Buhmann covariance functions, their compact supports, and their smoothness
BBuhmann covariance functions, their compactsupports, and their smoothness
E. PORCU ∗ , V.P. ZASTAVNYI † , and M. BEVILACQUA ‡ September 28, 2016
Abstract
We consider the Buhmann class of compactly supported radial basis functions, whihincludes a wealth of special cases that have been studied in both numerical analysis andspatial statistics literatures. In particular, the celebrated Wu, Wendland and MissingWendland functions are notable special cases of this class. We propose a very simpledifference operator and show the conditions for which the application of it to Buhmannfunctions preserves positive definiteness on m -dimensional Euclidean spaces. We alsoshow that the application of the difference operator increases smoothness at the ori-gin, whilst keeping positive definiteness in the same m -dimensional Euclidean space, aswell as compact support. Thus, our operator is a competitor of the celebrated Mont´eeoperator, which allows to increase the smoothness at the origin, at the expense of losingpositive definiteness in the space where the radial basis function is originally defined.The proofs of our results highlight surprising connections with past literatures on cele-brated class of functions. Amongst them, absolute and completely monotone functions. Keywords : Buhmann functions; Compact Support; Completely Monotonic; Fouriertransforms; Laplace transforms. ∗ Departamento de Matem´atica, Universidad Federico Santa Maria, † Department of Mathematics, Donetsk National University ‡ Department of Statistics, University of Valparaiso. a r X i v : . [ m a t h . S T ] S e p Introduction
Interpolation of data has a notable importance in both numerical analysis and geostatisti-cal communities: in geostatistics, the underlying structure of the data is assumed to be astochastic process, with the interpolation procedure known as kriging [23]. The method ismathematically equivalent to kernel interpolation, a method used in numerical analysis forthe same problem, but derived under completely different modelling assumptions.Radial basis functions are well-known and successful tools for the interpolation of data inmany dimensions [5]. Several radial basis functions of compact support that give rise to non-singular interpolation problems have been proposed, and we cite [19, 12, 5, 30, 20, 32, 33]and the impressive reviews in [8] and [21], amongst others. In particular, the motivation be-hind Buhmann’s [5] tour de force is to propose radial functions of compact support that alsogive positive definite matrices and have genuinely banded interpolation matrices (similarlyto the multiquadric and Gaussian kernels). Of such nature are those ones we will discuss inthe present paper. Early examples of radial functions with compact support that have a sim-ple piecewise polynomial structure are due to [32]. Then Schaback and Wu [22], Wendland[30] and finally Schaback [20] established several of their special properties, such as certainoptimality facts about their degree and smoothness. The functions proposed by Buhmannare closely related to the so-called multiply monotone radial basis functions as discussed in[15].Radial basis functions are known under the name of covariance (correlation) functions inthe geostatistical community: the use of compactly supported covariance functions has beenadvocated in a number of papers, and we refer the reader to [13], with the references therein,and to [7] for a recent effort under the framework on multivariate Gaussian fields. Covariancefunctions with compact support represent the building block for the construction of methodsallowing to overcome the big data problem ([11]). The recent work of [4] brought even moreattention on the role of some classes of compactly supported covariances for asymptoticallyoptimal prediction on a bounded set of R d . From the cited works it has become apparentthat the smoothness at the origin (intended as even extension) of a compactly supportedand isotropic covariance function plays a crucial role for both estimation and prediction.Wendland functions [30] have been especially popular, being compactly supported over ballsof R d with arbitrary radii, and additionally allowing for a continuous parameterization of2ifferentiability at the origin, in a similar