complex positive de niteness, including characteristic and
TRANSCRIPT
Universidade de LisboaFaculdade de Ciencias
Departamento de Matematica
Complex positive definiteness,including characteristic and moment
generating functions
Alexandra Symeonides
Dissertacao
Mestrado em Matematica
2013
Universidade de LisboaFaculdade de Ciencias
Departamento de Matematica
Complex positive definiteness,including characteristic and moment
generating functions
Alexandra Symeonides
Dissertacao
Mestrado em Matematica
Orientador: Professor Doutor Jorge Buescu
2013
Resumo
A partir do inıcio do seculo passado, as funcoes definidas positivas foram ob-jecto de estudos em muitas e diferentes areas da matematica como teoria daprobabilidade, teoria dos operadores, analise de Fourier etc. Foi por causadisto que notacoes e generalizacoes das funcoes definidas positivas prove-nientes das diversas areas nunca foram reunidas numa unica doutrina. Oproposito desta tese, e estudar com maior detalhe funcoes definidas positivasde variavel complexa em domınios particulares do plano complexo.
No Capıtulo 1, daremos a definicao de funcao definida positiva, algu-mas propriedades basicas, o teorema de representacao de Bochner e tambemalgumas propriedades diferenciais destas funcoes. Em particular, vamos con-siderar o caso de funcao definida positiva e analıtica sobre o eixo real e vamosver, como neste caso, e possıvel estender a funcao ao plano complexo, assimgeneralizando o conceito de funcao definida positiva no caso de funcao devariavel complexa. E a partir deste resultado devido a Z. Sasvari, see [2],que J. Buescu e A. C. Paixao deram a primeira definicao de funcao definidapositiva de variavel complexa, sem requerer nenhuma ulterior regularidadesobre a funcao. Veremos, como muitas das propriedas basicas e diferenciaisde funcoes definidas positivas reais sao validas tambem no caso complexocom generalizacoes oportunas. Alem disso, J. Buescu e A. C. Paixao carac-terizaram os conjuntos do plano complexo onde a definicao de funcao definidapositiva esta bem dada, e chamaram a estes conjuntos codifference sets. En-fim, neste Capıtulo 1, vamos apresentar tambem o conceito de funcao realco-definida positiva e vamos estudar relacoes e analogias desta funcao comas de uma funcao definida positiva classica. Por exemplo, enunciaremos oanalogo do teorema de Bochner, o teorema de Widder, que garante a ex-istencia de uma representacao integral para funcoes co-definidas positivas.
No Capıtulo 2, vamos estudar funcoes definidas positivas, mas a par-tir de um ponto de vista da teoria da probabilidade. De facto, a notacaoprobabilıstica revela-se particularmente util quando se trabalha com repre-sentacoes integrais de funcoes definidas positivas, sejam de variavel real oude variavel complexa. Os teoremas de representacao de Bochner e de Widder
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para funcoes respectivamente definidas e co-definidas positivas explicitam arelacao destas funcoes com a bem conhecida ferramenta da teoria da proba-bilidade, ou seja funcoes caracterısticas, funcoes geradoras dos momentos eproblema dos momentos. Portanto, iremos estudar estas funcoes na opticado nosso interesse acerca das funcoes definidas positivas, logo nao iremosfornecer uma classica revisao desta ferramenta, que de facto pode ser en-contrada em qualquer manual de teoria da probabilidade. Referimos porexemplo os livros de J. S. Rosenthal [13] e de R. Ash [1].
Enfim, no Capıtulo 3, vamos concentrar-nos sobre funcoes definidas pos-itivas de variavel complexa em faixas do plano complexo. De facto, veremoscomo as faixas parecem ser os unicos conjuntos onde faz sentido consideraruma funcao definida positiva que possui um mınimo de regularidade. Provaristo, foi um dos propositos, indirectos, desta tese. De facto, os resultadosdesta tese sugerem e nao refutam, mas ainda nao provam, a suposicao prece-dente. Daremos condicoes sobre funcoes complexas definidas positivas emfaixas para garantir a existencia e eventualmente a unicidade de uma repre-setacao integral. Observaremos, que a existencia ou nao desta representacaodepende da regularidade da funcao e que a regularidade da funcao em todaa faixa e dominada pela regularidade da funcao sobre o intervalo do eixoimaginario que intersecta a faixa considerada. Em particular, iremos provarque uma funcao complexa definida positiva numa faixa que seja pelo menoscontınua no intervalo do eixo imaginario que intersecta a faixa e de factouma funcao analıtica em toda a faixa. Tambem, demonstraremos que umafuncao analıtica definida positiva numa faixa do plano complexo possui umaunica representacao integral. Alem disso, daremos uma generalizacao no casocomplexo do problema de extensao para funcoes definidas positivas. Veremoscomo, dada uma funcao definida positiva num codifference set qualquer, nascomponentes conexas do codifference set que intersectam o eixo imaginarioe possıvel, comforme a regularidade da funcao, extender a funcao e a pro-priedade de ser definida positiva, a todas as faixas horizontais que contemas componentes conexas do codifference set original. Infelizmente, veremostambem como este conjunto de resultados resolve so parcialmente a questaode estabelecer as faixas como codifference sets por excelencia.
