheres and their Walks through Dimensions

We provide walks through dimensions for isotropic positive definite functions defined over complex spheres. We show that the analogues of Montée and Descente operators as proposed by Beatson and zu Castell [J. Approx. Theory 221 (2017), 22-37] on the basis of the original Matheron operator [Les vari...

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Datum:	2017
Hauptverfasser:	Massa, E., Peron, A.P., Porcu, E.
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spelling	irk-123456789-1492632019-02-23T01:23:14Z Positive Definite Functions on Complex Spheres and their Walks through Dimensions Massa, E. Peron, A.P. Porcu, E. We provide walks through dimensions for isotropic positive definite functions defined over complex spheres. We show that the analogues of Montée and Descente operators as proposed by Beatson and zu Castell [J. Approx. Theory 221 (2017), 22-37] on the basis of the original Matheron operator [Les variables régionalisées et leur estimation, Masson, Paris, 1965], allow for similar walks through dimensions. We show that the Montée operators also preserve, up to a constant, strict positive definiteness. For the Descente operators, we show that strict positive definiteness is preserved under some additional conditions, but we provide counterexamples showing that this is not true in general. We also provide a list of parametric families of (strictly) positive definite functions over complex spheres, which are important for several applications. 2017 Article Positive Definite Functions on Complex Spheres and their Walks through Dimensions / E. Massa, A.P. Peron, E. Porcu // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 49 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 42A82; 42C10; 42C05; 30E10; 62M30 DOI:10.3842/SIGMA.2017.088 http://dspace.nbuv.gov.ua/handle/123456789/149263 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We provide walks through dimensions for isotropic positive definite functions defined over complex spheres. We show that the analogues of Montée and Descente operators as proposed by Beatson and zu Castell [J. Approx. Theory 221 (2017), 22-37] on the basis of the original Matheron operator [Les variables régionalisées et leur estimation, Masson, Paris, 1965], allow for similar walks through dimensions. We show that the Montée operators also preserve, up to a constant, strict positive definiteness. For the Descente operators, we show that strict positive definiteness is preserved under some additional conditions, but we provide counterexamples showing that this is not true in general. We also provide a list of parametric families of (strictly) positive definite functions over complex spheres, which are important for several applications.
format	Article
author	Massa, E. Peron, A.P. Porcu, E.
spellingShingle	Massa, E. Peron, A.P. Porcu, E. Positive Definite Functions on Complex Spheres and their Walks through Dimensions Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Massa, E. Peron, A.P. Porcu, E.
author_sort	Massa, E.
title	Positive Definite Functions on Complex Spheres and their Walks through Dimensions
title_short	Positive Definite Functions on Complex Spheres and their Walks through Dimensions
title_full	Positive Definite Functions on Complex Spheres and their Walks through Dimensions
title_fullStr	Positive Definite Functions on Complex Spheres and their Walks through Dimensions
title_full_unstemmed	Positive Definite Functions on Complex Spheres and their Walks through Dimensions
title_sort	positive definite functions on complex spheres and their walks through dimensions
publisher	Інститут математики НАН України
publishDate	2017
url	http://dspace.nbuv.gov.ua/handle/123456789/149263
citation_txt	Positive Definite Functions on Complex Spheres and their Walks through Dimensions / E. Massa, A.P. Peron, E. Porcu // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 49 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT massae positivedefinitefunctionsoncomplexspheresandtheirwalksthroughdimensions AT peronap positivedefinitefunctionsoncomplexspheresandtheirwalksthroughdimensions AT porcue positivedefinitefunctionsoncomplexspheresandtheirwalksthroughdimensions
first_indexed	2025-07-12T21:11:15Z