way to the Mat´ern family ([25]).The tour de force in [30], [13] and [39] puts emphasis on linear operators, called Mont´ee ,that allow to increase the smoothness of a given radial function, being positive definite on R d . This is done at the expense of losing positive definiteness, which is only achieved on R d − , for d ≥ .Figure 1 depicts the following situation: Dashed lines report Wendland functions [30]with unit compact support, for k = 0 , , (from left to right) being the parameter that allowsto govern differentiability at the origin. Dashed-dotted lines report the respective Wendlandfunctions for a compact support equal to . . The functions depicted with continuous linesreport their weighted differences (see Equation (17)). We can clearly appreciate that the levelof differentiability changes for these last.This gives a substantial motivation for the present work: we consider the Buhmann func-tions as defined in Zastavnyi [36] on the basis of [5]. This class includes as special casemany other classes of compactly supported covariance functions, such as Askey [1] Wend-land [30] and Missing Wendland [20] functions, as well as the Zastavnyi [34, 35, 36] andTrigub [26, 27] classes. Finally, also Wu functions [32] and the celebrated spherical model[29] are included as special cases. We show that some parameterized differences of Buh-mann functions preserve positive definiteness in m -dimensional Euclidean spaces, and wethen determine the exact level of smoothness induced by such operation.We focus throughout on the class Φ m of continuous functions ϕ : [0 , ∞ ) → R such that ϕ ( (cid:107) x (cid:107) ) is positive definite on R m . Thus, ϕ ( (cid:107) · (cid:107) ) with ϕ (0) = 1 is the correlation functionof some Gaussian field.Apparently, the functions C ( · ) := ϕ ( (cid:107) · (cid:107) ) are radially symmetric and Schoenberg’s the-orem (1938, see [6] for a more recent discussion) uniquely identifies them as scale mixturesof the type ϕ ( t ) = (cid:90) [0 , ∞ ) Ω m ( rt ) F (d r ) , t ≥ , (1)with F a uniquely determined probability measure, and Ω m ( · ) being the characteristic func-tion of a random vector that is uniformly distributed on the spherical shell of R m . Daryland Porcu [6] put emphasis on the measure F , termed Schoenberg measure there. It is wellknown [13] that any random vector X of R m with characteristic function ϕ can be writtenas X = η R , with η having Ω m as characteristic function, R a positive random variable3istributed according to F , and η and R are stochastically independent. The identity aboveis intended as equality in distribution.The class Φ m is nested, with the following inclusion relation Φ ⊃ Φ ⊃ . . . ⊃ Φ ∞ := (cid:92) m ≥ Φ m , m ∈ N , being strict (see, for example, [6, 13]). A function f : (0 , ∞ ) → R is called completelymonotone if it is infinitely often differentiable and ( − n f ( n ) ( x ) ≥ , for all n ∈ Z + andfor all x > . The set of completely monotone functions on (0 , ∞ ) is denoted CM . BySchoenberg’s theorem, ϕ ( t ) ∈ Φ ∞ if and only if ϕ ( √ t ) ∈ CM with ϕ (0) = ϕ (+0) < ∞ .The plan of the paper is the following. Section 2 introduces the Buhmann class and offersa formal statement of the problem. Section 3 exposes the structure of the solution. We alsocharacterize the smoothness of differences of Buhmann functions in a neighborhood of theorigin. Section 4 illustrates connections with previous literature. We denote C ( R m ) the set of continuous functions on R m , for m = 1 , , . . . . Let δ, µ, ν ∈ C + := { z ∈ C : Re z > } and α ∈ C . Zastavnyi [36] (2006) proposed the following evenfunctions given on R : ϕ δ,µ,ν,α ( x ) := (cid:82) | x | ( s − x ) ν − (1 − s δ ) µ − s α − ν +1 d s, | x | < , | x | ≥ . (2)If δ, µ, ν ∈ C + , then arguments in Proposition 1 and Theorem 1 in [36] show, respec-tively, that ϕ δ,µ,ν,α ∈ C ( − , if and only if α ∈ C + and that ϕ δ,µ,ν,α ∈ C ( R ) if and only if α, µ + ν − ∈ C + If δ, µ, ν, α ∈ C + , then ϕ δ,µ,ν,α (0) = B ( α/δ, µ ) /δ , with B denoting theBeta function.The functions ϕ δ,µ,ν,α coincide (modulo some positive factors) with the functions φ δ,(cid:37),λ,α ( x ) ≡ ϕ δ,(cid:37) +1 ,λ +1 , α +2 ( x ) , x ∈ R , ϕ δ,µ,ν,α includes a wealth of interesting special cases. For instance, µδϕ δ,µ, ,δ ( x ) =(1 − | x | δ ) µ + , which implies that µϕ ,µ, , coincides with the Askey