Palavras-chave Funcoes definidas positivas, Analise complexa, Funcoescaracterısticas e outras transformadas.Mathematics Subject Classification (2010) Primario 42A82; Secundario 30A10,60E10.
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Abstract
In Chapter 1, we will give the definition of positive definite functions on Rand we will present some basic and differential properties of these functions.In particular, we will consider the case of analytic positive definite functionson R in order to construct continuations to the complex plane. In viewof this, we will present the definition of complex-variable positive definitefunction mainly due to J. Buescu and A. Paixao and we will see how severalof the differential properties valide in the real case can be generalized in thecomplex settings. Moreover, is given here the notion of codifference set as theset of the complex plane in which the definition of complex positive definitefunction is well-given. In Chapter 1, we will also introduce another similarproperty to positive definiteness, namely co-positive definiteness.
In Chapter 2, we will look at the concept of positive definite functionfrom a probabilistic point of view. In order to do that, we will recall thenotion of characteristic function and moment generating function and we willshow how, thanks to Bochner’s and Widder’s representation theorems, theseobjects respectively correspond to positive definite and co-positive definitefunctions. Furthermore, we will present the so-called moment problem.
In Chapter 3 we will focus on complex-variable positive definite functionson strips of the complex plane. We tried to understand under which condi-tions a complex positive definite function on a strip benefits of an integralrepresentation and eventually when it is unique. We found out that theexistence or not of such a representation depends on the regularity of thefunction; and that the regularity of a complex positive definite function on astrip is completely imposed by the regularity of the function on the intervalof the imaginary axis contained in the strip. Moreover, we will state a gen-eralization of the extension problem for complex positive definite function.
Keywords Positive definite functions, Complex analysis, Characteristic func-tions and other transforms.Mathematics Subject Classification (2010) Primary 42A82; Secondary 30A10,60E10.
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Acknowledgements
I would like to thank my advisor Jorge Buescu and the Professor AntonioCarlos Paixao that to all effects is co-advisor of this thesis. I want to thankthem for all the time spent speculating about complex positive definite func-tions, for the devotion to their and to this work. I really enjoyed to do myMaster thesis and I simply couldn’t do it without their support.Thanks to Sergio for his encouragements. Thanks to all the friends of Ruada Saudade. Grazie a mamma e papa, e a Sara, sempre tanto vicini.
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Contents
Resumo i
Abstract ii
1 Introduction 1
2 Positive definite functions 52.1 Real-variable positive definite functions . . . . . . . . . . . . . 5
2.1.1 Co-positive definite functions . . . . . . . . . . . . . . 82.2 Complex-variable positive definite functions . . . . . . . . . . 9
2.2.1 Codifference sets . . . . . . . . . . . . . . . . . . . . . 102.2.2 Properties of complex positive definite functions . . . . 12
3 Characteristic functions 233.1 Moment generating functions . . . . . . . . . . . . . . . . . . 293.2 Moment problem . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.1 Hamburger moment problem . . . . . . . . . . . . . . . 32
4 Complex positive definite functions on strips 374.1 Propagation of regularity . . . . . . . . . . . . . . . . . . . . . 374.2 Integral representations . . . . . . . . . . . . . . . . . . . . . . 404.3 The extension problem . . . . . . . . . . . . . . . . . . . . . . 45
5 Bibliography 49
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Chapter 1
Introduction
The concept of positive definiteness appears for the first time in 1907 in apaper of the mathematician Caratheodory. He was looking for necessary andsufficient conditions on the coefficients of the power series
1 +∞∑k=1
(ak + ibk)zk
analytic on the unit disc in order to have positive real part. Caratheodorycharacterized these point, (a1, b1, . . . , an, bn) for n ∈ N, as the points that liein the smallest convex set containing the points
2(cosϕ, sinϕ, . . . , cosnϕ, sinnϕ), with 0 ≤ ϕ ≤ 2π.
In 1911 Toepliz noticed that Caratheodory’s condition is equivalent to
n∑k,l=1
dk−lckcl ≥ 0, ∀n ∈ N, ∀ cknk=1 ∈ C (1.1)
where d0 = 2, dk = ak − ibk, d−k = dk. That is, if and only if dk is a positivedefinite sequence. In the same year, thanks to Toepliz’s deduction, Herglotzsolved the so-called trigonometric moment problem. Indeed, he stated that asequence dn satisfies (1.1) if and only if there exists a unique non-negativeand finite Borel measure µ such that
dn =
∫ 2π
0
eintdµ(t), n ∈ Z.
That is, such that dn is a solution of the trigonometric moment problem.
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In 1923 Mathias introduced the notion of positive definite function. Afunction f : R→ C is positive definite if
f(−x) = f(x), x ∈ R (1.2)
andm∑
j,k=1
ξjξkf(xj − xk) ≥ 0 (1.3)
for all m ∈ N, ξkmk=1 ⊂ C and xkmk=1 ⊂ R. That is, if every squarematrix [f(xj − xk)]
mj,k=1 is positive semi-definite. Remark that, condition
(1.2) remained part of the definition of a positive definite function until 1933,when F. Riesz pointed out that it follows easily from (1.3).