last_indexed	2025-07-12T21:11:15Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 088, 16 pages Positive Definite Functions on Complex Spheres and their Walks through Dimensions Eugenio MASSA †, Ana Paula PERON † and Emilio PORCU ‡§ † Departamento de Matemática, ICMC-USP - São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil E-mail: eug.massa@gmail.com, apperon@icmc.usp.br ‡ School of Mathematics and Statistics, Chair of Spatial Analytics Methods, University of Newcastle, UK E-mail: emilio.porcu@newcastle.ac.edu § Department of Mathematics, Universidad Técnica Federico Santa Maria, Avenida España 1680, Valparáıso, 230123, Chile Received April 06, 2017, in final form October 30, 2017; Published online November 08, 2017 https://doi.org/10.3842/SIGMA.2017.088 Abstract. We provide walks through dimensions for isotropic positive definite functions defined over complex spheres. We show that the analogues of Montée and Descente operators as proposed by Beatson and zu Castell [J. Approx. Theory 221 (2017), 22–37] on the basis of the original Matheron operator [Les variables régionalisées et leur estimation, Masson, Paris, 1965], allow for similar walks through dimensions. We show that the Montée operators also preserve, up to a constant, strict positive definiteness. For the Descente operators, we show that strict positive definiteness is preserved under some additional conditions, but we provide counterexamples showing that this is not true in general. We also provide a list of parametric families of (strictly) positive definite functions over complex spheres, which are important for several applications. Key words: Descente; disk polynomials; Montée; positive definite functions 2010 Mathematics Subject Classification: 42A82; 42C10; 42C05; 30E10; 62M30 1 Introduction and main results Positive definite functions have a long history which can be traced back to papers by Carathéo- dory, Herglotz, Bernstein and Matthias, culminating in Bochner’s theorem from 1932–1933. See Berg [6] for details. In the last twenty years several results related to this topic were obtained in fields as diverse as mathematical analysis, numerical analysis, potential theory, probability theory and geostatistics: we refer the reader to the surveys in Schaback [35, 36], Berg [6] and Fasshauer [14] for a complete list of references in this direction. Positive definite radial functions have been known since the two seminal papers by Schoen- berg [39, 40]. The former is devoted to radially symmetric functions depending on the Euclidean distance, and the latter to isotropic functions on unit spheres Sd of Rd+1. Literature on radially symmetric functions on Euclidean spaces has been especially fervent. In his essay devoted to the clavier spherique, Matheron [24] proposed operators called Montée and Descente that preserve the property of positive definiteness but changing the dimension of the space initially considered. Such a property has been called walk through dimensions. It is worth noting that the walk through dimensions is achieved at the expense of modifying the differentiability at the origin of a given candidate function. Wendland [45] used the Montée operator with a class of compactly supported radial basis functions, termed Wendland’s functions after his works. Schaback [37] mailto:eug.massa@gmail.com mailto:apperon@icmc.usp.br mailto:emilio.porcu@newcastle.ac.edu https://doi.org/10.3842/SIGMA.2017.088 2 E. Massa, A.P. Peron and E. Porcu covered the missing cases of walks through dimensions. Porcu et al. [30] used a fractional version of the Montée operator to obtain generalized versions of Wendland’s functions. For a reference on walks through dimensions in the geostatistical setting, the reader is referred to Gneiting [16] and to the more recent work of Porcu and Zastavnyi [29]. Positive definite functions as well as strictly positive definite functions in several contexts have been deeply studied by the mathematical analysis literature, and the reader is referred to the works by Menegatto et al. (see Chen et al. [11], Menegatto and Peron [26], Guella et al. [19], and references therein). The use of positive definite functions on real spheres for geostatisticians has arrived recently, thanks to the survey by Gneiting [17] and the recent developments by Berg and Porcu [7] and Porcu et al. [28]. In particular, Berg and Porcu [7] characterized the class of the positive definite functions on the product of Sd with a locally compact group, extending the Schoenberg’s class Ψd of the positive definite functions on Sd (Schoenberg [40]). A continuous function f : [−1, 1]→ R belongs to the class Ψd when the kernel K : Sd × Sd → R : K(ξ, η) = f(〈ξ, η〉) is positive definite. Schoenberg [40] proved that f ∈ Ψd if, and only if, f(x) = ∑ k≥0 adkck(d, x), ∑ k≥0 adk <∞, adk ≥ 0, ∀ k ≥ 0, (1.1) where ck(d, ·) are the normalized Gegenbauer polynomials associated to the index d (see Szegő [44, p. 80]). The coefficients in the above series are called d-Schoenberg coefficients. On the other hand, the subclass Ψ+ d of Ψd of the strict positive definite functions on Sd, d ≥ 2, was characterized by Chen et al. [11]: f ∈ Ψ+ d if, and only if, the set {k : adk > 0} contains infinitely many odd and infinitely many even integers. The class