functions [1]. Also, wehave that ϕ ,µ,ν, ν − ( x ) ≡ h µ,ν ( x ) ≡ ν − Γ( ν ) µ ψ µ,ν − ( x ) , x ∈ R , (3)with the functions h µ,ν being introduced by Zastavnyi (2002) [35, 34] and defined as follows: h µ,ν ( x ) := 0 for | x | ≥ and h µ,ν ( x ) := (cid:90) | x | (2 u − | x | ) g µ,ν ( u ) g µ,ν ( u − | x | ) du, | x | < , where g µ,ν ( u ) := u µ − (1 − u ) ν − , u ∈ (0 , , µ, ν ∈ C + . (4)Functions of the form (4) arise in the study of exponential type entire functions without zerosin the lower half-plane [37, Proposition 5.1]. The functions ψ µ,ν − , with µ > , ν ∈ N , havebeen introduced in 1995 by Wendland [30], and they have been termed Wendland functionsin both numerical analysis and geostatistical literatures: for µ > , k ∈ Z + , we have ψ µ, ( x ) := ψ µ ( x ) := (1 − | x | ) µ + , ψ µ,k := I k ψ µ ( k ∈ N ) , where I ( f )( x ) := (cid:82) + ∞| x | sf ( s ) d s is the Matheron’s [14] Mont´ee operator (provided theintegral is well defined), and where I k is the k -fold application of the operator I . Argumentsin [30] and subsequently [13] show that Iϕ belongs to the class Φ m − whenever ϕ ∈ Φ m ,for m ≥ . For k < m , the k -fold application of the Mont´ee operators shows that I k ϕ ∈ Φ m − k , k ∈ N .Gneiting [13, Equation (17)] has proposed a generalization of Wendland functions onthe basis of the fractional Mont´ee operator, which coincides with the normalized Buh-mann functions ϕ ,µ +1 ,ν, ν ( x ) /ϕ ,µ +1 ,ν, ν (0) , µ, ν > , as well as with the functions h µ,ν +1 ( x ) /h µ,ν +1 (0) ≡ ψ µ,ν ( x ) /ψ µ,ν (0) (see Equation (6)). Arguments in [5] show thatWu functions [32] and consequently the spherical model are special cases of the Buhmannclass.For r ∈ Z + and k ∈ N , we have h r + k,r +1 ( x ) ≡ B ( r + k, r + 1) A r, k − ( x ) , with the splines A r, k − introduced by Trigub (1987), and we refer to [26], [27, § A r, k − ( x ) ≡ ψ r + k,r ( x ) /ψ r + k,r (0) , for r ∈ Z + and k ∈ N .For a proof of the identities above, the reader is referred to Zastavnyi and Trigub [35,Remarks 10 and 11], to [34, Theorems 12 and 13], [36] and [38, § δ, µ, ν ∈ C + and x ∈ R ,ϕ , µ , µ + ν, ν − ( x ) ≡ µ − Γ (cid:0) µ (cid:1) Γ (cid:0) µ + ν (cid:1) Γ( µ ) ϕ ,µ,ν, ν − ( x ) , νϕ δ,µ +1 ,ν, ν ( x ) ≡ δµϕ δ,µ,ν +1 , ν + δ ( x ) , (5)and, for µ, ν ∈ C + and x ∈ R , we also have the obvious identities: ϕ ,µ +1 ,ν, ν ( x ) ≡ µ ν ϕ ,µ,ν +1 , ν +1 ( x ) ≡ µ ν h µ,ν +1 ( x ) ≡ ν − Γ( ν ) ψ µ,ν ( x ) . (6) After the illustration of the relation between Buhmann and other celebrated classes of radialbasis function, we need some preliminary material in order to provide a better description ofthe results coming subsequently. For a function h defined on (0 , ∞ ) and m ∈ C , we definethe Hankel transform F m as follows: F m ( h )( t ) := t − m (cid:90) ∞ h ( u ) u m J m − ( tu ) d u = (cid:90) ∞ h ( u ) u m − j m − ( tu ) d u, t > , (7)where J λ is the Bessel function of the first kind (see [28, Sec. 3.1]) and j λ ( x ) := J λ ( x ) x λ = 12 λ ∞ (cid:88) k =0 k + λ + 1) · (cid:16) − x (cid:17) k k ! , x ∈ C , λ ∈ C . (8) Remark 1
For m ∈ N the transform F m is connected with the Fourier transforms F m ofradial functions through the identity F m ( h ( (cid:107) · (cid:107) ))( x ) = (2 π ) m F m ( h )( (cid:107) x (cid:107) ) , x ∈ R m . These facts and Bochner-Khintchine theorem (see, for example [18, 24, 27]) imply that, if h is a continuous functions on [0 , ∞ ) and (cid:82) ∞ t m − | h ( t ) | dt < ∞ , then h ∈ Φ m if and only if F m ( h ) is nonnegative on the positive real line.6or δ, µ, α + 1 ∈ C + and ν ∈ C , we define the function I δ,µ,ν,α : R + → C through I δ,µ,ν,α ( t ) := t − α − − δ ( µ − (cid:90) t ( t δ − u δ ) µ − u α − ν + J ν − ( u ) d u = (cid:90) (1 − x δ ) µ − x α j ν − ( tx )d x , t > . (9)The following result reports succinctly a collection of useful results from [36] (Theo-rems 2, 3 and Proposition 4 (Assertions 1,3)). Theorem 1 (Zastavnyi [36])
Let the functions I and F m as being defined through Equa-tions (9) and (7), respectively. Denote with I (cid:48) the first derivative of I . Then, the followingassertions are true:1. Let δ, µ, ν, m, α + m ∈ C + . Then F m ( ϕ δ,µ,ν,α )( t ) = 2 ν − Γ( ν ) I δ,µ, m − + ν,m − α ( t ) . Moreover, if n, m − n + 2 ν ∈ C + , then F m ( ϕ δ,µ,ν,α )( t ) = 2 n − m Γ( ν )Γ( m − n + ν ) F n ( ϕ δ,µ, m − n + ν,m − n + α )( t ) , t ≥ .
