Strictly and non-strictly positive definite functions on spheres

# Strictly and non-strictly positive definite functions on spheres

\fnmsTilmann \snmGneiting\correflabel=e1]t.gneiting@uni-heidelberg.delabel=e2 [[    url]www.math.uni-heidelberg.de/spatial Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 294, D-69120 Heidelberg, Germany. \printeade1
###### Abstract

Isotropic positive definite functions on spheres play important roles in spatial statistics, where they occur as the correlation functions of homogeneous random fields and star-shaped random particles. In approximation theory, strictly positive definite functions serve as radial basis functions for interpolating scattered data on spherical domains. We review characterizations of positive definite functions on spheres in terms of Gegenbauer expansions and apply them to dimension walks, where monotonicity properties of the Gegenbauer coefficients guarantee positive definiteness in higher dimensions. Subject to a natural support condition, isotropic positive definite functions on the Euclidean space , such as Askey’s and Wendland’s functions, allow for the direct substitution of the Euclidean distance by the great circle distance on a one-, two- or three-dimensional sphere, as opposed to the traditional approach, where the distances are transformed into each other. Completely monotone functions are positive definite on spheres of any dimension and provide rich parametric classes of such functions, including members of the powered exponential, Matérn, generalized Cauchy and Dagum families. The sine power family permits a continuous parameterization of the roughness of the sample paths of a Gaussian process. A collection of research problems provides challenges for future work in mathematical analysis, probability theory and spatial statistics.

\kwd
\aid

0 \volume19 \issue4 2013 \firstpage1327 \lastpage1349 \doi10.3150/12-BEJSP06 \newremarkexampleExample

\runtitle

Positive definite functions on spheres

{aug}

completely monotone \kwdcovariance localization \kwdfractal index \kwdinterpolation of scattered data \kwdisotropic \kwdlocally supported \kwdmultiquadric \kwdPólya criterion \kwdradial basis function \kwdSchoenberg coefficients

\pdfkeywords

completely monotone, covariance localization, fractal index, interpolation of scattered data, isotropic, locally supported, multiquadric, Polya criterion, radial basis function, Schoenberg coefficients

## 1 Introduction

Recently, there has been renewed interest in the study of positive definite functions on spheres, motivated in part by applications in spatial statistics, where these functions play crucial roles as the correlation functions of random fields on spheres (Banerjee, 2005; Huang, Zhang and Robeson, 2011), including the case of star-shaped random particles (Hansen, Thorarinsdottir and Gneiting, 2011). In a related development in approximation theory, strictly positive definite functions arise as radial basis functions for interpolating scattered data on spherical domains (Fasshauer and Schumaker, 1998; Cavoretto and De Rossi, 2010; Le Gia, Sloan and Wendland, 2010).

Specifically, let be an integer, and let denote the unit sphere in the Euclidean space , where we write for the Euclidean norm of . A function is positive definite if

 n∑i=1n∑j=1cicjh(xi,xj)≥0 (1)

for all finite systems of pairwise distinct points and constants . A positive definite function is strictly positive definite if the inequality in (1) is strict, unless , and is non-strictly positive definite otherwise. The function is isotropic if there exists a function such that

 h(x,y)=ψ(θ(x,y))for allx,y∈Sd, (2)

where is the great circle, spherical or geodesic distance on , and denotes the scalar or inner product in . Thus, an isotropic function on a sphere depends on its arguments via the great circle distance or, equivalently, via the inner product only.

For we write for the class of the continuous functions with which are such that the associated isotropic function in (2) is positive definite. Hence, we may identify the class with the correlation functions of the mean-square continuous, stationary and isotropic random fields on the sphere (Jones, 1963). We write for the class of the continuous functions with such that the isotropic function in (2) is strictly positive definite.