In 1932 Bochner proved a celebrated theorem on positive definite func-tions: if f is a continuous positive definite function on R, then there exists abounded non-decreasing function µ on R such that f is the Fourier-Stieltjestransform of µ, that is
f(x) =
∫ +∞
−∞eitxdµ(t)
holds for all x.Positive definite functions have a lot of generalizations, as for example,
positive definite kernels that in the context of reproducing kernel Hilbertspaces have several applications to the theory of integral equations. Actually,positive definite kernels were introduced by Mercer in 1909, that is beforepositive definite functions. We call k a positive definite kernel if k(x, y) isany complex-valued function on R2 such that
n∑i,j=1
k(xi, xj)ξiξj ≥ 0.
for all ξi ∈ C and (xi, xj) ∈ R2 and for i, j = 1, . . . , n.This is only one of the numerous applications of positive definite func-
tions. After Bochner stated his theorem, Riesz pointed out that it couldbe used to prove an important theorem on one-parameter groups of unitaryoperators, namely Stone Theorem; and with the appearance of harmonicanalysis on groups in 1940’s the role of positive definite functions in Fourieranalysis became apparent.
Perhaps the area of mathematics in which most people are familiar withpositive definite function is that of probability theory. In fact, the Fourier-Stieltjes transform of a probability distribution is called a characteristic func-tion, and thus, by virtue of Bochner’s theorem, f is a characteristic function
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if and only if f is continuous, positive definite and f(0) = 1. Even if char-acteristic functions hail as far as Laplace and Cauchy, it was Levy who firstrecognized that in general it is easier to work with characteristic functionsinstead of probability distributions. It is not surprising that the central limitproblem (the problem of convergence of sums of laws of probability) was infact solved with the aid of positive definite functions.
It is because of the concept of positive definite function being such acentral notion in so many different theories that it never had been unified ina unique doctrine; and that still today there is a big disparity between thenotations from distinct mathematical areas.
In Chapter 1, after recalling the definition of positive definite functionson R, we will present some basic properties and some differential propertiesof these functions. In particular, we will consider the case of analytic positivedefinite functions on R in order to construct continuations to the complexplane, see Sasvari [2], and thus, in order to extend the condition of positivedefiniteness to analytic functions of the complex variable. In view of this,we will present the a priori definition, that is requiring no further regularityon the function, of complex-variable positive definite function mainly due J.Buescu and A. Paixao, see [10] and we will see how several of the differentialproperties valid in the real case can be generalized in the complex settings.Also is given here the notion of codifference set, also due to J. Buescu andA. Paixao, see [10], as the set of the complex plane in which the definitionof complex positive definite function is well-given. In Chapter 1, we will alsointroduce another property very similar to positive definiteness, namely co-positive definiteness, and we will show that even for such functions exists arepresentation theorem of 1933 due to Widder, see [18].
In Chapter 2, we will look at the concept of positive definite function froma probabilistic point of view, since the notation of the probability theory re-sulted pretty useful when dealing with integral representations of positivedefinite functions both of real or complex variable. In order to do that,we will recall the notion of characteristic function and moment generatingfunction and we will show how, thanks to Bochner’s and Widder’s repre-sentation theorems, these objects respectively correspond to positive definiteand co-positive definite functions. However, we will explore these tools inthe perspective of what we are interested in, thus we will not offer a commonexposition of characteristic and moment generating functions as can be foundin any manual of probability theory, see for example J. S. Rosenthal [13] andR. Ash [1]. Furthermore, we will present the so-called moment problem inorder to show its relations with positive definite functions.
Finally, in Chapter 3 we will focus on complex-variable positive definitefunctions on strips of the complex plane. In fact, our interest is particularly
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focused on this kind of codifference sets, since at a first sight they seemsto be the only sets in which it makes sense to consider a complex positivedefinite function. To prove this was one of the purposes, an indirect one,of this thesis. Indeed, we tried to understand under which conditions acomplex positive definite function benefits of an integral representation andeventually when it is unique. We found out that the existence or not ofsuch a representation depends on the regularity of the function; and thatthe regularity of a complex positive definite function on a strip is completelyimposed by the regularity of the function on the interval of the imaginaryaxis contained in the strip. In particular, we will see that a complex positivedefinite function on a strip, that is at least continuous on the imaginary axiswill result holomorphic in the whole strip; and that holomorphy will ensurethe existence of a unique integral representation in the strip. Moreover,we will state a generalization of the extension problem for complex positivedefinite functions. In fact, we will show when and how a positive definitefunction on an arbitrary codifference set can be extended to strips of thecomplex plane. However, this results accomplish only in part the problemof establishing the strips as the only set in which make sense to considerpositive definite functions.
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Chapter 2
Positive definite functions
The purpose of this chapter is to introduce the theory of positive definitefunctions of real variable and to extend, in case of analyticity, this conceptto complex-variable positive definite functions, see Z. Sasvari [2]. Moreover,we will present the recent a priori description of positive definite functionsof complex variable, that is without requiring further regularity on the func-tions, mainly due to J. Buescu and A. C. Paixao, see [10] and [9].
2.1 Real-variable positive definite functions
Definition 2.1. A function f : R→ C is positive definite if
m∑j,k=1
ξjξkf(xj − xk) ≥ 0 (2.1)
∀m ∈ N, ∀ ξkmk=1 ⊂ C and ∀ xkmk=1 ⊂ R, that is, if every square matrix[f(xj − xk)]mj,k=1 is positive semi-definite.