2. Let δ, µ, α + 1 ∈ C + and ν ∈ C . Then I (cid:48) δ,µ,ν,α ( t ) = − tI δ,µ,ν +1 ,α +2 ( t ) , t > . (10)
3. If µ, ν ∈ C + , t > , then I ,µ,ν, ν − ( t ) = 2 − ν Γ( µ )Γ(2 ν )Γ (cid:0) ν + (cid:1) Γ( µ + 2 ν ) F (cid:18) ν ; µ + 2 ν , µ + 2 ν + 12 ; − t (cid:19) . (11)
4. Let δ, µ, ν ∈ C + and α ∈ C . Then ϕ δ,µ,ν +1 ,α +2 ( t ) = 2 ν (cid:90) ∞| t | u ϕ δ,µ,ν,α ( u ) d u , t (cid:54) = 0 . (12)A relevant remark is that Equation (11) describes the spectral density of the Wendlandfunctions. Another remarkable consequence of Theorem 1 is that, for µ, ν, m, ν − m ∈ C + , F m ( h µ,ν )( t ) = F m ( ϕ ,µ,ν, ν − )( t ) = 2 − m Γ( ν )Γ( m − + ν ) F ( h µ, m − + ν )( t ) , t ≥ , (13)7hich in turn shows, in concert with [35, Lemma 12], that in some cases the Hankel trans-forms above can be written in closed form. Specifically, we have F m ( h µ,ν )( t ) = F m ( ϕ ,µ,ν, ν − )( t ) = 2 ν − Γ( ν ) I ,µ, m − + ν,m − ν − ( t ) = D ( m, µ, ν ) · F (cid:18) m −
12 + ν ; m −
12 + ν + µ , m −
12 + ν + µ + 12 ; − t (cid:19) , with D ( m, µ, ν ) := 2 − m Γ( ν )Γ( µ )Γ( m − ν )Γ (cid:0) m + ν (cid:1) Γ( µ + m − ν ) , µ, ν, m, ν − m ∈ C + . (14)Let us use the abuse of notation (cid:98) h µ,ν for the one dimensional Fourier transform of thefunction h µ,ν . We also denote with L the Laplace transform operator. For µ, ν ∈ C + ,arguments in Zastavnyi and Trigub [35, Equation (44)] show that L (cid:16) t ν + µ − (cid:98) h µ,ν ( t ) (cid:17) ( x ) := (cid:90) ∞ e − tx t ν + µ − (cid:98) h µ,ν ( t ) d t = Γ ( ν )Γ( µ )2 ν − x µ (1 + x ) ν , x > . (15)Thus, for µ, ν, m, ν − m ∈ C + and x > , we have L (cid:0) t m − ν + µ − F m ( h µ,ν )( t ) (cid:1) ( x )= 2 − m Γ( ν )Γ( m − + ν ) L (cid:16) t m − ν + µ − F ( h µ, m − + ν )( t ) (cid:17) ( x )= 2 − m Γ( ν )Γ( m − + ν ) · Γ ( m − + ν )Γ( µ )2 m − ν − (2 π ) · x µ (1 + x ) m − + ν = C ( m, µ, ν ) x µ (1 + x ) m − + ν , with C ( m, µ, ν ) := 2 − m Γ( ν )Γ( µ )Γ( m − ν )Γ (cid:0) m + ν (cid:1) = D ( m, µ, ν )Γ( µ + m − ν ) . (16) Remark 2
Equation (16) is the crux of the proof of the main part of Theorem 11 in [34]: ( i ) If ν > and µ ≥ max { ν, } , then h µ,ν ∈ Φ . If, additionally, ( µ ; ν ) (cid:54) = (1; 1) , then thereexist constants c i > , i = 1 , , depending on µ and ν only, such that c ≤ (1 + t ) ν · (cid:98) h µ,ν ( t ) ≤ c , t ∈ R . ( ii ) If ν ≥ , then h µ,ν ∈ Φ ⇐⇒ µ ≥ ν . ( iii ) If m ≥ , then h µ,ν ∈ Φ m ⇐⇒ ν > and µ ≥ m − + ν . In this case, there exist twoconstants c i > , i = 1 , , depending on µ , ν and m , and such that c ≤ (1 + t ) m − + ν · F m ( h µ,ν )( t ) ≤ c , t ≥ . I ,µ,ν, ν − ( t ) for all t > . The-orems on positiveness of the functions I δ,µ,ν,α ( t ) are obtained in [36, Theorems 4,5,6] (thewell-known cases given before Theorem 4 from [36]). We are now able to state our main
Problem 1