We study the classes and with particular attention to the practically most relevant case of a two-dimensional sphere, such as planet Earth. Applications abound, with atmospheric data assimilation and the reconstruction of the global temperature and green house gas concentration records being key examples. Section 2 summarizes related results for isotropic positive definite functions on Euclidean spaces and studies their restrictions to spheres. In Section 3, we review characterizations of strictly and non-strictly positive definite functions on spheres in terms of expansions in ultraspherical or Gegenbauer polynomials. Monotonicity properties of the Gegenbauer coefficients in any given dimension guarantee positive definiteness in higher dimensions.

The core of the paper is Section 4, where we supply easily applicable conditions for membership in the classes and , and use them to derive rich parametric families of correlation functions and radial basis functions on spheres. In particular, compactly supported, isotropic positive definite functions on the Euclidean space remain positive definite with the great circle distance on a one-, two- or three-dimensional sphere as argument. Consequently, compactly supported radial basis functions on , such as Askey’s and Wendland’s functions, translate directly into locally supported radial basis functions on the sphere. Criteria of Pólya type guarantee membership in the classes , and we supplement recent results of Beatson, zu Castell and Xu (2011) in the case of spheres of dimension . Completely monotone functions are strictly positive definite on spheres of any dimension, and include members of the powered exponential, Matérn, generalized Cauchy and Dagum families. Some of the key results in the cases of one-, two- and three-dimensional spheres are summarized in Table 1.

The paper ends in Section 5, where we refer to a collection of research problems in mathematical analysis, probability theory and statistics that offer challenges of diverse difficulty and scope, with details given in a supplemental article (Gneiting, 2013).

## 2 Isotropic positive definite functions on Euclidean spaces

In this expository section, we review basic results about isotropic positive definite functions on Euclidean spaces and study their restrictions to spheres.

Recall that a function is positive definite if the inequality (1) holds for all finite systems of pairwise distinct points and constants . It is strictly positive definite if the inequality is strict unless . The function is radial, spherically symmetric or isotropic if there exists a function such that

 h(x,y)=φ(∥x−y∥)for allx,y∈Rd. (3)

For an integer we denote by the class of the continuous functions with such that the function in (3) is positive definite. Thus, we may identify the members of the class with the characteristic functions of spherically symmetric probability distributions, or with the correlation functions of mean-square continuous, stationary and isotropic random fields on .

Schoenberg (1938) showed that a function belongs to the class if, and only if, it is of the form

 φ(t)=∫[0,∞)Ωd(rt)dF(r)fort≥0, (4)

where is a uniquely determined probability measure on and

 Ωd(t)=Γ(d/2)(2t)(d−2)/2J(d−2)/2(t),

with being a Bessel function (Digital Library of Mathematical Functions, 2011, Section 10.2). The classes are nonincreasing in ,

 Φ1⊃Φ2⊃⋯⊃Φ∞=∞⋂d=1Φd,

with the inclusions being strict. If , any nonconstant member of the class corresponds to a strictly positive definite function on (Sun, 1993, Theorem 3.8) and thus it belongs to the class . Finally, as shown by Schoenberg (1938), the class consists of the functions of the form

 φ(t)=∫[0,∞)exp(−r2t2)dF(r)fort≥0, (5)

where is a uniquely determined probability measure on .

From the point of view of isotropic random fields, the asymptotic decay of the function at the origin determines the smoothness of the associated Gaussian sample path (Adler, 2009). Specifically, if

 φ(0)−φ(t)=O(tα)ast↓0 (6)

for some , which we refer to as the fractal index, the graph of a Gaussian sample path has fractal or Hausdorff dimension almost surely. If , the sample path is smooth and differentiable and its Hausdorff dimension, , equals its topological dimension, . If , the sample path is non-differentiable. Table 2 shows some well known parametric families within the class , namely the powered exponential family (Yaglom, 1987), the Matérn class (Guttorp and Gneiting, 2006), the generalized Cauchy family (Gneiting and Schlather, 2004), and the Dagum class (Berg, Mateu and Porcu, 2008), along with the corresponding fractal indices. Table 3 shows compactly supported families within the class that have been introduced and studied by Wendland (1995) and Gneiting (1999a, 2002).