Positive definite functions verify some basic properties that simply followfrom the definition considering the cases n = 1, 2 with a suitable choice ofthe sequences xkmk=1 and ξkmk=1, namely
1. f(0) ≥ 0;
2. f(−x) = f(x), ∀x ∈ R;
3. |f(x)| ≤ f(0) , ∀x ∈ R.
Theorem 2.1. Let f1(x), f2(x) be positive definite functions. Then thefunctions f1, f1(−x), Re(f1), |f1|2 and f1f2 are positive definite. Moreover,p1f1 + p2f2 is positive definite for all p1, p2 ≥ 0.
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Proof. See Theorem 1.3.2 of Sasvari [14].
However, the most significant result that holds for positive definite func-tions is the following representation theorem due to Bochner (1932).
Theorem 2.2 (Bochner’s theorem). A continuous function f : R → C ispositive definite if and only if it is the Fourier-Stieltjes transform of a finiteand non-negative measure µ on R, that is
f(x) =
∫ +∞
−∞eitxdµ(t). (2.2)
Proof. We will only prove that for a function f to be a Fourier-Stieltjestransform of a finite non-negative measure µ on R is sufficient to be a positivedefinite function.
m∑j,k=1
ξjξkf(xj − xk) =
∫ +∞
−∞
m∑j,k=1
ξjξkei(xj−xk)tdµ(t)
=
∫ +∞
−∞
m∑j,k=1
ξjξkeixjteixktdµ(t)
=
∫ +∞
−∞
∣∣∣∣∣m∑j=1
ξjeixjt
∣∣∣∣∣2
dµ(t) ≥ 0.
For the other implication we refer to [6].
Another characteristic property of positive definite functions is a kind of“propagation of regularity”. In fact, as a consequence of Bochner’s theorem,we have that a positive definite function that is continuous in a neighborhoodof the origin is uniformly continuous on R.
Theorem 2.3 (Propagation of regularity). Let f : R → C be a positivedefinite function of class C2n in some neighborhood of the origin for somepositive integer n, then f ∈ C2n(R).
Proof. Using Bochner’s representation (2.2) and standard tools from Har-monic Analysis, Donoghue [4] pag.186 proves the statement.
The above result holds even for C∞ or analytic functions, see remark inBuescu and Paixao [8] and the corresponding literature. Note that propa-gation of regularity only occurs for even-order derivatives, in fact even-orderderivatives play a central role in the theory of positive definite functions, asfollows from the next results too.
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Proposition 2.1. Let f : R → C be a positive definite function of classC2n in some neighborhood of the origin for some positive integer n. Thenf ∈ C2n(R) and for all integers 0 ≤ m ≤ n, the function (−1)mf 2m(x) ispositive definite.
Proof. See [8].
This result gives rise to a two-parameter family of differential inequalitiesfor positive definite functions which is very useful when dealing for examplewith integral equations. In the context of positive definite kernel Hilbertspaces, these inequalities may be interpreted as a generalized Cauchy-Schwarzinequality.
Proposition 2.2. Let f : R → C be a positive definite function of classC2n in some neighborhood of the origin for some positive integer n. Thenf ∈ C2n(R) and for all integers m1,m2 with 0 ≤ m1 ≤ n, 0 ≤ m2 ≤ n andevery x ∈ R we have
|f (m1+m2)(x)|2 ≤ (−1)m1+m2f (2m1)(0)f (2m2)(0). (2.3)
Proof. See [8].
Remark 2.1. Observe that since (−1)mf (2m)(x) is positive definite for every0 ≤ m ≤ n, the right hand-side of (2.3) is positive because of basic property1 of positive definite functions, thus the inequality is meaningful.
Theorem 2.4. Let f : R→ C be a positive definite function of class C2n insome neighborhood of the origin for some positive integer n. If f (2m)(0) = 0for some non-negative integer m ≤ n, then f is constant on R.
Proof. The statement of this theorem trivially follows in the case m = 0,since |f(x)| ≤ f(0) for every x ∈ R, and in the case m = 1 because of (2.3)with m1 = 1 and m2 = 0, that implies |f ′(x)|2 ≤ −f(0)f ′′(0) for every x ∈ R.Using (2.3) it is possible to complete the proof, see [8].
We will recall a characterization of real analytic functions before statingthe next result.
Lemma 2.1. Let f be a real function in C∞(I) for some open interval I.Then f is real analytic if and only if, for each α ∈ I, there are an openinterval J , with α ∈ J ⊂ I, and constants C > 0 and R > 0 such that thederivatives satisfy
|f (k)(x)| ≤ Ck!
Rk, ∀x ∈ J. (2.4)
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Theorem 2.5. Let f : R→ C be a positive definite function of class C∞ insome neighborhood of the origin. Then, if there exist positive constants Mand D such that
0 ≤ (−1)nf (2n)(0) ≤ D(2n)!