Let µ > , ν > , µ + ν > . Then h µ,ν ∈ C ( R ) (see [35, 34]). Let ε > and f µ,ν,ε,β ,β ( x ) := β ε h µ,ν (cid:18) xβ (cid:19) − β ε h µ,ν (cid:18) xβ (cid:19) , x ∈ R . (17)Let m ∈ N . Show the conditions on ( ε, µ, ν ) such that, for any β > β > , we have f µ,ν,ε,β ,β ∈ Φ m . (18)Next section details the structure of the solution and offers the exact smoothness of thenew covariance resulting from the differences of Buhmann functions. Let us start with a general assertion regarding the structure of Problem 1.
Proposition 1
The following conditions are equivalent:1. Condition (18) is satisfied.2. For any β > β > , the function t (cid:55)→ β ε + m F m ( h µ,ν )( β t ) − β ε + m F m ( h µ,ν )( β t ) isnonnegative in interval (0 , ∞ ) .3. The function t ε + m F m ( h µ,ν )( t ) = 2 ν − Γ( ν ) t ε + m I ,µ, m − + ν,m − ν − ( t ) increases in theinterval (0 , ∞ ) .4. The following inequality is true: ( ε + m ) I ,µ, m − + ν,m − ν − ( t ) + tI (cid:48) ,µ, m − + ν,m − ν − ( t ) =( ε + m ) I ,µ, m − + ν,m − ν − ( t ) − t I ,µ, m − + ν +1 ,m − ν +1 ( t ) ≥ , ∀ t > . Let n := m − + ν . Then, ( ε + m ) F ( h µ,n )( t ) − t n F ( h µ,n +1 )( t ) ≥ , ∀ t > .
6. We have L (cid:18) t n + µ − (cid:16) ( ε + m ) (cid:98) h µ,n ( t ) − t n (cid:98) h µ,n +1 ( t ) (cid:17)(cid:19) ( x ) =Γ ( n )Γ( µ )2 n − (cid:18) ε + mx µ (1 + x ) n − nx µ (1 + x ) n +1 (cid:19) ∈ CM . ε − ν + 1 + ( ε + m ) x x µ (1 + x ) m − + ν +1 ∈ CM (19) Remark 3
It follows from Hausdorff-Bernstein-Widder theorem (see, for example, [9, 18,24, 31]) that if g ∈ C [0 , + ∞ ) and its Laplace transform Lg ( x ) := (cid:90) + ∞ e − xs g ( s ) ds converges for all x > , then g ( s ) ≥ for s ≥ if and only if Lg ∈ C M .The proof Proposition 1 is an easy consequence of Remark 1 in concert with the Hausdorff-Bernstein-Widder theorem (see Remark 3), Theorem 1 (statements 1 and 2), and equali-ties (13) and (15).Note that, if µ, ν > , then x − µ (1 + x ) − ν ∈ CM if and only if (cid:98) h µ,ν ( t ) ≥ for all t > ,if and only if I ,µ,ν, ν − ( t ) ≥ for all t > (see Equations (15) and (14)).Proposition 1 will now be combined with the following facts:1. If ν ≥ , then x − µ (1 + x ) − ν ∈ CM if and only if µ ≥ ν . The sufficiency of this resultcan be found in [10], and the necessity has been proved in [16], [33, Lemma 8]).2. If < ν < , µ ≥ , then x − µ (1 + x ) − ν ∈ CM [16] and [37, Example 5.4], [38, § ν > , µ ≥ ν , then x − µ (1 + x ) − ν ∈ CM [2].4. If n = 1 , , , then ( a + x ) / ( x n (1 + x ) n ) ∈ CM if and only if a ≥ / (2 n − + 1) [35, § ν > and µ ≥ min { ν ; max { , ν }} , then x − µ (1 + x ) − ν ∈ CM .The combination of these facts with Proposition 1 has just offered the proof of the fol-lowing Theorem 2
The following assertions are true:1. If (18) is true, then ε ≥ ν − .2. If m ∈ N , ν > , ε ≥ ν − and µ ≥ ( m − / ν + 3 , then condition (18) is true.If, in addition, ε = 2 ν − , then (18) is true if and only if µ ≥ ( m − / ν + 3 .3. Suppose that for some n = 1 , , , we have ε ≥ − n ( m + (2 ν − n − + 1)) , ( m − / ν + 1 − n > and µ − n ≥ min (cid:26) m − ν + 2 − n ; max { m −
12 + ν + 1 − n } (cid:27) . Then, condition (18) is true.
