## Stieltjes-Pick-Bernstein-Schoenberg and their connection to complete monotonicity (2007)

Citations: | 3 - 1 self |

### BibTeX

@MISC{Berg07stieltjes-pick-bernstein-schoenbergand,

author = {Christian Berg},

title = {Stieltjes-Pick-Bernstein-Schoenberg and their connection to complete monotonicity},

year = {2007}

}

### OpenURL

### Abstract

This paper is mainly a survey of published results. We recall the definition of positive definite and (conditionally) negative definite functions on abelian semigroups with involution, and we consider three main examples: Rk, [0, ∞ [ k, N0–the first with the inverse involution and the two others with the identical involution. Schoenberg’s theorem explains the possibility of constructing rotation invariant positive definite and conditionally negative definite functions on euclidean spaces via completely monotonic functions and Bernstein functions. It is therefore important to be able to decide complete monotonicity of a given function. We combine complete monotonicity with complex analysis via the relation to Stieltjes functions and Pick functions and we give a survey of the many interesting relations between these classes of functions and completely monotonic functions, logarithmically completely monotonic functions and Bernstein functions. In Section 6 it is proved that log x − Ψ(x) and Ψ ′ (x) are logarithmically completely monotonic (where Ψ(x) = Γ ′ (x)/Γ(x)), and these results are new as far as we know. We end with a list of completely monotonic functions related to the Gamma function.

### Citations

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Citation Context ... ∞. 1 + ξ 0 See [14] for details and compare with Theorem 2.6. It follows easily that f(x) α = lim x→∞ x , f ′ (x) = L [ αδ0 + ξ1]0,∞[(ξ) dν(ξ) ] (x), (18) and in particular f(x) = O(x) for x → ∞. In =-=[18]-=- Bernstein functions are called Laplace exponents because of the following result, which shall be compared with Theorem 2.2: Theorem 5.2 For a function f : ]0, ∞[ → [0, ∞[ the following are equivalent... |

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Citation Context ... kernels and functions play an important role in probability theory, Dirichlet spaces, potential theory, and in relation with isometric embeddings of metric spaces into Hilbert spaces, see Schoenberg =-=[42]-=- and the treatment in [13]. Let (S, +, ∗) be an abelian semigroup with neutral element and involution ∗ like in Section 1. Definition 2.1 A function ψ : S → C is called (conditionally) negative defini... |

125 |
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49 |
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Citation Context ...f positive measures on [0, ∞[, see [17]. Stieltjes moment sequences, i.e. moment sequences of positive measures on [0, ∞[ were introduced and characterized by Stieltjes in 1894 in the pioneering work =-=[45]-=-, where he introduced the Stieltjes integral with respect to a distribution of mass. His results were extended to a characterization of moment sequences of measures on the real line by Hamburger in 19... |

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Citation Context ... functions are also called Nevanlinna functions. This is reflected in the notation N . (R. Nevanlinna (1895-1980), Finnish). Pick and Nevanlinna studied interpolation problems for Pick functions, see =-=[35]-=-,[33] and Nevanlinna related this class to the indeterminate moment problem, see [33]. Pick functions are closely related to operator monotone functions studied by Loewner, see [22]. We return to this... |

28 |
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Citation Context ...ν(]0, ∞[) < ∞ by the monotonicity theorem of Lebesgue. Since ν is a finite measure we can integrate term by term in (17), and the result follows. □ 6 Logarithmically completely monotonic functions In =-=[24]-=- the authors call a function f :]0, ∞[→]0, ∞[ logarithmically completely monotonic if it is C ∞ and (−1) k [log f(x)] (k) ≥ 0, for k = 1, 2, . . . . (24) If we denote the class of logarithmically comp... |

24 |
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Citation Context ...ourier transforms of positive finite Radon measures on the dual group. Completely monotonic functions were characterized by Bernstein in 1928 as Laplace transforms of positive measures on [0, ∞[, see =-=[17]-=-. Stieltjes moment sequences, i.e. moment sequences of positive measures on [0, ∞[ were introduced and characterized by Stieltjes in 1894 in the pioneering work [45], where he introduced the Stieltjes... |

18 |
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(Show Context)
Citation Context ...amj j=1 ) Then Ln ∈ L. For n = 2 the theorem states that for any a1, a2 > 0 ga1,a2(x) = Γ(x)Γ(x + a1 + a2) Γ(x + a1)Γ(x + a2) ∈ L. It was earlier established by Bustoz and Ismail that ga1,a2 ∈ C, see =-=[20]-=-. In [30, Theorem 6.1] is given a list of functions in L. 7 Some completely monotonic functions related to the Gamma function There are many completely monotonic functions related to the Γ-function, s... |