M2n(2.5)
for every non-negative integer n, we have:
1. f is analytic in R;
2. let l = lim sup 2n
√|f (2n)(0)|
(2n)!, then l < ∞. Defining h = 1/l if l 6= 0
and h = ∞ if l = 0, there exist α, β ∈ [h,+∞] such that f extendsholomorphically to the complex strip S = z ∈ C : −α < Im(z) < β,where α and β are maximal with this property. Moreover, if h < ∞,f cannot be holomorphically extended to both the points z = ih andz = −ih simultaneously, implying in particular that h = minα, β.
Proof. See [8].
Remark 2.2. The statement of this theorem is slightly different from othersalready known in the literature, for example Z. Sasvari [2] using strongerhypothesis, that is including statement 1, concluding that the holomorphicextension of f to the maximal strip S must present singularities in bothz = −iα and z = iβ whenever α and β are finite.
2.1.1 Co-positive definite functions
Next we state the definition of a co-positive definite function. It is convenientto observe that a different sign in the definition with respect to positivedefinite functions will lead to a completely different, but analogous, varietyof properties for these functions.
Definition 2.2. A function f : R→ C is co-positive definite if
m∑j,k=1
ξjξkf(xj + xk) ≥ 0 (2.6)
∀m ∈ N, ∀ ξkmk=0 ⊂ C and ∀ xkmk=1 ⊂ R, that is, if every square matrix[f(xj + xk)]
mj,k=1 is positive semi-definite.
Co-positive definite functions do not verify the basic properties of positivedefinite functions. However, considering the case n = 1 in (2.6) we concludethat f(x) ≥ 0 for every x ∈ R, thus f has real values. Moreover, even forthis kind of function there exists a representation theorem due to Widder(1933).
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Theorem 2.6. A function f can be represented in the form
f(x) =
∫ +∞
−∞e−xtdα(t) (2.7)
where α(t) is a non-decreasing function and the integral converges for a <x < b if and only if f is continuous co-positive definite in the interval (a, b).
Proof. The proof of the sufficient condition is analogous to the part of The-orem 2.2 that we proved, for the other implication we refer to [19, 18].
2.2 Complex-variable positive definite func-
tions
Complex-variable positive definite functions naturally arise from real-variablepositive definite functions in the conditions of Theorem 2.5. Indeed, ananalytic real-variable positive definite function extends holomorphically to ahorizontal strip of the complex plane S = z ∈ C : −α < Im(z) < β, withα, β > 0 and maximal with this property. Bochner’s integral representation(2.2) extends holomorphically to S too, so that
f(z) =
∫ +∞
−∞eitzdµ(t), ∀ z ∈ S. (2.8)
Z. Sasvari, see [2], proved that a function with an integral representation(2.8) verifies the property
m∑j,k=1
ξjξkf(zj − zk) ≥ 0 (2.9)
∀m ∈ N, ∀ξkmk=1 ⊂ C, ∀zj, zk ∈ S such that zj − zk ∈ S. In fact,
m∑j,k=1
ξjξkf(zj − zk) =
∫ +∞
−∞
m∑j,k=1
ξjξkei(zj−zk)tdµ(t)
=
∫ +∞
−∞
m∑j,k=1
ξjξkeizjteizktdµ(t)
=
∫ +∞
−∞
∣∣∣∣∣m∑j=1
ξjeizjt
∣∣∣∣∣2
dµ(t) ≥ 0.
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That is, a function f with the integral representation (2.8) is a complex-variable positive definite function.
In their recent work, J. Buescu and A. C. Paixao [10] give a definitionof complex-variable positive definite function that naturally arises from theabove observation of Z. Sasvari , but that requires no further assumption onthe regularity of the function. From this new definition of complex positivedefinite function, Buescu and Paixao deduce a list of properties for that kindof function and they figure out on which kind of complex set make sense toconsider a complex positive definite function. In the following, I will reportthe main results (and their proofs) of this paper [10].
Definition 2.3. A function f : C → C is positive definite in the open setS ⊂ C if
m∑j,k=0
ξj ξkf(zj − zk) ≥ 0 (2.10)
∀m ∈ N, ∀ξkmk=0 ⊂ C, ∀zj, zk ∈ S such that zj − zk ∈ S.
Remark that Definition 2.3 does not require any regularity on the functionf and that in the complex case Bochner’s representation theorem is notvalid. Thus a complex function as in Definition 2.3 does not have an integralrepresentation (2.8). However, we already saw that holomorphic extensions ofreal analytic positive definite functions have an integral representation (2.8)and provide examples of complex positive definite functions on a complexstrip containing the real axis.
Moreover, another matter is now open: which kind of set S is such thatfor every zj, zk ∈ S, then zj− zk ∈ S? On which kind of set S is then possibleto define a complex-variable positive definite function?
2.2.1 Codifference sets
In order to answer the problem of defining a suitable set such that the defi-nition of complex-variable positive definite function is well-given, J. Buescuand A. C. Paixao, [10], introduce the codifference sets.
Definition 2.4. A set S ⊂ C is a codifference set if there exists a set Ω ⊂ Csuch that S may be written as
S = Ω− Ω ≡ z ∈ C : ∃ z1, z2 ∈ Ω : z = z1 − z2. (2.11)
We shall say that S =codiff(Ω).
Remark 2.3. Note that the set operation used in (2.11) is not the usual setdifference.