Ψd has received special interest in the last twenty years, while walks through dimensions for positive definite functions on real spheres have been studied in the recent tour de force by Beatson and zu Castell [3, 4]. In particular, Beatson and zu Castell [4] define the Montée operator (If)(x) = ∫ x −1 f(u)du, x ∈ [−1, 1], for f integrable in [−1, 1], and the Descente operator (Df)(x) = d dx f(x), x ∈ [−1, 1], for f absolutely continuous in [−1, 1]. They prove that, for d ≥ 2: (i) if f ∈ Ψd+2, then there exists a constant c such that c+ If ∈ Ψd; (ii) if f ∈ Ψ+ d+2, then there exists a constant c such that c+ If ∈ Ψ+ d ; (iii) if f ∈ Ψd+2, f ≥ 0 and all (d+ 2)-Schoenberg coefficients are positive, then If ∈ Ψd and all its d-Schoenberg coefficients are positive; (iv) if f ∈ Ψd and Df is continuous, then Df ∈ Ψd+2; (v) if f ∈ Ψ+ d and Df is continuous, then Df ∈ Ψ+ d+2. Observe that the property of (strict) positive definiteness of f is preserved by the operators Montée I and Descente D. In this paper, inspired by the work of Beatson and zu Castell [4], we study positive definite functions on complex unit spheres Ω2q of Cq. In particular, we provide walks through dimensions over complex spheres. Positive Definite Functions on Complex Spheres and their Walks through Dimensions 3 Below, we state our main results and we refer to Section 2 for the necessary background. We denote the class of positive definite functions on Ω2q by Ψ(Ω2q). A characterization of such functions was proposed in Menegatto and Peron [26]: let D := {z ∈ C : \|z\| ≤ 1} ⊂ C, when a continuous function f : D→ C belongs to Ψ(Ω2q), an expansion similar to (1.1) exists, namely f(z) = ∑ m,n≥0 aq−2 m,nR q−2 m,n(z), z ∈ D, (see equation (2.3) and Theorem 2.1). We will call the coefficients aq−2 m,n as (2q)-complex Schoen- berg coefficients. In order to make the statements clear, it is convenient to introduce the Descente and Montée operators in the complex context. Given f : D→ C, we say that f is differentiable if, writing z = x+ iy ∈ D, f is differentiable as a function of x and y. Then, we denote by Dxf and Dyf the partial derivatives with respect to x and y, respectively, and we define the Descente operators through the following Wirtinger derivatives: Dzf = 1 2 (Dxf − iDyf), Dzf = 1 2 (Dxf + iDyf). (1.2) We observe that f might not be complex differentiable, actually it is so only when Dzf = 0, and in this case Dzf = f ′, the complex derivative of f . If f admits a z-primitive F and a z-primitive G in D, that is, DzF = DzG = f , then we can define the Montée operators I and I by I(f)(z) := F (z)− F (0) and I(f)(z) := G(z)−G(0), z ∈ D. By definition, Dz(If) = f and Dz(If) = f. (1.3) Moreover, I(Dz(f))(z) = f(z)− f(0) and I(Dz(f))(z) = f(z)− f(0), z ∈ D. Our main results are related with walks through dimensions for Descente and Montée opera- tors over complex spheres: Theorem 1.1. Let f : D→ C be continuously differentiable. (i) If f belongs to the class Ψ(Ω2q), then Dzf , Dzf and Dxf belong to the class Ψ(Ω2q+2). (ii) If f belongs to the class Ψ(Ω2q) and has all positive (2q)-complex Schoenberg coefficients, then Dzf , Dzf and Dxf belong to the class Ψ+(Ω2q+2). Theorem 1.2. Let f : D→ C be a continuous function admitting a z-primitive and a z-primitive in D. (i) If f belongs to the class Ψ(Ω2q+2), then there exist real constants c and C such that c+If and C + If belong to the class Ψ(Ω2q). (ii) If f belongs to the class Ψ+(Ω2q+2), then there exist real constants c and C such that c+If and C + If belong to the class Ψ+(Ω2q). 4 E. Massa, A.P. Peron and E. Porcu Observe that in Theorem 1.1(ii) we assumed the additional condition that all (2q)-complex Schoenberg coefficients are positive. This condition can be weakened (see Remark 1.4 below), but not completely removed. In fact, the following counterexamples show that the Descente operators over complex spheres do not preserve, in general, strict positive definiteness, in contrast to the real case of Beatson and zu Castell. Counterexample 1.3. Let q ≥ 2 be an integer. (i) If f(z) = ∞∑ m=0 aq−2 m,0R q−2 m,0 (z), where ∞∑ m=0 aq−2 m,0 <∞ and aq−2 m,0 > 0 for all m, then f ∈ Ψ+(Ω2q) and Dzf,Dxf ∈ Ψ+(Ω2q+2) but Dzf 6∈ Ψ+(Ω2q+2). (ii) If f(z) = ∞∑ n=0 aq−2 0,n R q−2 0,n (z), where ∞∑ n=0 aq−2 0,n <∞ and aq−2 0,n > 0 for all n, then f ∈ Ψ+(Ω2q) and Dzf,Dxf ∈ Ψ+(Ω2q+2) but Dzf 6∈ Ψ+(Ω2q+2). (iii) If f(z) = ∞∑ n=0 aq−2 0,n R q−2 0,n (z) + ∞∑ m=0 aq−2 m,0R q−2 m,0 (z), where aq−2 0,n , a q−2 m,0 ≥ 0 for all m, n, and aq−2 0,n > 0⇐⇒ n ∈ 5Z+ + 4, aq−2 m,0 > 0⇐⇒ m ∈ (5Z+ \ {0}) ∪ (5Z+ + 2) ∪ (5Z+ + 3) ∪ (5Z+ + 4), then f ∈ Ψ+(Ω2q) but Dzf,Dzf,Dxf /∈ Ψ+(Ω2q+2). Remark 1.4. In the real case, the condition that all d-Schoenberg coefficients are positive is satisfied by most of the functions in the class