Yadrenko (1983) pointed out that if is a member of the class for some integer , then the function defined by

 ψ(θ)=φ(2sinθ2)forθ∈[0,π] (7)

corresponds to the restriction of the isotropic function from to , with the chordal or Euclidean distance expressed in terms of the great circle distance, , on the sphere . Thus, given any nonconstant member of the class , the function defined by (7) belongs to the class .

Various authors have argued in favor of this construction, including Fasshauer and Schumaker (1998), Gneiting (1999a), Narcowich and Ward (2002) and Banerjee (2005), as it readily generates parametric families of isotropic, strictly positive definite functions on spheres and retains the interpretation of scale, support, shape and smoothness parameters. In particular, the mapping (7) from to preserves the fractal index, in the sense that

 ψ(0)−ψ(θ)=O(θα)asθ↓0 (8)

if has fractal index , as defined in (6). Nevertheless, the approach is of limited flexibility. For example, if it is readily seen from (4) that the function defined in (7) does not admit values less than . Furthermore, the mapping in the argument of in (7), while being essentially linear for small , is counter to spherical geometry for larger values of the great circle distance, and thus may result in physically unrealistic distortions. In this light, we now turn to characterizations and constructions of positive definite functions that operate directly on a sphere.

## 3 Isotropic positive definite functions on spheres

Let be an integer. As defined in Section 1, the classes and consist of the functions with which are such that the isotropic function in (2) is positive definite or strictly positive definite, respectively. Furthermore, we consider the class of the non-strictly positive definite functions, and we define

 Ψ∞=∞⋂d=1Ψd,Ψ+∞=∞⋂d=1Ψ+dandΨ−∞=∞⋂d=1Ψ−d.

Schoenberg (1942) noted that the classes and are convex, closed under products, and closed under limits, provided the limit function is continuous. The classes and are convex and closed under products, but not under limits.

We proceed to review characterizations in terms of ultraspherical or Gegenbauer expansions, for which we require classical results on orthogonal polynomials (Digital Library of Mathematical Functions (2011), Section 18.3). Given and an integer , the function is defined by the expansion

 1(1+r2−2rcosθ)λ=∞∑n=0rnCλn(cosθ)forθ∈[0,π], (9)

where , and is the ultraspherical or Gegenbauer polynomial of degree , which is even if is even and odd if is odd. For reference later on, we note that . If , we follow Schoenberg (1942) and set

 C0n(cosθ)=cos(nθ)forθ∈[0,π].

By equation (3.42) of Askey and Fitch (1969),

 Cλn(cosθ)=Γ(ν)Γ(λ)Γ(λ−ν)[n/2]∑k=0(n−2k+ν)Γ(k+λ−ν)Γ(n−k+λ)k!Γ(n−k+ν+1)Cνn−2k(cosθ) (10)

whenever and , where the coefficients on the right-hand side are strictly positive. This classical result of Gegenbauer will be used repeatedly in the sequel.

The following theorem summarizes Schoenberg’s (1942) classical characterization of the classes and , along with more recent results of Menegatto (1994), Chen, Menegatto and Sun (2003) and Menegatto, Oliveira and Peron (2006) for the classes and .

###### Theorem 1

Let be an integer.

• [(a)]

• The class consists of the functions of the form

 ψ(θ)=∞∑n=0bn,dC(d−1)/2n(cosθ)C(d−1)/2n(1)forθ∈[0,π], (11)

with nonnegative, uniquely determined coefficients such that . If , the class consists of the functions in with the coefficients being strictly positive for infinitely many even and infinitely many odd integers . The class consists of the functions in which are such that, given any two integers , there exists an integer with the coefficient being strictly positive.