We now provide a characterization result for the following problem: wether the condi-tion (18) is satisfied for fixed β , β > (and not for any β > β > as in the problem 1). Theorem 3
Let µ > , ν > , µ + ν > and m ∈ N . Let ε ∈ R , β , β > and a := β /β .Then f µ,ν,ε,β ,β ∈ Φ m if and only if x µ (1 + x ) m − + ν − a ν − − ε x µ (1 + a x ) m − + ν ∈ CM . (20) If f µ,ν,ε,β ,β ∈ Φ m and β > β > , then ε ≥ ν − . Proof.
From Remarks 1 and 3, it follows that the following conditions are equivalent:1. f µ,ν,ε,β ,β ∈ Φ m .2. β ε + m F m ( h µ,ν )( β t ) − β ε + m F m ( h µ,ν )( β t ) ≥ for all t ∈ (0 , ∞ ) .3. L (cid:0) t m − ν + µ − (cid:0) β ε + m F m ( h µ,ν )( β t ) − β ε + m F m ( h µ,ν )( β t ) (cid:1)(cid:1) ( x ) ∈ CM .Furthermore, we have (see (16)): β ε + m L (cid:0) t m − ν + µ − F m ( h µ,ν )( βt ) (cid:1) ( x ) = β ε − ν +1 x µ (1 + x /β ) m − + ν , β > , x > . (0 , + ∞ ) , we have that − a ν − − ε ≥ . If, additionally, a = β /β > ,then ε ≥ ν − . The proof is completed. (cid:4) Direct inspection of the proof of Proposition 1 as well as the proof of Theorem 3 showsthat the following proposition is true.
Proposition 2
Let µ, ν, m, ν − m > and ε ∈ R . Then following conditions areequivalent: . ε − ν + 1 + ( ε + m ) x x µ (1 + x ) m − + ν +1 ∈ CM . . x µ (1 + x ) m − + ν − a ν − − ε x µ (1 + a x ) m − + ν ∈ CM , ∀ a > . . x µ (1 + x ) m − + ν − a ν − − εn x µ (1 + a n x ) m − + ν ∈ CM for some sequence a n > , a n → . If the first condition is satisfied, or if the second condition is satisfied for some a > , then ε ≥ ν − . Proof
The crux of the proof is in the following equality, which holds for for a > : (cid:90) a ( ε − ν + 1 + ( ε + m ) x t ) t ν − ε − x µ (1 + x t ) m − + ν +1 dt = 1 x µ (1 + x ) m − + ν − a ν − − ε x µ (1 + a x ) m − + ν . Since completely monotone functions are closed under scale mixtures, the equality aboveprovides the implication ⇒ . Assertion ⇒ is obvious. To prove assertion ⇒ , it is necessary to take in the last equation a = a n , divide both sides by a n − andtake the limit as n → ∞ , Finally, we make use of the well-known fact: if a sequence ofcompletely monotone functions converges pointwise on (0 , + ∞ ) , then the limit function isalso completely monotone. (cid:4) We conclude this section detailing the exact smoothness of the differences of Buhmannfunctions.