Ψ+ d which appear in applications such as in statistics and geostatistics. In the complex case, among the examples that we provide in Section 2.1, only the exponential function satisfies this condition. On the other hand, the Aktaş–Taşdelen–Yavuz, Horn and Lauricella families, satisfy the following simple weaker condition, which is also sufficient to obtain the conclusion of Theorem 1.1(ii): • if aq−2 m,n are the (2q)-complex Schoenberg coefficients of f , then for some c, d ∈ N, the set{ m− n : aq−2 m,n > 0, m, n ≥ c } contains (d+ Z+) or (−d− Z+). In fact, the weakest possible condition to be used in Theorem 1.1(ii) follows from Guella and Menegatto [18] and reads as follows:{ m− n : aq−2 m,n > 0, m, n ≥ 1 } ∩ (NZ + j) 6= ∅, (1.4) for every N ≥ 1, j = 0, 1, . . . , N − 1. We will prove Theorem 1.1 with this last condition, since the previous ones are stronger. This paper is organized as follows: in Section 2, we provide the necessary background about positive definite functions on complex spheres and we give a list of parametric families of these functions, which are of interest for both numerical analysis and geostatistical communities. Finally, in Section 3, we obtain all necessary technical lemmas, we give the proofs of Theorems 1.1 and 1.2, and we show the Counterexample 1.3. Positive Definite Functions on Complex Spheres and their Walks through Dimensions 5 2 The classes Ψ(Ω2q) and Ψ+(Ω2q): a brief survey This section is largely expository and presents some basic facts and background needed for a self contained exposition. For q being a positive integer, we denote by Ω2q the unit sphere of Cq and by B2q := {z ∈ Cq : \|z\| ≤ 1} the closed disk in Cq. Also, we define the Pochhammer symbol (a)n := a(a+ 1) · · · (a+ n− 1), with (a)0 := 1. Let A be a nonempty set. A continuous kernel K : A2 → C is positive definite if and only if l∑ µ,ν=1 cµcνK(ξµ, ξν) ≥ 0, (2.1) for all l ∈ Z+ := {0, 1, 2, . . .}, {ξ1, ξ2, . . . , ξl} ⊂ A and {c1, c2, . . . , cl} ⊂ C. If the inequality in (2.1) is strict when at least one cµ is nonzero, then K is called strictly positive definite. For q a strictly positive integer, we define Aq := Ω2 when q = 1 and Aq := D for q > 1. Throughout we shall work with the class Ψ(Ω2q) of continuous functions f : Aq → C such that the kernel K : Ω2q × Ω2q → C defined as K(ξ, η) = f(〈ξ, η〉), (ξ, η) ∈ Ω2q × Ω2q, (2.2) where the symbol 〈·, ·〉 denotes the usual inner product in Cq, is positive definite. Observe that an immediate consequence of the definition is that f satisfies f(z) = f(z). We shall use the notation Ψ+(Ω2q) if the kernel K associated to f through (2.2) is strictly positive definite. Positive definite kernels satisfying the identity above are called isotropic. The class Ψ(Ω2q) is parenthetical to the class Ψd introduced by Schoenberg [40], and we refer the reader to the recent review in Gneiting [17] for a thorough description of the properties of this class. Further, the class Ψd represents the building block for extension to product spaces, and the reader is referred to Berg and Porcu [7] as well as to Guella et al. [19] for recent efforts in this direction. The classes Ψ(Ω2q) are nested, with the following inclusion relation being strict: Ψ(Ω4) ⊃ Ψ(Ω6) ⊃ · · · ⊃ Ψ(Ω∞), where Ω∞ is the unit sphere in the Hilbert space `2(C). Analogous relations apply to Ψ+(Ω2q). Observe that the class Ψ(Ω2) is a different class and it can not be added to the inclusions above (see Menegatto and Peron [26]). For this reason, in this work we always consider q ≥ 2. Actually the main purpose here is to study the walks through dimensions considering functions in the classes Ψ(Ω2q). Characterization theorems for the classes Ψ(Ω2q) are available in recent literature, and some ingredients are needed for a detailed exposition. We refer to Boyd and Raychowdhury [10], Dreseler and Hrach [13], and Koornwinder [22, 23] for more information concerning this necessary material. The disc polynomial Rαm,n of degree m + n in x and y associated to a real number α > −1 was introduced by Zernike [47] and Zernike and Brinkman [48], see also Koornwinder [22], as the polynomial given by Rαm,n(z) := r\|m−n\|ei(m−n)θR (α,\|m−n\|) min{m,n} ( 2r2 − 1 ) , z = reiθ = x+ iy ∈ D, (2.3) where R (α,β) k is the usual Jacobi polynomial of degree k associated to the numbers α, β > −1 and normalized by R (α,β) k (1) = 1 (see Szegő [44, p. 58]). Note that the function Rαm,n is a polynomial of degrees m and n with respect to the arguments z and z, respectively. Moreover it satisfies Rαm,n(z) = Rαm,n(z). 