• The class consists of the functions of the form

 ψ(θ)=∞∑n=0bn(cosθ)nforθ∈[0,π], (12)

with nonnegative, uniquely determined coefficients such that . The class consists of the functions in with the coefficients being strictly positive for infinitely many even and infinitely many odd integers .

If , Schoenberg’s representation (11) reduces to the general form,

 ψ(θ)=∞∑n=0bn,1cos(nθ)forθ∈[0,π], (13)

of a member of the class , that is, a standardized, continuous positive definite function on the circle. By basic Fourier calculus,

 b0,1=1π∫π0ψ(θ)dθ% andbn,1=2π∫π0cos(nθ)ψ(θ)dθ (14)

for integers . If , the representation (11) yields the general form,

 ψ(θ)=∞∑n=0bn,2n+1Pn(cosθ)%forθ∈[0,π], (15)

of a member of the class in terms of the Legendre polynomial of integer order . As Bingham (1973) noted, the classes considered here are convex, and so (11), (12), (13) and (15) can be interpreted as Choquet representations in terms of extremal members, which are non-strictly positive definite.

The general form (12) of the members of the class can be viewed as a power series in the variable with nonnegative coefficients. To give an example, a standard Taylor expansion shows that the function defined by

 ψ(θ)=(1−δ)2τ(1+δ2−2δcosθ)τforθ∈[0,π] (16)

admits such a representation if and ; furthermore, the coefficients are strictly positive, whence belongs to the class . When , the special cases in (16) with the parameter fixed at and have been known as the inverse multiquadric and the Poisson spline, respectively (Cavoretto and De Rossi, 2010). In this light, we refer to (16) as the multiquadric family.

The sine power function of Soubeyrand, Enjalbert and Sache (2008),

 ψ(θ)=1−(sinθ2)αforθ∈[0,π], (17)

also admits the representation (12) if . Moreover, the coefficients are strictly positive if , whence belongs to the class . The parameter corresponds to the fractal index in the relationship (8) and thus parameterizes the roughness of the sample paths of the associated Gaussian process.

The classes are nonincreasing in , with the following result revealing details about their structure. The statement in part (a) might be surprising, in that a function that is non-strictly positive definite on the sphere cannot be strictly positive definite on a lower-dimensional sphere, including the case of the circle.

###### Corollary 1

• [(a)]

• If for positive integers and , then . Similarly, if for positive integers and , then . In particular,

 Ψ∞=Ψ+∞∪Ψ−∞, (18)

where the union is disjoint.

• The classes are nonincreasing in ,

 Ψ+1⊃Ψ+2⊃⋯⊃Ψ+∞=⋂d≥1Ψ+d,

with the inclusions being strict.

• The classes are nonincreasing in ,

 Ψ−1⊃Ψ−2⊃⋯⊃Ψ−∞=⋂d≥1Ψ−d,

with the inclusions being strict.

{pf}

In part (a) we use an argument of Narcowich (1995), and our key tool in proving that the inclusions in parts (b) and (c) are strict is Gegenbauer’s relationship (10).

• [(a)]

• If , it is trivially true that implies . If , suppose, for a contradiction, that . By part (a) of Theorem 1 applied in dimension , is either an even function plus an odd polynomial in , or an odd function plus an even polynomial in . By Theorem 2.2 of Menegatto (1995), we conclude that , for the desired contradiction to the assumption that . The proof of the second claim is analogous, and the statement in (18) then is immediate.

• The inclusion is trivially true. To see that the inclusion is strict, note that by part (a) of Theorem 1 and the relationship (10) any admits the representation (11) with being strictly positive for all . However, there are members of the class with for at least one integer , which thus do not belong to the class .

• The inclusion is immediate from part (a). To demonstrate that the inclusion is strict, we show that if then the function belongs to but not to . For a contradiction, suppose that . Then by the relationship (10), is of the form (11) with at least two distinct coefficients being strictly positive, contrary to the uniqueness of the Gegenbauer coefficients.\qed

\noqed

The following classical result is a consequence of Schoenberg’s (1942) representation (11) and the orthogonality properties of the ultraspherical or Gegenbauer polynomials.