18 |
Complete monotonicities of functions involving the gamma and digamma functions
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(Show Context)
Citation Context ... C ∞ -function such that −(log f) ′ ∈ C. The functions of class L have been implicitly studied in [3], and Lemma 2.4(ii) in that paper can be stated as the inclusion L ⊂ C, a fact also established in =-=[23]-=-. The class L can be characterized in the following way, established by Horn[29, Theorem 4.4]: Theorem 6.1 For a function f : ]0, ∞[ →]0, ∞[ the following are equivalent: 17(i) f ∈ L, (ii) f α ∈ C fo... |

16 |
Subordination in the sense of Bochner and a related functional calculus
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(Show Context)
Citation Context ...f(x) = β − + αx + − dˇτ(t) + x 0 t 0 t x + t 0 x + t dˇτ(t), which shows (iii) with σ = ( ∫ ∞ β − 0 ) ( dˇτ(t) δ0 + t + t 1 ) dˇτ(t).□ t Remark 4.11 A proof of Theorem 4.10 is also given in Schilling =-=[39]-=- as well as in his Dissertation [38]. The functions of Theorem 4.10 are called complete Bernstein functions by Schilling. 5 Bernstein functions The terminology Bernstein function is not universally ac... |

15 | A class of completely monotonic functions
- ALZER, BERG
(Show Context)
Citation Context ... ∫ b a (x + ξ)−1 dξ, 0 ≤ a < b. (c) 1/ log(1 + x); (log(1 + x))/x (Apply (i) above to example (b) with a = 0, b = 1 and apply then (ii)). (d) (x log x)/ log Γ(x + 1), cf. [15]. (e) (1 + 1/x) −x , cf. =-=[2]-=-. (f) See [2] [ ( (x + 1) e − 1 + 1 ) x] = x e ∫ 1 1 ξ + 2 π 0 ξ (1 − ξ) 1−ξ sin(πξ) dξ. x + ξ (g) [Γ(1 + 1/x)] x , cf. [3]. (h) See [9] [Γ(x + 1)]1/x Φ(x) = x 9 ( 1 + 1 ) x ; log Φ. x4 Pick function... |

15 | Integral representation of some functions related to the gamma function
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(Show Context)
Citation Context .... (d) (x log x)/ log Γ(x + 1), cf. [15]. (e) (1 + 1/x) −x , cf. [2]. (f) See [2] [ ( (x + 1) e − 1 + 1 ) x] = x e ∫ 1 1 ξ + 2 π 0 ξ (1 − ξ) 1−ξ sin(πξ) dξ. x + ξ (g) [Γ(1 + 1/x)] x , cf. [3]. (h) See =-=[9]-=- [Γ(x + 1)]1/x Φ(x) = x 9 ( 1 + 1 ) x ; log Φ. x4 Pick functions Pick functions have a long history and a whole book is dedicated to them, see [22]. The upper half-plane is denoted H, i.e. H = {z ∈ C... |

13 |
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Citation Context ...n [30, Theorem 6.1] is given a list of functions in L. 7 Some completely monotonic functions related to the Gamma function There are many completely monotonic functions related to the Γ-function, see =-=[30]-=-. For the Digamma function Ψ(x) = Γ ′ (x)/Γ(x) we have the classical formula, see [25] (−1) n+1 Ψ (n) (x) = n! ∞∑ k=0 [ 1 = L (x + k) n+1 ξ n 1 − e −ξ which show that these functions are completely mo... |

12 | A completely monotone function related to the Gamma function
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(Show Context)
Citation Context ...1. (b) log((x + b)/(x + a)) = ∫ b a (x + ξ)−1 dξ, 0 ≤ a < b. (c) 1/ log(1 + x); (log(1 + x))/x (Apply (i) above to example (b) with a = 0, b = 1 and apply then (ii)). (d) (x log x)/ log Γ(x + 1), cf. =-=[15]-=-. (e) (1 + 1/x) −x , cf. [2]. (f) See [2] [ ( (x + 1) e − 1 + 1 ) x] = x e ∫ 1 1 ξ + 2 π 0 ξ (1 − ξ) 1−ξ sin(πξ) dξ. x + ξ (g) [Γ(1 + 1/x)] x , cf. [3]. (h) See [9] [Γ(x + 1)]1/x Φ(x) = x 9 ( 1 + 1 ) ... |