6 E. Massa, A.P. Peron and E. Porcu Let dνα be the positive measure having total mass identically equal to one on D, and given by dνα(z) = α+ 1 π ( 1− x2 − y2 )α dxdy, z = x+ iy. (2.4) Due to the orthogonality relations for Jacobi polynomials, the set {Rαm,n : 0 ≤ m,n <∞} forms a complete orthogonal system in L2(D,dνα) with∫ D Rαm,n(z)Rαk,l(z)dνα(z) = 1 hαm,n δm,kδn,l, (2.5) where hαm,n = m+ n+ α+ 1 α+ 1 ( α+m α )( α+ n α ) , (2.6) and δn,l denotes the Kronecker delta. Thus, a function f ∈ L1(D, να), α ≥ 0, has an expansion in terms of disc polynomials Rαm,n defined through f(z) ∼ ∑ m,n≥0 aαm,nR α m,n(z), (2.7) where aαm,n = hαm,n ∫ D f(z)Rαm,n(z)dνa(z). (2.8) The Poisson–Szegő kernel will be a fundamental tool for the proof of Theorem 2.1(1) below: the characterization of the class Ψ(Ω2q). We give here a brief presentation of it, since this kernel will also be used ahead. The Poisson–Szegő kernel is defined by Pq(rξ, η) := 1 σ2q (1− \|rξ\|2)q \|1− 〈rξ, η〉\|2q , r ∈ [0, 1), ξ, η ∈ Ω2q, (2.9) where σ2q is the total surface of Ω2q. Folland [15] proved that it has an expansion in terms of disc polynomials as Pq(rξ, η) = ∑ m,n≥0 hq−2 m,n σ2q Sqm,n(r)Rq−2 m,n(〈ξ, η〉), ξ, η ∈ Ω2q, r ∈ [0, 1), (2.10) where Sqm,n(r) ≥ 0, lim r→1− Sqm,n(r) = 1 and the series converges absolutely and uniformly for ξ, η ∈ Ω2q and 0 ≤ r ≤ R, for each R < 1. The Poisson–Szegő kernel also appears in the solution of the following Dirichlet problem for the Laplace–Beltrami operator ∆2q (see Stein [43]): given a continuous function h : Ω2q → C, there exists a continuous function u : B2q → C such that ∆2qu = 0 and u\|Ω2q = h. The solution u can be computed through u(z) = ∫ Ω2q Pq(z, ρ)h(ρ)dω2q(ρ), z ∈ B2q, (2.11) where dω2q denotes the rotation-invariant surface element on Ω2q. In fact, using this, if α = q − 2 ≥ 0 is an integer and f is a continuous function on D, the coefficients in the series in (2.7), can be written as (see Menegatto and Peron [26]): aq−2 m,n = hq−2 m,n σ2q ∫ Ω2q f(〈ρ, e1〉)Rq−2 m,n(〈e1, ρ〉)dω2q(ρ), (2.12) where e1 = (1, 0, . . . , 0) ∈ Ω2q. We give now the representations for the elements of the classes Ψ(Ω2q) and Ψ+(Ω2q) that were proved by Menegatto and Peron [25, 26] and Guella and Menegatto [18]: Positive Definite Functions on Complex Spheres and their Walks through Dimensions 7 Theorem 2.1. Let f : D→ C be a continuous function. The following assertions are true: (1) f ∈ Ψ(Ω2q) if, and only if, f(z) = ∑ m,n≥0 aq−2 m,nR q−2 m,n(z), z ∈ D, (2.13) where ∑ m,n≥0 aq−2 m,n <∞ and aq−2 m,n ≥ 0 for all (m,n); (2) f ∈ Ψ+(Ω2q) if, and only if, f ∈ Ψ(Ω2q) and{ m− n : aq−2 m,n > 0, m, n ≥ 0 } ∩ (NZ + j) 6= ∅, (2.14) for every N ≥ 1, j = 0, 1, . . . , N − 1. Note that the index α = q − 2 of the disc polynomials is related to the sphere Ω2q and consequently α+ 1 = q − 1 is related to Ω2q+2. The coefficients aq−2 m,n are the analogue of the d-Schoenberg coefficients adk as in Daley and Porcu [12] and Ziegel [49], referring to the expansion of the members of the Schoenberg class Ψd. In analogy, we will call aq−2 m,n as (2q)-complex Schoenberg coefficients. 2.1 Families within the classes Ψ(Ω2q) and Ψ+(Ω2q) It is well known that there exist many examples of functions in the class Ψd, some of them widely used in applications (see for example Gneiting [17] and Porcu et al. [28]). In the literature it is also possible to find examples of functions that satisfy the condi- tions in Theorem 2.1, or those in Remark 1.4, and therefore they belong to the classes Ψ(Ω2q) and Ψ+(Ω2q). Some of them, as well as their use in applications, appeared recently, probably originated by the work of Wünsche [46], that deals with disc polynomials: a fundamental tool for studying the functions in these classes. We give below a collection of such functions. 1. Disk Polynomials and related families. The product kernel (Boyd and Raychowd- hury [10]), fm,n(z) = zmzn = min{m,n}∑ j=0 cjq,m,nR q−2 m−j,n−j(z), cjq,m,n ≥ 0, z ∈ D, is an element of the class Ψ(Ω2q), for each m,n ≥ 0. 2. Poisson–Szegő kernel and related families. An application of (2.9) and (2.10) shows that fr(z) := 1 σ2q (1− r2)q \|1− rz\|2q = ∑ m,n≥0 hq−2 m,n σ2q Sqm,n(r)Rq−2 m,n(z), z ∈ D, and hence it is a member of the class Ψ(Ω2q), for each r ∈ [0, 1). 3. Exponential function. The function (Menegatto et al. [27]) ez+z = ∞∑ m+n=0 (m+ 1)q−2(n+ 1)q−2 (q − 2)!  ∞∑ j=0 1 j!(m+ n+ q − 1)j Rq−2 m,n(z), z ∈ D, belongs to the class Ψ+(Ω2q). 8 E. Massa, A.P. Peron and E. Porcu 4. Aktaş, Taşdelen and Yavuz family. The function (Aktaş et al. [2]) ft(z) := 1 R ( 2 1− t+R )q−2 e(2tz)/(1+t+R) = ∑ m,n≥0 (q − 1)n tm+n m!n! Rq−2 m+n,n(z), z ∈ D, where R := ( 1− 2 ( 2\|z\|2 − 1 ) t+ t2 )1/2 , is a member of Ψ+(Ω2q), for each t ∈ (0, 1). 