###### Corollary 2

Let be an integer. A continuous function with belongs to the class if and only if

 bn,d=2n+d−123−dπ(Γ(d−12))2Γ(d−1)∫π0C(d−1)/2n(cosθ)(sinθ)d−1ψ(θ)dθ≥0 (19)

for all integers . Furthermore, the coefficient in the Gegenbauer expansion (11) of a function equals the above value.

Interesting related results include Theorem 4.1 of Narcowich and Ward (2002), which expresses the Gegenbauer coefficient of a function of Yadrenko’s form (7), where , in terms of the Fourier transform of the respective member of the class , Theorem 6.2 of Le Gia, Sloan and Wendland (2010), which deduces asymptotic estimates, and the recent findings of Ziegel (2013) on convolution roots and smoothness properties of isotropic positive definite functions on spheres.

Adopting terminology introduced by Daley and Porcu (2013) and Ziegel (2013), we refer to the Gegenbauer coefficient as a -Schoenberg coefficient and to the sequence as a -Schoenberg sequence. We now provide formulas that express the -Schoenberg coefficient in terms of and . In special cases, closely related results were used by Beatson, zu Castell and Xu (2011).

###### Corollary 3

Consider the -Schoenberg coefficients in the Gegenbauer expansion (11) of the members of the class .

• [(a)]

• It is true that

 b0,3=b0,1−12b2,1 (20)

and

 bn,3=12(n+1)(bn,1−bn+2,1) (21)

for all integers .

• If , then

 bn,d+2=(n+d−1)(n+d)d(2n+d−1)bn,d−(n+1)(n+2)d(2n+d+3)bn+2,d (22)

for all integers .

{pf}

We take advantage of well known recurrence relations for trigonometric functions and Gegenbauer polynomials.

• [(a)]

• The identity in (20) holds true because for , so that

 b0,3=2π∫π0(sinθ)2ψ(θ)dθ=1π∫π0(1−cos(2θ))ψ(θ)dθ=b0,1−12b2,1.

To establish (21), we note that

for . In view of the formulas (19) and (14) for the Gegenbauer coefficients and the Fourier cosine coefficients , respectively, the claim follows easily if we integrate the above equality over .

• By equation (18.9.8) of Digital Library of Mathematical Functions (2011),

 C(d+1)/2n(cosθ)(sinθ)2 = (n+d−1)(n+d)(d−1)(2n+d+1)C(d−1)/2n(cosθ) −(n+1)(n+2)(d−1)(2n+d+1)C(d−1)/2n+2(cosθ).

In view of the formula (19) for the -Schoenberg coefficients , we establish (22) by integrating the above equality over and consolidating coefficients.\qed

\noqed

If is an integer, so that is odd, the recursion (22) allows us to express the -Schoenberg coefficient in terms of the Fourier cosine coefficients . For instance,

 bn,5=(n+2)(n+3)3⋅4(b∗n,1−2n+2n+3bn+2,1+n+1n+3bn+4,1) (23)

for all integers , where and for . Similarly, if is even, we can express the -Schoenberg coefficient in terms of the Legendre coefficients . Furthermore, it is possible to relate the Legendre coefficients to the Fourier cosine coefficients, in that, subject to weak regularity conditions,

 bn,2=∞∑k=0cnk(b∗n+2k,1−bn+2k+2,1), (24)

where

 cnk=22n(n!)2(2n)!1⋅3⋯(2k−1)k!(n+1)⋅(n+2)⋯(n+k)(2n+3)⋅(2n+5)⋯(2n+2k+1)

for integers and , as used by Huang, Zhang and Robeson (2011).

The recursions in Corollary 3 allow for dimension walks, where monotonicity properties of the -Schoenberg coefficients guarantee positive definiteness in higher dimensions, as follows.