12 |
Completely monotonic functions involving the gamma and q-gamma functions
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Citation Context ...x), (27) so Ψ ′ ∈ C. As before we also have Ψ ′ ∈ L since 1/(1 − e−x ) ∈ C. The latter holds because ∞∑ 1 = e 1 − e−x −kx [ ∞∑ ] = L δk (x). k=0 The following result was given in Grinshpan and Ismail =-=[26]-=-: Lemma 6.3 Let αk, βk, k = 1, . . . , n be real numbers such that ∑ n k=1 αk = 0 and βk ≥ 0 for all k. Then u(x) := k=0 n∏ Γ αk (x + βk) ∈ L k=1 if and only if v(ξ) = n∑ αkξ βk ≥ 0 for ξ ∈ ]0, 1] . k... |

10 |
Quelques remarques sur le cône de Stieltjes
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(Show Context)
Citation Context ...cone S is logarithmically convex, was established in [6]. Property (vi) is a special case of (v) with g = 1. All the above properties of S can easily be deduced from Theorem 3.2 as was pointed out in =-=[7]-=-. Examples of Stieltjes functions (a) x−α , 0 < α ≤ 1. (b) log((x + b)/(x + a)) = ∫ b a (x + ξ)−1 dξ, 0 ≤ a < b. (c) 1/ log(1 + x); (log(1 + x))/x (Apply (i) above to example (b) with a = 0, b = 1 and... |

9 |
Monotone Matrix Functions and Analytic Continuation
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Citation Context ...(πξ) dξ. x + ξ (g) [Γ(1 + 1/x)] x , cf. [3]. (h) See [9] [Γ(x + 1)]1/x Φ(x) = x 9 ( 1 + 1 ) x ; log Φ. x4 Pick functions Pick functions have a long history and a whole book is dedicated to them, see =-=[22]-=-. The upper half-plane is denoted H, i.e. H = {z ∈ C | ℑz > 0}, and the set of holomorphic functions in a domain G is denoted H(G). Definition 4.1 A function f ∈ H(H) is called a Pick function if ℑf(z... |

8 | Correspondance d'Hermite et de Stieltjes I,II - Baillaud, Bourget - 1905 |

8 | On powers of Stieltjes moment sequences
- Berg
(Show Context)
Citation Context ...0 for all n is infinitely divisible if and only if (− log sn) is conditionally negative definite on (N0, +). Tyan’s result was reexamined in the case of Stieltjes moment sequences in the recent paper =-=[11]-=-, which contains several examples of infinitely divisible Stieltjes moment sequences. Coming back to rotation invariant functions on Rk we have the following counterpart of Theorem 1.6. It follows by ... |

8 | Generation of generators of holomorphic semigroups - Berg, Boyadzhiev, et al. - 1993 |

6 | Inequalities involving gamma and psi function
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(Show Context)
Citation Context ...gamma function Ψ(x) = Γ ′ (x)/Γ(x) we have the classical formula, see [25] (−1) n+1 Ψ (n) (x) = n! ∞∑ k=0 [ 1 = L (x + k) n+1 ξ n 1 − e −ξ which show that these functions are completely monotonic. In =-=[21]-=- Clark and Ismail introduced the functions Gm(x) = x m Ψ(x) and Φm(x) = −x m Ψ (m) (x). ] (x), n = 1, 2, . . . , (28) They proved that G (m+1) m is completely monotonic for m = 1, 2, ... and that Φ (m... |

6 |
Positive Definite and definitizable functions, Akademie
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(Show Context)
Citation Context ...s an important chapter 1in all treatments of harmonic analysis. They can be traced back to papers by Carathéodory, Herglotz, Bernstein, Matthias, culminating in Bochner’s theorem from 1932-1933. See =-=[37]-=- for details. Shortly after Bochner’s characterization of continuous positive definite functions on R or R k , his theorem was extended to the framework of locally compact abelian groups: The continuo... |

6 |
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(Show Context)
Citation Context ... that the sequence stops being completely monotonic when m becomes very large. For α, β > 0 the generalized Mittag-Leffler function is defined by Fα,β(x) = Γ(β) ∞∑ k=0 It was established by Schneider =-=[40]-=- that (−x) k , x > 0. (29) Γ(αk + β) Fα,β ∈ C ⇔ 0 < α ≤ 1, α ≤ β. Note that F1,1(x) = e −x . The complete monotonicity of Fα,β for 0 < α < 1, β = 1 was established by Feller and Pollard. 20References... |

5 | Pick functions related to the gamma function
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(Show Context)
Citation Context ...thm, i.e. Log z = log |z| + i Arg z, since the principal argument Arg z ∈ ]0, π[ for z ∈ H. (d) zα = eα Log z for 0 < α ≤ 1. (e) tan z, since sinh y cosh y ℑ tan(x + iy) = cos2 x + sinh 2 y . (f) See =-=[16]-=- log Γ(x + 1) ; x log x log Γ(x + 1) . x (g) See [34] for Pick functions related to canonical products with negative zeros. In 1911 G. Herglotz gave an integral representation of the holomorphic funct... |