5. Horn family. Let r, R be positive integers such that 4r = (R− 1)2. Horn’s function H4 is defined on p. 57 of Srivastava and Manocha [42] by H4(a, b; c, d;x, y) = ∞∑ m,n=0 (a)2m+n(b)n (c)m(d)n xmyn m!n! , where \|x\| < r and \|y\| < R. An application of Theorem 2.2 in Aktaş et al. [2] shows that ft,s,b(z) := 1 (1− s)q−1 H4 ( q − 1, b; q − 1, q − 1; s(\|z\|2 − 1) (1− s)2 , tz 1− s ) = ∑ m,n≥0 (q + n− 1)m(b)n tnsm m!n! Rq−2 m,m+n(z), z ∈ D. Hence it is a member of Ψ+(Ω2q), for each b, a positive integer, and t, s positive numbers satisfying \|s\| < 1, \|s\| (1− s)2 < r, and \|t\| 1− s < R. 6. Lauricella family. Let r1, r2 and r3 be positive integers such that r1r2 = (1−r2)(r2−r3). The Lauricella hypergeometric function of three variables F14 (Saran’s notation FF is also used (Saran [32])) is defined by (see p. 67 of Srivastava and Manocha [42]) F14(a1, a1, a1, b1, b2, b1; c1, c2, c2;x1, x2, x3) = ∞∑ m,n,p=0 (a1)m+n+p(b1)m+p(b2)n (c1)m(c2)n+p xm1 x n 2x p 3 m!n!p! , where \|x1\| < r1, \|x2\| < r2 and \|x3\| < r3. For t, s ∈ R such that \|s\| < r1 and \|t\| < r2, where r1 = r2(1− r2), define ft,s,b(z) := F14 ( 1, 1, 1, q − 1, b, q − 1; q − 1, 1, 1; s ( \|z\|2 − 1 ) , tz, s\|z\|2 ) , z ∈ D. From Theorem 2.3 in Aktaş et al. [2] we get ft,s,b(z) = ∑ m,n≥0 (q − 1)n(b)m tmsn m!n! Rq−2 m+n,n(z), z ∈ D, and hence, ft,s,b is a member of Ψ+(Ω2q), for each b, a positive integer, and t, s positive numbers satisfying the relevant conditions above. Some comments are in order. Lauricella functions are generalizations of the Gauss hyper- geometric functions to multiple variables and were introduced by Lauricella in 1893. Recursion formulas and integral representation for Lauricella functions, including F14 (FF ), have been studied and can be found, for example, in Sahai and Verma [31] and Saran [33, 34]. In 1873, Schwarz [41] found a list of 15 cases where hypergeometric functions can be expressed alge- braically. More precisely, Schwarz gave a list of parameters determining the cases where the hypergeometric differential equation has two independent solutions that are algebraic functions. Between 1989 and 2009 several researchers extended this list: to general one-variable hypergeo- metric functions p+1Fp (Beukers and Heckman [8]), the Appell–Lauricella functions F1 and FD (Beazley Cohen and Wolfart [5]), the Appell functions F2 and F4 (Kato [20, 21]), and the Horn function G3 (Schipper [38]). In 2012, Bod [9] extended Schwarz’ list to the four classes of Appell–Lauricella functions and the 14 complete Horn functions, including H4. Positive Definite Functions on Complex Spheres and their Walks through Dimensions 9 3 Proof of the results In this section we first prove some technical lemmas. Then, we shall be able to give the proof of our main results and to present the counterexamples. The first lemma contains recurrence formulas connecting disc polynomials of different indexes and degrees. They are obtained from equation (5.5) in Aharmim et al. [1] and the following properties of the disc polynomials Rαm,n(z) = Rαn,m(z), DzRαm,n(z) = DzRαn,m(z), α > −1, m, n ≥ 0, z ∈ D. We observe that the normalization adopted in Aharmim et al. [1] for the disc polynomials is different from the one we use here. Lemma 3.1. Let m, n be non negative integers and α > −1 be a real number. Then, for any z ∈ D, we have (α+ 1)Rαm,n+1(z) = (α+ 1)zRα+1 m,n (z)− ( 1− \|z\|2 ) DzRα+1 m,n (z), (3.1) and (α+ 1)Rαn+1,m(z) = (α+ 1)zRα+1 n,m (z)− ( 1− \|z\|2 ) DzRα+1 n,m (z). (3.2) Below we prove an important technical result, that connects the expansion of a continuously differentiable function f in terms of the disc polynomials Rαm,n with the expansion of its deriva- tives in terms of the disc polynomials Rα+1 m,n . Since Rq−2 m,n belongs to Ψ(Ω2q) when q ≥ 2 is an integer, this connection will be the main ingredient in order to obtain preservation of positive definiteness for the Descente operators, when walks through dimensions over complex spheres are provided. Lemma 3.2. Let f : D → C be continuously differentiable and let α > −1 be a real number. Consider the expansion of f in terms of the disc polynomials Rαm,n and the expansions of Dzf and Dzf in terms of the disc polynomials Rα+1 m,n f(z) ∼ ∞∑ m,n=0 aαm,nR α m,n(z), z ∈ D, Dzf(z) ∼ ∞∑ m,n=0 bα+1 m,nR α+1 m,n (z) and Dzf(z) ∼ ∞∑ m,n=0 b̃α+1 m,nR α+1 m,n (z), z ∈ D. Then, bα+1 m,n = (m+ 1)(n+ α+ 1) (α+ 1) aαm+1,n, m, n ≥ 0, and b̃α+1 m,n = (n+ 1)(m+ α+ 1) (α+ 1) aαm,n+1, m, n ≥ 0. It is worth noting that this result is not surprising if we consider the identities obtained in Koornwinder [23]: for α > −1, DzRαm,n = cα(m,n)Rα+1 m−1,n and DzRαm,n = cα(n,m)Rα+1 m,n−1, (3.3) where cα(m,n) := (m(n+ α+ 1))/(α+ 1). These are, in the complex case, the analogue of the identities for the derivative of the Gegenbauer polynomials (see Szegő [44, equation (4.7.14)]). Actually, Lemma 3.2 shows that the coefficients in the expansions are linked as if the series could be derived term by term. 10 E. Massa, A.P. Peron and E. Porcu Proof of Lemma 3.2. The coefficients bα+1 m,n are given by