###### Corollary 4

Suppose that the function is continuous with . For integers and , let denote the Fourier cosine and Gegenbauer coefficients (14) and (19) of , respectively.

• [(a)]

• The function belongs to the class if, and only if, and for all integers . It belongs to the class if, and only if, furthermore, the inequality is strict for infinitely many even and infinitely many odd integers.

• If , the function belongs to the class if, and only if,

 bn+2,d≤2n+d+32n+d−1(n+d−1)(n+d)(n+1)(n+2)bn,d (25)

for all integers . It belongs to the class if, and only if, furthermore, the inequality is strict for infinitely many even and infinitely many odd integers.

• If , the function belongs to the class if for all integers .

## 4 Criteria for positive definiteness and applications

In this section, we provide easily applicable tests for membership in the classes and and apply them to construct rich parametric families of correlation functions and radial basis functions on spheres.

### 4.1 Locally supported strictly positive definite functions on two- and three-dimensional spheres

There are huge computational savings in the interpolation of scattered data on spheres if the strictly positive definite function used as the correlation or radial basis function admits a simple closed form, and is supported on a spherical cap. Thus, various authors have sought such functions (Schreiner, 1997, Fasshauer and Schumaker, 1998, Narcowich and Ward, 2002), with particular emphasis on the practically most relevant case of the two-dimensional sphere.

The following result of Lévy (1961) constructs a positive definite function on the circle from a positive definite function on the real line, in the form of a compactly supported member of the class . Here and in the following, we write for the restriction of a function to the interval . For essentially identical results, see Exercise 1.10.25 of Sasvári (1994) and Wood (1995).

###### Theorem 2

Suppose that the function is such that for . Then the restriction belongs to the class .

On the two-dimensional sphere, a particularly simple way of constructing a locally supported member of the corresponding class is to take a compactly supported member of the class , such as any of the functions in Table 3, and apply Yadrenko’s recipe (7). If is such that for , where , then the function in (7) belongs to the class and satisfies for . However, the approach is subject to the problems described in Section 2, in that the use of the chordal distance may result in physically unrealistic representations.

Perhaps surprisingly, the following result shows that compactly supported, isotropic positive definite functions on the Euclidean space remain positive definite with the great circle distance on a one-, two- or three-dimensional sphere as argument. Thus, they can be used either with the chordal distance or with the great circle distance as argument.

###### Theorem 3

Suppose that the function is such that for . Then the restriction belongs to the class .

{pf}

If the function is such that for , the Fourier cosine coefficient (14) of its restriction can be written as

 bn,1=2π∫π0cos(nθ)ψ(θ)dθ=1π∫Rcos(nt)ϕ(t)dt=2^ϕ(n)

for all integers . Here, the function is defined by for , and denotes the inverse Fourier transform of evaluated at . Similarly, . By equation (36) of Gneiting (1998a) in concert with the Paley–Wiener theorem (Paley and Wiener, 1934), the function is strictly decreasing in . Thus, the sequence is strictly decreasing in ; furthermore, . By part (a) of Corollary 4, we conclude that .

Theorem 3 permits us to use any of the functions in Table 3 with support parameter as a correlation function or radial basis function with the spherical or great circle distance on the sphere as argument, where . This gives rise to flexible parametric families of locally supported members of the class , where , that admit particularly simple closed form expressions. A first example is the spherical family,

 ψ(θ)=(1+12θc)(1−θc)2+forθ∈[0,π], (26)

which derives from a popular correlation model in geostatistics.

Using families of compactly supported members of the class described by Wendland (1995) and Gneiting (1999a), Theorem 3 confirms that Askey’s (1973) truncated power function

 ψ(θ)=(1−θc)τ+forθ∈[0,π] (27)

belongs to the class if and , as shown by Beatson, zu Castell and Xu (2011). If smoother functions are desired, the -Wendland function

 ψ(θ)=(1+τθc)(1−θc)τ+forθ∈[0,π] (28)

s in if and ; similarly, the -Wendland function defined by

 ψ(t)=(1+τθc+τ2−13θ2c2)(1−θc)τ+forθ∈[0,π] (29)

belongs to the class if and .