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Here are some properties of codifference sets that directly follow fromDefinition 2.4. Let S ⊂ C be a codifference set such that S =codiff(Ω), then
1. the set Ω is not uniquely determined. In particular, S is invariant underany translation of the codifference-generating set Ω parallel to the realaxis.
2. If z ∈ S, there exist z1, z2 ∈ Ω such that z = z1 − z2, obviouslyz2 − z1 = −z ∈ S. Hence any codifference set is symmetric withrespect to the imaginary axis.
3. Any non-empty codifference set intersects the imaginary axis.If S =codiff(Ω) and z = a+ ib = z1 − z2 ∈ S for some z1, z2 ∈ Ω, thenthere exists β ∈ R such that z1−z1 = b+β ∈ S and z2−z2 = b−β ∈ S.
4. If Ω is open, then S is also an open set since it is union of open sets.
The simplest examples of codifference sets are the horizontal strips
S(r, α1, α2) = z = a+ ib ∈ C : |a| < r, α1 < b < α2
with r, α1, α2 positive real or infinite.So S(r, α1, α2) =codiff(S(r/2, α1/2, α2/2)). Another example of codifferenceset are
S1 = codiff(Q1(0) ∪Q1(3 + 3i)),
S2 = codiff(Q1(0) ∪Q1(5 + 5i));
where
Qr(z) = w ∈ C : |Re(w − z)| < r and |Im(w − z)| < r.
See Figure 2.1 and note that a codifference set need not to be simply con-nected or even connected.
11
Figure 2.1: Codifference sets
2.2.2 Properties of complex positive definite functions
We now present some basic properties of complex-variable positive definitefunctions directly derived from Definition 2.3 by J. Buescu and A. C. Paixao[10]. Observe that most of the following properties are the complex analogof the corresponding properties of real-variable positive definite functions.
Proposition 2.3 (Positivity on the imaginary axis). Let f be a complexpositive definite function on a codifference set S, f(ib) ≥ 0, ∀ib ∈ S, whereb ∈ R.
12
Proof. Let b ∈ R such that ib ∈ S and let z = ib2
such that z − z ∈ S,
then from Definition 2.3 with m = 1 follows that ξξf(z − z) ≥ 0, thusf(ib) ≥ 0.
Therefore positive definite functions are always real and non-negative onthe imaginary axis.
Proposition 2.4 (Basic properties). Let f be a complex positive definitefunction on a codifference set S, ∀a, b, β ∈ R such that ±a+ ib and i(b± β)are in S
1. f(−a+ ib) = f(a+ ib), ∀x ∈ R;
2. |f(a+ ib)|2 ≤ f(i(b− β))f(i(b+ β)).
Proof. Let z1 = a2
+ i(b−β2
)and z2 = −a
2+ i(b+β2
)such that zi − zj ∈ S for
i, j = 1, 2. From Definition 2.3 with m = 2 follows that
2∑i,j=1
ξiξjf(zi − zj) ≥ 0. (2.12)
Therefore the complex matrix
A =
(f(i(b+ β)) f(a+ ib)f(−a+ ib) f(i(b− β))
)is positive semi-definite, which implies statements 1 and 2.
Let us now explicitly prove a basic property for complex variable positivedefinite functions on strips of the complex plane, that directly follow fromthe definitions of positive and co-positive definiteness.
Proposition 2.5. Let f be a complex-variable positive definite function onthe open strip S = z ∈ C : a < Im(z) < b with a, b ∈ R. Then
1. Fy(x) = f(x+ iy) for some y ∈ (a, b) is a real-variable positive definitefunction on R,
2. G(y) = f(iy) is a real-variable co-positive definite function on (a, b).
Proof. Remark that the open strip S is a codifference set for some openset Ω, that is S =codiff(Ω). Therefore, in order to prove statement 1, let
13
zk = xk + iy2
and zj = xj + iy2
be in Ω such that zk − zj ∈ S. Observe that,such zk and zj exist by virtue of property 2 of codifference sets. Then
n∑k,j=1
ξkξjFy(xk − xj) =n∑
k,j=1
ξkξjf(xk − xj + iy)
=n∑
k,j=1
ξkξjf(xk + iy
2− xj + i
y
2) =
n∑k,j=1
ξkξjf(zk − zj) ≥ 0,
and statement 1 is proved. Similarly, to prove statement 2, let zk = iyk andzj = iyj in Ω such that zk − zj ∈ S. Then
n∑k,j=1
ξkξjG(yk + yj) =n∑
k,j=1
ξkξjf(i(yk + yj)) =n∑
k,j=1
ξkξjf(zk − zj) ≥ 0.
Thus G(y) is a co-positive definite function on (a, b), that is statement 2 isproved.
Complex positive definite functions are controlled by their behaviour onthe imaginary axis as real positive definite function are controlled by theirbehaviour at the origin. Indeed, this is the content of the following results.
Lemma 2.2. Let S be a codifference set such that S ∩ Im(z) = iI for somereal interval I, and let f : S → C be a positive definite function, then:
1. if f(iu) = 0 for some u ∈ I, then f(ic) = 0 for every c ∈ int(I);
2. if f(iu) 6= 0 for every u ∈ I, then logf is mid-point convex on iI.