the formula bα+1 m,n = hα+1 m,n ∫ D Dzf(z)Rα+1 m,n (z)dνα+1(z), where the constants hα+1 m,n are given in (2.6). Define I := ∫ D Dzf(z)Rα+1 n,m (z)dνα+1(z) = α+ 2 π ∫ D Dzf(z)Rα+1 n,m (z) ( 1− x2 − y2 )α+1 dxdy. Integration by parts and direct inspection shows that I = α+ 2 π {∫ D Dz [ f(z)Rα+1 n,m (z) ( 1− \|z\|2 )α+1] dxdy − ∫ D f(z)Dz [ Rα+1 n,m (z) ( 1− \|z\|2 )α+1] dxdy } . Using Green’s theorem and (1.2) we have∫ Ω2 g(z)dz = −2i ∫ D Dz(g)(z)dxdy, for any continuously differentiable function g. Thus, I = α+ 2 π { i 2 ∫ Ω2 f(z)Rα+1 n,m (z) ( 1− \|z\|2 )α+1 dz − ∫ D f(z)Dz [ Rα+1 n,m (z) ( 1− \|z\|2 )α+1 ] dxdy } = −α+ 2 π ∫ D f(z)Dz [ Rα+1 n,m (z) ( 1− \|z\|2 )α+1 ] dxdy. Now, by noting that Dz [ Rα+1 n,m (z) ( 1− \|z\|2 )α+1] = DzRα+1 n,m (z) ( 1− \|z\|2 )α+1 − (α+ 1) ( 1− \|z\|2 )α zRα+1 n,m (z), we get I = α+ 2 π ∫ D f(z) ( 1− \|z\|2 )α [ (α+ 1)zRα+1 n,m (z)− ( 1− \|z\|2 ) DzRα+1 n,m (z) ] dxdy. Hence, using Lemma 3.1, we have I = α+ 2 π ∫ D f(z) ( 1− \|z\|2 )α (α+ 1)Rαn,m+1(z)dxdy = (α+ 2) ∫ D f(z)Rαm+1,n(z)dνα(z). Thus, bα+1 m,n = hα+1 m,n I = (α+ 2)hα+1 m,n 1 hαm+1,n aαm+1,n. Replacing the values of hα+1 m,n and hαm+1,n given in equation (2.6), we obtain bα+1 m,n = (α+ 2) (m+ 1)(α+ n+ 1) (α+ 2)(α+ 1) aαm+1,n = (m+ 1)(α+ n+ 1) (α+ 1) aαm+1,n. The proof for the case of the operator Dz is analogous observing that∫ Ω2 g(z)dz = 2i ∫ D Dz(g)(z)dxdy. � Positive Definite Functions on Complex Spheres and their Walks through Dimensions 11 The last technical lemma gives a condition for the expansion of a continuous function in terms of the disc polynomials to be uniformly convergent. Lemma 3.3. Let g : D→ C be a continuous function and consider its expansion g(z) ∼ ∑ m,n≥0 dq−2 m,nR q−2 m,n(z), z ∈ D, (3.4) where dq−2 m,n are given as in (2.12). If dq−2 m,n ≥ 0 for all m,n ≥ 0, then ∑ m,n≥0 dq−2 m,n < ∞. In particular, the series in (3.4) converges uniformly in D. Proof. The argument is similar to the one used in the proof of Theorem 4.1 in Menegatto and Peron [26]. Given ξ ∈ Ω2q, consider the continuous function h(ρ) := g(〈ρ, ξ〉), ρ ∈ Ω2q. By equation (2.11), the solution of the Dirichlet problem ∆2qu = 0 in the interior of B2q with boundary condition h, evaluated on the segment rξ, r ∈ [0, 1), is u(rξ) = ∫ Ω2q Pq(rξ, ρ)g(〈ρ, ξ〉)dω2q(ρ) = ∑ m,n≥0 Sqm,n(r)dq−2 m,n, where the last equality is obtained from (2.10), (2.12). Since u is continuous up to the boundary and coincides with h on Ω2q, we obtain lim r→1− ∑ m,n≥0 dq−2 m,nS q m,n(r) = lim r→1− u(rξ) = u(ξ) = g(〈ξ, ξ〉) = g(1). Now, note that 0 ≤ k∑ m=0 l∑ n=0 dq−2 m,nS q m,n(r) ≤ ∑ m,n≥0 dq−2 m,nS q m,n(r), 0 ≤ r < 1. Letting r → 1−, we get 0 ≤ sk,l := k∑ m=0 l∑ n=0 dq−2 m,n ≤ lim r→1− ∑ m,n≥0 dq−2 m,nS q m,n(r) = g(1), k, l ∈ Z+. Hence, the sequence {sk,l}k,l∈Z+ is bounded and increasing. Thus, the series ∑ m,n≥0 dq−2 m,n is con- vergent. Using the fact that \|Rq−2 m,n(z)\| ≤ 1 for all z ∈ D and using the Weierstrass M-Test, the proof is completed. � At this point, we are able to prove our main results. Proof of Theorem 1.1. Let f be a function in the class Ψ(Ω2q). Then, by Theorem 2.1(1), f(z) = ∑ m,n≥0 aαm,nR α m,n(z), z ∈ D, where α = q − 2, aαm,n ≥ 0, for all m,n ≥ 0, and ∑ m,n≥0 aαm,n < ∞. Consider the expansion in terms of disc polynomials of Dzf : Dzf(z) ∼ ∑ m,n≥0 bα+1 m,nR α+1 m,n (z), z ∈ D. 12 E. Massa, A.P. Peron and E. Porcu By Lemma 3.2 and equation (3.3), bα+1 m,n = cα(m+ 1, n)aαm+1,n, m, n ≥ 0. (3.5) Roughly speaking, (3.5) means that the coefficients { bα+1 m,n } are obtained from the {aαm,n} by suppressing the aα0,n, translating in the first index and multiplying by the positive constants {cα(m+ 1, n)}. Then, by equation (3.3), we have∑ m,n≥0 aαm,nDzRαm,n(z) = ∑ m≥−1 ∑ n≥0 aαm+1,nDzRαm+1,n(z) = ∑ m,n≥0 bα+1 m,nR α+1 m,n (z). Now, since cα(m + 1, n) are positive constants, we have that bα+1 m,n ≥ 0 for all m,n ≥ 0. By Lemma 3.3, the series∑ m,n≥0 aαm,nDzRαm,n(1) = ∑ m,n≥0 bα+1 m,n is convergent and the series ∑ m,n≥0 bα+1 m,nR α+1 m,n (z) converges uniformly in D. It follows, by term by term differentiation, that Dzf(z) = ∑ m,n≥0 bα+1 m,nR α+1 m,n (z). Hence, by Theorem 2.1(1), Dzf belongs to the class Ψ(Ω2q+2). Similarly, we can conclude the same for the operator Dz. For the item (ii), observe that, as a consequence of (3.5), if the (2q)-complex Schoenberg coefficients aαm,n of f satisfy (1.4), then the (2q + 2)-complex Schoenberg coefficients bα+1 m,n of Dzf (and similarly for Dzf) satisfy (2.14). Actually, the condition m,n ≥ 1 in the set considered in (1.4) guarantees that the intersections with the arithmetic progressions in Z do not depend on the coefficients aαm,0 or aα0,n, which are suppressed by the Descente operators. The results for Dxf follow immediately by (1.2). � Proof of Theorem 1.2. Suppose that f belongs to the class Ψ(Ω2q+2). By Theorem 2.1, f(z) = ∑ m,n≥0 aα+1 m,nR α+1 m,n (z), z ∈ D, where α = q− 2 and aα+1 m,n ≥ 0 for all m,n ≥ 0 and ∑ m,n≥0 aα+1 m,n <∞. By equation (3.3), we have I(Rα+1 m−1,n)(z) = 1 cα(m,n) ( Rαm,n(z)−Rαm,n(0) ) , (3.6) where Rαn,n(0) = (−1)nn!