In atmospheric data assimilation, locally supported positive definite functions are used for the distance-dependent filtering of spatial covariance estimates on planet Earth (Hamill, Whitaker and Snyder, 2001). The traditional construction relies on the Gaspari and Cohn (1999) function,

 φGC(t)=⎧⎪⎨⎪⎩1−203t2+5t3+8t4−8t5,0≤t≤12,13t−1(8t2+8t−1)(1−t)4,12≤t≤1,0,t≥1,

which belongs to the class , along with all functions of the form , where is a constant. If , Yadrenko’s construction (7) yields the localization function

 ψ1(θ)=φGC(sinθ2sinc2)forθ∈[0,π], (30)

which is a member of the class with support . Theorem 3 suggests a natural alternative, in that, for every , the function defined by

 ψ2(θ)=φGC(θc)forθ∈[0,π] (31)

is also a member of the class with support . Clearly, for , as illustrated in Figure 1, where . This suggests that might be a more effective localization function

than the traditional choice, . Similar comments apply to covariance tapers in spatial statistics, as proposed by Furrer, Genton and Nychka (2006).

### 4.2 Criteria of Pólya type

Criteria of Pólya type provide simple sufficient conditions for positive definiteness, by imposing convexity conditions on a candidate function and/or its higher derivatives. Gneiting (2001) reviews and develops Pólya criteria on Euclidean spaces. In what follows, we are concerned with analogues on spheres that complement the recent work of Beatson, zu Castell and Xu (2011).

The following theorem summarizes results of Wood (1995), Gneiting (1998b) and Beatson, zu Castell and Xu (2011) in the case of the circle .

###### Theorem 4

Suppose that the function is continuous, nonincreasing and convex with and . Then belongs to the class . It belongs to the class if it is not piecewise linear.

Our next result is a Pólya criterion that applies on spheres of dimension . Its proof relies on the subsequent lemmas, the first of which is well known.

###### Theorem 5

Suppose that is a continuous function with , and a continuous derivative such that is convex for . Then the restriction belongs to the class .

To give an example, we may conclude from Theorem 5 that the truncated power function (27) with shape parameter belongs to the class for all values of the support parameter , including the case , in which it is supported globally.

###### Lemma 1

Let be an integer. Suppose that is a subset of a Euclidean space, and let be a Borel probability measure on . If belongs to the class for every , then the function defined by

 ψ(θ)=∫Cψc(θ)dP(c)forθ∈[0,π]

belongs to the class , too. If furthermore belongs to for every in a set of positive -measure, then belongs to the class .

###### Lemma 2

For all , the function defined by

 ψc(θ)=(1+12θc)(1−θc)2+forθ∈[0,π] (32)

belongs to the class .

{pf}

By Theorem 3, the function belongs to the class if . If , the formulas in (14) yield the Fourier cosine coefficients

 b0,1=18c3(8c3−6πc2+π3)% andb2k,1=3π4c31k2

for all integers . Similarly, we find that

 b2k+1,1=3πc3(2c2−π2(2k+1)2+4(2k+1)4)

for all integers and conclude from part (a) of Corollary 4 that belongs to .

{pf*}

Proof of Theorem 5 By Theorem 3.1 of Gneiting (1999b), the function admits a representation of the form

 φ(t)=∫(0,∞)φc(t)dF(c)fort≥0,

where is defined by

 φc(t)=(1+12tc)(1−tc)2+fort≥0, (33)

and is a probability measure on . Therefore, the restriction is of the form

 ψ(θ)=∫(0,∞)ψc(θ)dF(c)forθ∈[0,π],

where