Proof. Let c ∈int(I) and define a sequence un recursively by
un+1 =
c+un
2if 2c− un /∈ I
c if 2c− un ∈ I
with u1 = u. Observe that there exists p ∈ N such that un = c for n ≥ p.Then, we will show that f(iun) = 0 for all n ∈ N since this implies thatf(ic) = 0. The statement is true for n = 1 by hypothesis. For each n we setun = b− β, un+1 = b, a = 0 and b+ β = 2un+1 − un, then using statement 2of Proposition 2.4
|f(iun+1)|2 ≤ f(iun)f(i(2un+1 − un)).
Hence, f(iun) = 0 implies f(iun+1) = 0 for all n ∈ N. By induction andsince c is arbitrary in int(I) we complete the proof of 1. To prove statement
14
2, observe that since by hypothesis f(ib) 6= 0 for every b ∈ I, then becauseof Property 2.3, f(ib) > 0 for all b ∈ I. Thus g = log(f) is well-defined oniI. Taking a = 0, b1 = b + β and b2 = b − β in statement 2 of Proposition2.4 it follows that
g
(ib1 + ib2
2
)≤ g(ib1) + g(ib2)
2
for every b1, b2 ∈ I, and thus g is midpoint convex in iI.
Remark 2.4. The convexity of logf on the imaginary axis was already provedby Dugue [5] under the further assumption that f is holomorphic.
Theorem 2.7. Let S be a codifference set in C and f : S → C a positivedefinite function. If f is zero on every connected component of S ∩ Im(z),then f is identically zero on S.
Proof. Let z = a + ib ∈ S. Property 3 of codifference sets establish theexistence of β such that b ± β ∈ S, while statement 2 of Proposition 2.4asserts that |f(a+ ib)|2 ≤ f(i(b−β))f(i(b+β)). By virtue of Lemma 2.2 thehypothesis on the zeros of f implies that f vanishes identically on S∩Im(z),leading to the conclusion that f ≡ 0 on S.
In order to do something similar to what was done for with real-variablepositive definite functions, J. Buescu and A. C. Paixao, [10], state a collec-tion of differential properties for complex positive definite functions. How-ever, this time the use of positive definite kernels in two complex variableis mandatory since without further assumption of regularity a complex posi-tive definite function does not possess of an integral representation. Positivedefinite functions are related with positive definite kernels in two complexvariables in the following way. Suppose f is positive definite in S ⊂ Cand that V = (z, u) ∈ C2 : z − u ∈ S. Define k : V → C such thatk(z, u) := f(z − u). Let Ω ⊂ C such that Ω2 ⊂ V , that is such thatcodiff(Ω) ⊂ S. Therefore
n∑i,j=1
k(zi, zj)ξiξj ≥ 0 (2.13)
for all ξi ∈ C for i = 1, . . . , n. That is, k is a positive definite kernel in Ω.Moreover, if f is holomorphic in S, then k is a sesquiholomorphic function(i.e. analytic in the first variable and anti-analytic in the second variable) inΩ2, thus k is a holomorphic positive definite kernel in Ω. Under this furtherassumption of regularity much more can be said. The following results areproved in [7] for holomorphic positive definite kernels of several complexvariable.
15
Theorem 2.8. Let Ω ⊂ C be an open set and k : Ω2 → C be a holomorphicpositive definite kernel on Ω. Then for any m ∈ N
km(z, u) :=∂2m
∂um∂zmk(z, u)
is a holomorphic positive definite kernel on Ω.
Corollary 2.1. Let Ω ⊂ C be an open set and k : Ω2 → C be a holomorphicpositive definite kernel on Ω. Then for all z, u ∈ Ω and all m ∈ N we have
∂2m
∂um∂zmk(z, z) ≥ 0 and∣∣∣∣ ∂2m
∂um∂zmk(z, u)
∣∣∣∣2 ≤ ∂2m
∂um∂zmk(z, z)
∂2m
∂um∂zmk(u, u).
Theorem 2.9. Let Ω ⊂ C be an open set and k : Ω2 → C be a holomorphicpositive definite kernel on Ω. Then for all m1,m2 ∈ N and for all z, u ∈ Ωwe have ∣∣∣∣ ∂m1+m2
∂um1∂zm2k(z, u)
∣∣∣∣2 ≤ ∂2m1
∂um1∂zm1k(z, z)
∂2m2
∂um2∂zm2k(u, u).
The relation between complex positive definite functions and complexpositive definite kernels allow us to state similar results for holomorphic pos-itive definite functions.
Theorem 2.10. Let S ⊂ C be an open codifference set and suppose thatf : S → C is positive definite and holomorphic in S. Then (−1)mf (2m)(z) isa positive definite function in S for every m ∈ N.
Proof. We want to show that
n∑i,j=1
(−1)mf (2m)(zi − zj)ξiξj ≥ 0 (2.14)
for every n ∈ N, for every ξi ∈ C with i = 1, . . . , n and for every zi ∈ Cfor i = 1, . . . , n such that codiff(zi) ∈ S, that is zij := zi − zj ∈ S for alli, j = 1, . . . , n. Consider Ω as the union of n squares Qr(z), that is Ω =⋃i=1,...,nQr/2(zi), and choose r such that U :=codiff(Ω) =
⋃i=1,...,nQr(zij) is
contained in S. Then k(z, u) := f(z − u) is a positive definite kernel in Ωand because of Theorem 2.8
∂2m
∂um∂zmk(z, u) = (−1)mf (2m)(z − u)
16
is a positive definite kernel in Ω, that is (2.14) is verified.