5 |
Über eine Volterrasche Integralgleichung mit totalmonotonem
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(Show Context)
Citation Context ... f ◦ 1 1 , g f◦g ∈ S (v) f, g ∈ S, 0 < α < 1 ⇒ f α g 1−α ∈ S. (vi) f ∈ S, 0 < α < 1 ⇒ f α ∈ S. The properties (i),(iii) and (iv) were proved by Hirsch in [28]. Property (ii) was established by Reuter =-=[36]-=- and independently by Itô [31], but already Stieltjes noticed it in a letter to Hermite, see [5, Lettre 426]. The property (v), showing that the cone S is logarithmically convex, was established in [6... |

4 |
On a generalized gamma convolution related to the q-calculus
- Berg
- 2005
(Show Context)
Citation Context ...(x) = log Γ(x + 1)/x log(x), cf. [15]. (f) f(x) = (1 + 1/x) x , f(x) = (1 + x) 1+1/x , cf. [2]. (g) f(x) = ∑∞ n=0 log(1 + xqn ), where 0 < q < 1. The corresponding convolution semigroup is studied in =-=[10]-=-. (h) (log Pρ(x))/x1/ρ , where ρ > 1 and Pρ is the canonical product (cf. [34]) Pρ(x) = ∞∏ n=1 ( 1 + x nρ ) . The paper [32] contains a supplementary list of Bernstein functions. As an important appli... |

4 |
les cônes convexes de Riesz et les noyaux complètement sousharmoniques
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(Show Context)
Citation Context ...∈ S, 0 < α < 1 ⇒ f α g 1−α ∈ S. (vi) f ∈ S, 0 < α < 1 ⇒ f α ∈ S. The properties (i),(iii) and (iv) were proved by Hirsch in [28]. Property (ii) was established by Reuter [36] and independently by Itô =-=[31]-=-, but already Stieltjes noticed it in a letter to Hermite, see [5, Lettre 426]. The property (v), showing that the cone S is logarithmically convex, was established in [6]. Property (vi) is a special ... |

4 |
Asymptotische Entwicklungen beschränkter Funktionen und das Stieltjessche Momentenproblem
- Nevanlinna
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(Show Context)
Citation Context ...tions are also called Nevanlinna functions. This is reflected in the notation N . (R. Nevanlinna (1895-1980), Finnish). Pick and Nevanlinna studied interpolation problems for Pick functions, see [35],=-=[33]-=- and Nevanlinna related this class to the indeterminate moment problem, see [33]. Pick functions are closely related to operator monotone functions studied by Loewner, see [22]. We return to this in T... |

4 |
The structure of Bivariate distribution functions and their relation to Markov processes
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(Show Context)
Citation Context ...g ∫ (1 − x) 2 ∫ dν(x) < ∞, |x| n dν(x)∞, ∀n ∈ N0. 0<|x−1|<1 |x−1|≥1 For the proof of Theorem 2.7 we refer to [13]. Theorem 2.7 is related to infinitely divisible moment sequences in the sense of Tyan =-=[46]-=-. He defined a moment sequence (sn) to be infinitely divisible if (i) sn ≥ 0 for all n ≥ 0, (ii) (s c n) is a moment sequence for all c > 0. The main result of Tyan can be formulated: Theorem 2.8 A mo... |

2 |
On a conjecture of
- Alzer, Berg, et al.
(Show Context)
Citation Context ...ic for m = 1, 2, ... and that Φ (m) m is completely monotonic for m = 1, 2, ..., 16. Clark and Ismail conjectured that Φ (m) m is completely monotonic for all natural numbers m. This was disproved in =-=[4]-=-, where it was shown that the sequence stops being completely monotonic when m becomes very large. For α, β > 0 the generalized Mittag-Leffler function is defined by Fα,β(x) = Γ(β) ∞∑ k=0 It was estab... |

2 |
The Stieltjes cone is logarithmically convex
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(Show Context)
Citation Context ...6] and independently by Itô [31], but already Stieltjes noticed it in a letter to Hermite, see [5, Lettre 426]. The property (v), showing that the cone S is logarithmically convex, was established in =-=[6]-=-. Property (vi) is a special case of (v) with g = 1. All the above properties of S can easily be deduced from Theorem 3.2 as was pointed out in [7]. Examples of Stieltjes functions (a) x−α , 0 < α ≤ 1... |