Theorem 4
Let ν ∈ N , µ > , ε ∈ R , β , β > , and β (cid:54) = β . Let q := min( β , β ) . Then,1. If ε (cid:54) = 2 ν − , then f µ,ν,ε,β ,β ∈ C ν − ( − q, q ) , and f µ,ν,ε,β ,β (cid:54)∈ C ν − ( − q, q ) .2. If ε = 2 ν − , µ / ∈ { , } , then f µ,ν,ε,β ,β ∈ C ν ( − q, q ) , and f µ,ν,ε,β ,β (cid:54)∈ C ν +1 ( − q, q ) . . If ε = 2 ν − , µ = 1 or µ = 2 , then f µ,ν,ε,β ,β is a even polynomial of degree at most µ + 2 ν − on [ − q, q ] , and therefore f µ,ν,ε,β ,β ∈ C ∞ ( − q, q ) . Proof. If µ, ν > , then arguments in [35, Equality (40)] show that h µ,ν ( x ) = (1 − x ) µ + ν − (cid:90) t µ − (1 − t ) ν − (1 − t + (1 + t ) x ) ν − d t , x ∈ (0 , . (21)Let ν ∈ N . Then from Proposition 1 in [36] we have that h µ,ν ∈ C ν − ( − , and h µ,ν (cid:54)∈ C ν − ( − , . Thus, from (21) it follows that h µ,ν ( x ) = ∞ (cid:88) k =0 a k ( µ, ν ) x k + | x | ν − ∞ (cid:88) k =0 b k ( µ, ν ) x k , | x | < ,b ( µ, ν ) (cid:54) = 0 , b ( µ, ν ) = h (2 ν +1) µ,ν (+0)(2 ν + 1)! . (22)From (12) we have that h (cid:48) µ,ν ( x ) = − ν − x h µ,ν − ( x ) for ν ≥ , x > , and h ( k +1) µ,ν ( x ) = − ν − x h ( k ) µ,ν − ( x )+ k h ( k − µ,ν − ( x )) for k ≥ , < x < . From the last equation it followsthat h ( k +1) µ,ν (+0) = − ν − k h ( k − µ,ν − (+0) for ν ≥ , k ≥ , and for ν ≥ , k ≥ ν − , wehave (for convenience, we consider that and ( − ) h ( k +1) µ,ν (+0) = ( − ν − ( ν − · k !!( k − ν + 2)!! h ( k − ν +3) µ, (+0) . (23)The equality (23) is true for ν = 1 , k ≥ − . It’s obvious that h µ, ( x ) = (1 − | x | ) µ + /µ and h ( p ) µ, (+0) = ( − p Γ( µ ) / Γ( µ − p + 1) , p ∈ Z + . Therefore, for k ≥ ν − , with ν ∈ N , wehave h (2 k +1) µ,ν (+0) = ( − ν − ( ν − · k !( k − ν + 1)! · h (2 k − ν +3) µ, (+0)= − ( − ν − ( ν − · k !( k − ν + 1)! · Γ( µ )Γ( µ − k + 2 ν −
2) ; h (2 ν +1) µ,ν (+0) = − ( − ν − ( ν − ν !( µ − µ − . On the other hand, (22) shows that, for | x | < q = min( β , β ) , f µ,ν,ε,β ,β ( x ) = ∞ (cid:88) k =0 a k ( µ, ν ) (cid:0) β ε − k − β ε − k (cid:1) x k ++ | x | ν − ∞ (cid:88) k =0 b k ( µ, ν ) (cid:0) β ε − ν +1 − k − β ε − ν +1 − k (cid:1) x k . Assertions 1. and 2. are thus proved. 13f, in addition, µ ∈ N , then h µ,ν is a polynomial of degree µ + 2 ν − on the compactinterval [0 , . If µ = 1 or µ = 2 , then in the second sum of (22), the terms with k ≥ vanish. Therefore, if ε = 2 ν − , µ = 1 or µ = 2 , then f µ,ν,ε,β ,β on the interval [ − q, q ] is aeven polynomial of degree ≤ µ + 2 ν − . Assertion 3. is proved. The proof is completed. (cid:4) The k -fold application of the Mont´ee operator to the Askey function A µ ( t ) = (1 − t ) µ + , for µ ≥ ( m + 1) / , results in Wendland functions ψ µ,k as described in previous section. Table1 depicts the role of the mapping f µ,ν,ε,β ,β as defined through Equation (17). In particular,we consider f µ,k +1 , k +1 ,β ,β , for k = 0 , , . Expressions of the corresponding Wendlandfunctions are reported in the second column of the same table. According to Theorem 2, f µ,k +1 , k +1 ,β ,β ∈ Φ m if and only if µ ≥ ( m + 7) / k . The third and fourth column allowto describe the action of the mapping f µ,k +1 , k +1 ,β ,β . When µ / ∈ { , } , one can clearlyappreciate the increase in terms of differentiability at the origin. k ψ µ,k D before D after (1 − x ) µ + C ( { } ) C ( { } ) (1 − x ) µ +1+ (1 + ( µ + 1) x ) C ( { } ) C ( { } )2 (1 − x ) µ +2+ (1 + ( µ + 2) x + (( µ + 2) − C ( { } ) C ( { } ) Table 1: Examples of the Wendland functions ψ µ,k for k = 0 , , and µ / ∈ { , } . For allcases, x ≥ . The first column reports the values of k , and the corresponding expressionin the second column reports the analytic expression of ψ µ,k . In the third column, D