α!/(n + α)! and Rαm,n(0) = 0, m 6= n (Wünsche [46, equation (2.9)]). Thus consider F (z) := ∑ m,n≥0 aα+1 m,nI(Rα+1 m,n )(z) = ∑ m≥1 ∑ n≥0 aα+1 m−1,n cα(m,n) ( Rαm,n(z)−Rαm,n(0) ) , z ∈ D. (3.7) Since cα(m,n) ≥ 1 for all m ≥ 1, n ≥ 0 and \|Rαm,n(0)\| ≤ 1, for all m, n, we have that the series ∑ m≥1 ∑ n≥0 aα+1 m−1,n cα(m,n) and c := ∑ m≥1 ∑ n≥0 aα+1 m−1,n cα(m,n) Rαm,n(0) Positive Definite Functions on Complex Spheres and their Walks through Dimensions 13 are convergent. Furthermore, since∣∣∣∣∣ a α+1 m−1,n cα(m,n) ( Rαm,n(z)−Rαm,n(0) )∣∣∣∣∣ ≤ 2 aα+1 m−1,n cα(m,n) , m, n ≥ 0, z ∈ D, the series in (3.7) converges uniformly in D. On the other hand, by applying the derivation operator Dz term by term in (3.7), one obtains the uniformly convergent series of f . Then F is a z-primitive of f . Since F (0)=0, we conclude that (3.7) converges to I(f)(z). We can now write I(f)(z) = ∑ m,n≥0 bαm,nR α m,n(z), where bα0,0 := −c; bα0,n := 0, n ≥ 1; bαm,n := aα+1 m−1,n cα(m,n) , m ≥ 1, n ≥ 0; (3.8) and ∑ m,n≥0 bαm,n <∞. Now we can write c+ If(z) = ∑ m,n≥0 b̂αm,nR α m,n(z), where b̂α0,0 := 0; b̂α0,n := bα0,n, n ≥ 1 and b̂αm,n := bαm,n, m ≥ 1, n ≥ 0 (3.9) are nonnegative constants and ∑ m,n≥0 b̂αm,n <∞. Equations (3.8) and (3.9) mean that the coefficients { b̂αm,n } are obtained from the {aα+1 m,n }, by translating in the first index, adding the new coefficients b̂α0,n = 0, and dividing by the positive constants {cα(m,n)}. Hence, applying Theorem 2.1(1) again, we have that c+ If belongs to the class Ψ(Ω2q). For the item (ii), it is enough to observe that the (2q+2)-complex Schoenberg coefficients aα+1 m,n of f satisfy (2.14) by the assumption f ∈ Ψ+(Ω2q+2), then, as a consequence of (3.8), (3.9), also the (2q)-complex Schoenberg coefficients b̂αm,n of c+If satisfy (2.14), implying c+If ∈ Ψ+(Ω2q). For the operator I, one uses the relation I(Rα+1 m,n−1)(z) = 1 cα(n,m) ( Rαm,n(z)−Rαm,n(0) ) , and follows the same arguments. In fact, the (2q)-complex Schoenberg coefficients of C + If are given by b̌α0,0 := C − ∑ µ≥1 ∑ ν≥0 aα+1 µ,ν−1 cα(ν, µ) Rαµ,ν(0); b̌αm,0 := 0, m ≥ 1; b̌αm,n := aα+1 m,n−1 cα(m,n) , m ≥ 0, n ≥ 1. � 14 E. Massa, A.P. Peron and E. Porcu Proof of Counterexample 1.3. Let us denote by aq−2 m,n(g) the (2q)-complex Schoenberg coeffi- cients of a positive definite function g. Theorem 2.1(2) is required. (i) For a function f as in the statement, we have Dxf = Dzf and{ m− n : aq−1 m,n(Dzf) > 0 } = { m− n : aq−2 m,n(f) > 0 } = Z+. Hence the above set intercepts every arithmetic progression in Z, that is f ∈ Ψ+(Ω2q) and Dzf,Dxf ∈ Ψ+(Ω2q+2). However, Dzf ≡ 0, so that Dzf 6∈ Ψ+(Ω2q+2). (ii) Analogous to (i). (iii) For a function f as in the statement, we have { m− n : aq−2 m,n(f) > 0, m, n ≥ 0 } =  5⋃ j=2 5Z+ + j  ∪ (−5Z+ − 4), which intercepts every arithmetic progression in Z and then f ∈ Ψ+(Ω2q). However{ m− n : aq−1 m,n(Dzf) > 0, m, n ≥ 0 } = Z+ \ 5Z and { m− n : aq−1 m,n(Dzf) > 0, m, n ≥ 0 } = −5Z+ − 3, that is, Dzf,Dzf 6∈ Ψ+(Ω2q+2). To see that Dxf 6∈ Ψ(Ω2q+2), note that {m−n : aq−1 m,n(Dxf) > 0, m,n ≥ 0} is the union of the previous two sets, so it does not intersect the progression 5Z. � Acknowledgement The authors gratefully thank the anonymous referees for the constructive comments and rec- ommendations which helped to greatly improve the paper. Eugenio Massa was supported by grant #2014/25398-0, São Paulo Research Foundation (FAPESP) and grant #308354/2014-1, CNPq/Brazil. Ana P. Peron was supported by grants #2016/03015-7 and #2014/25796-5, São Paulo Research Foundation (FAPESP). Emilio Porcu was supported by grant FONDECYT #1170290 from the Chilean government. 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Soc. 142 (2014), 2063–2077, arXiv:1201.5833. https://doi.org/10.1016/S0377-0427(00)00345-9 https://doi.org/10.1007/s10444-009-9142-7 http://www.joachimschipper.nl/publications/bsc.pdf https://doi.org/10.2307/1968466 https://doi.org/10.1215/S0012-7094-42-00908-6 https://doi.org/10.1515/crll.1873.75.292 https://doi.org/10.1007/BF02123482 https://doi.org/10.1016/j.cam.2004.04.004 https://doi.org/10.1016/S0031-8914(34)80259-5 https://doi.org/10.1090/S0002-9939-2014-11989-7 https://doi.org/10.1090/S0002-9939-2014-11989-7 https://arxiv.org/abs/1201.5833 1 Introduction and main results 2 The classes (2q) and +(2q): a brief survey 2.1 Families within the classes (2q) and +(2q) 3 Proof of the results References

Positive Definite Functions on Complex Spheres and their Walks through Dimensions

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