2 |
Function spaces as Dirichlet spaces (about a paper by W. Maz ′ ya and
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(Show Context)
Citation Context ...n ), where 0 < q < 1. The corresponding convolution semigroup is studied in [10]. (h) (log Pρ(x))/x1/ρ , where ρ > 1 and Pρ is the canonical product (cf. [34]) Pρ(x) = ∞∏ n=1 ( 1 + x nρ ) . The paper =-=[32]-=- contains a supplementary list of Bernstein functions. As an important application of Corollary 5.3 we note that f ∈ B, 0 < α < 1 ⇒ f(x) α , f(x α ) ∈ B, (20) f ∈ C, 0 < α < 1 ⇒ f(x α ) ∈ C. (21) We m... |

1 |
Fonctions définies négatives et majoration de Schur. In: Colloque de Théorie du Potentiel Jacques Deny–Orsay
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(Show Context)
Citation Context ...and ν (the Lévy-measure) is a positive measure on Rk \ {0} satisfying ∫ ||ξ|| 2 ∫ dν(ξ) < ∞, e 〈x,ξ〉 dν(ξ) < ∞, ∀x ∈ [0, ∞[ k . 0<||ξ||≤1 ||ξ||>1 A complete proof of the above theorem can be found in =-=[8]-=-. In the case of k = 1 and nonnegative functions, Theorem 2.5 reduces to the following result, see [13, Th. 4.4.3]: Theorem 2.6 A function ψ : [0, ∞[→ [0, ∞[ is conditionally negative definite and con... |

1 |
Fonctions opérant sur les fonctions définies-négatives, Ann. Inst. Fourier 17,1
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(Show Context)
Citation Context ...nuous, conditionally negative definite function. Conversely, any bounded, continuous, conditionally negative definite function on R k has this form. This was proved (even for LCA-groups) by Harzallah =-=[27]-=-, cf. [14, Prop. 7.13]. In the case of S = [0, ∞[ k considered as an abelian semigroup with the identical involution x ∗ = x, we have the following result, which is the counterpart of Theorem 1.2 (not... |

1 |
Intégrales de résolventes et calcul symbolique, Ann. Inst. Fourier 22
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- 1972
(Show Context)
Citation Context ... an iterated Laplace transform. This also shows that f ∈ S is completely monotonic. The cone S is closed under pointwise convergence. For the proof of this and applications of Stieltjes functions see =-=[28]-=-. Formula (11) also shows that a Stieltjes function has a holomorphic extension to the cut plane A := C\] − ∞, 0]. The formula 1 x(1 + x2 ) = ∫ ∞ e −xξ (1 − cos(ξ)) dξ 0 shows that 1/x(1 + x 2 ) is co... |

1 |
Canonical products of small order and related Pick functions
- Pedersen
(Show Context)
Citation Context ...pal argument Arg z ∈ ]0, π[ for z ∈ H. (d) zα = eα Log z for 0 < α ≤ 1. (e) tan z, since sinh y cosh y ℑ tan(x + iy) = cos2 x + sinh 2 y . (f) See [16] log Γ(x + 1) ; x log x log Γ(x + 1) . x (g) See =-=[34]-=- for Pick functions related to canonical products with negative zeros. In 1911 G. Herglotz gave an integral representation of the holomorphic functions defined in the unit disc and having nonnegative ... |

1 |
Zum Pfadverhalten von Markovschen Prozessen, die mit Lévy-Prozessen vergleichbar sind. Dissertation, Universität Erlangen
- Schilling
- 1994
(Show Context)
Citation Context ... t x + t 0 x + t dˇτ(t), which shows (iii) with σ = ( ∫ ∞ β − 0 ) ( dˇτ(t) δ0 + t + t 1 ) dˇτ(t).□ t Remark 4.11 A proof of Theorem 4.10 is also given in Schilling [39] as well as in his Dissertation =-=[38]-=-. The functions of Theorem 4.10 are called complete Bernstein functions by Schilling. 5 Bernstein functions The terminology Bernstein function is not universally accepted. It has been used in potentia... |

1 |
Spherical distributions: Schoenberg (1938) revisited, Expo
- Steerneman, Kleij, et al.
- 2004
(Show Context)
Citation Context ...e have appeared many proofs of Schoenberg’s Theorem. We mention the abstract approach via Schoenberg triples in the monograph [13], which also works for non-continuous positive definite functions. In =-=[43]-=- there is a beautiful analytical proof based on Scheffé’s Lemma. 42 Conditionally negative definite functions Conditionally negative definite kernels and functions play an important role in probabili... |