before depicts the differentiability of ψ µ,k . In the last column, D after stays for the differentiabilityat the origin of f µ,k +1 , k +1 ,β ,β . Observe that when µ = 1 or µ = 2 , then D after = ∞ .14 .0 . . . . . . . . . . . . . . . . . . Figure 1: Continuous Lines: f ( d +1) / k +3 ,k +1 , k +1 , . , , defined according to Equation (17)for k = 0 , , (from left to right). Dashed and Dashed-dot Lines report Wendland func-tions with k = 0 , , and compact supports β = 0 . and β = 1 . All the functions arenormalized with their value at the origin.The Wendland radial basis functions are piecewise polynomial compactly supported re-producing kernels in Hilbert spaces which are normequivalent to Sobolev spaces. But theyonly cover the Sobolev spaces H d/ k +1 / ( R d ) , when k ∈ N . Motivated by this fact, RobertSchaback [20] covered the case of the integer order spaces in even dimensions. Namely, hederived the missing Wendland functions working for half-integer k and even dimensions, re-producing integer-order Sobolev spaces in even dimensions, and showing that they turn outto have two additional non-polynomial terms: a logarithm and a square root.Other walks through dimensions have been recently proposed by [17] through the Gen-eralized Askey functions ϕ n,k,m : [0 , ∞ ) R defined through ϕ n,k,m ( t ) = t k − n (1 − t ) n + m +1+ F ( n − k, n + 1 , n + m + 2 , − /t ) , t ≥ , with F being the Gauss hypergeometric function. The parameters ( n, m, k ) are then shownto be crucial in order to determine when Φ m (see their Proposition 2.3).To our knowledge, the only case of compactly supported correlation functions whichis not covered by this work is the case of the Euclid’s hat ([19, 12]), which is the self-convolution of an indicator function supported on the unit ball in R d . As noted by [20], whileEuclids hat is not differentiable and Wus functions have zeros in their Fourier transform,Wendlands functions have no such drawbacks. They are polynomials on [0 , and yieldpositive definite k -times differentiable radial basis functions on R d . Given these properties,their polynomial degree (cid:98) d/ (cid:99) + 3 k + 1 is minimal.15he proof of Theorem 2 highlights explicit connections with previous literature devotedto (sub) classes of completely monotone functions. A function f : [0 , ∞ ) → R is calledLogarithmically completely monotonic on (0 , ∞ ) , and denoted f ∈ L (0 , ∞ ) if and only if itis infinitely often differentiable on (0 , ∞ ) and ( − n [log f ( x )] ( n ) ≥ , x ≥ . By well known results (see [3], with the references therein) f ∈ L ⇔ f α ∈ CM ⇔ ( f ) /n ∈CM , for all α > and n ∈ N .Berg, Porcu and Mateu [3] introduced the so called Auxiliary family f α,β ( x ) = 1 x α (1 + x β ) , x > , with α, β positive parameters. They show that f α,β ∈ L for α ≥ and ≤ β ≤ .Additionally, f α, ∈ L if and only if α ≥ . This resut is one of the crux for describing thecomplete monotonicity of the Dagum family. Known cases, when x − µ (1 + x ) − ν / ∈ CM (if µ, ν > , then this is equivalent to the inequality I ,µ,ν, ν − ≥ is not true for all t > ): 1) µ < or µ = 0 , ν (cid:54) = 0 (it is obvious); 2) µ < ν ; 3) < µ = ν < . Proof of the latter twocases, see, for example, [16], [33, Lemma 8] and [36, Theorem 5]. Acknowledgement
We are grateful to Professor R. Furrer for helpful remarks through the preparation of themanuscript. Research work of Moreno Bevilacqua was partially supported by grant FONDE-CYT 11121408 from Chilean government. Research work of Emilio Porcu was partiallysupported by grant FONDECYT 1130647 from Chilean government.
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