Theorem 2.11. Let S ⊂ C be an open codifference set and suppose thatf : S → C is positive definite and holomorphic in S. Suppose that S containsthe points ±a+ ib and (b± β)i for a, b, β ∈ R. Then, for every non-negativeintegers m1,m2 we have
|f (m1+m2)(a+ ib)|2 ≤ (−1)m1+m2f (2m1)(i(b+ β))f (2m2)(i(b− β)). (2.15)
Proof. Using the notation of the squares Qr(z), let a+ib = z12, −a+ib = z21,(b+β) = z11 and (b−β) = z22. Choose r > 0 such that U =
⋃i,j=1,2Q2r(zij) ⊂
S. Consider the points z1 = a2
+i(b+β2
)and z2 = −a
2+i(b−β2
)such that zij =
zi − zj ∈ S for i, j = 1, 2. Defining Ω = Qr(z1) ∩ Qr(z2), U =codiff(Ω) ⊂ Sand then z − u ∈ U ⊂ S for all z, u ∈ Ω. Therefore k(z, u) := f(z − u) is apositive definite kernel in Ω and because of Theorem 2.9 applied to the point(z, u) = (z1, z2) it is possible to obtain (2.15) by successive application of thechain rule.
Let’s see now what it means for a meromorphic function to be positivedefinite. In particular, the interest of J. Buescu and A. C. Paixao in [10] is tounderstand if, for example, under the assumption of being positive definitethe poles of a meromorphic function can be easily found.
Theorem 2.12. Let Ω ⊂ C be an open set such that S =codiff(Ω). Supposef is meromorphic in S and positive definite in S ∩D(f), where D(f) is thedomain of f . Then f is holomorphic in S.
Proof. Observe that, since Ω is open, S is open. Let z = a+ ib ∈ S, then asproved in property 2 of codifference sets, −z ∈ S. Moreover, from property3 it follows that z11 := z1 − z1 and z22 := z2 − z2 lie in S whenever we writez = z1 − z2 for some z1, z2 ∈ Ω and β = Im(z1 − z2). Since S is an open setand the singularities of f are isolated, we may choose z1, z2, and thereforeβ, such that z11, z22 are points where f is analytic. Let zn := an + ib bea sequence converging to z, that is, such that limn→+∞ an = a. Since f ismeromorphic the set of singularities of f has no accumulation points, thereexists p ∈ N such that f is analytic in both zn and −zn for all n ≥ p. Foreach such n we apply inequality 2 of Proposition 2.4
|f(zn)|2 ≤ f(z11)f(z22).
Suppose that z is a pole of f , then taking the limit we have thatlimn→+∞ |f(zn)| = +∞, contradicting the previous inequality. Therefore zcannot be a pole of f . Since z is an arbitrary point of S, then f is holomorphicin S.
17
Corollary 2.2. Suppose f is meromorphic in C and positive definite in itsdomain. Then f is entire.
Proof. Consider the strip S = z = a + ib ∈ C : |a| < r and α1 < |b| <α2. When r, α1, α2 are infinite, then S ≡ C. Taking f meromorphic in Swith infinite r, α1, α2, it follows immediately from Theorem 2.12 that f isentire.
Theorem 2.13. Suppose S is an open codifference set and let f : S → Cbe a positive definite holomorphic function. If f (2m)(ib) = 0 for some non-negative integer m and some b ∈ R such that z = ib ∈ S, then f is constanton the open connected component of S containing ib.
Proof. Since f is holomorphic in a neighborhood of ib, F (x) = f(x + ib)defines an analytic real-variable function on an interval I = (−ε, ε) for somepositive ε. Moreover, F (x) is positive definite as proved in Proposition 2.5and such that F (k)(x) = f (n)(x + ib) for every x ∈ I and any non-negativeinteger k. By virtue of Proposition 2.1 with m1 = 0 and m2 = m we have
|F (m)(x)|2 ≤ (−1)(m)F (0)F (2m)(0) ∀x ∈ I.
Using the inequality with m = 1, we obtain
|F ′(x)|2 ≤ −F (0)F ′′(0) ∀x ∈ I. (2.16)
If F (m)(0) = 0 for m = 0 or m = 1 the thesis is trivially true. Consequentlywe will consider that m > 1. The idea of the proof is to show that F (2m)(0) =0 implies F ′′(0) = 0 for m > 1 since under that hypothesis it is possible toconclude from (2.16) that F ′ vanishes identically on I and, consequently,that f ′(x + ib) = 0 for every x ∈ I. Since f is holomorphic on S, analyticcontinuation of f ensures that f ′(z) = 0 on the open connected componentof S containing ib, implying that f is constant on this set and proving thestatement. To prove the implication, suppose m > 1 and define by recurrencea sequence of even numbers, with k1 = 2m and
ki+1 =
ki2
if ki/2 is evenki2
+ 1 if ki/2 is odd.
Notice that ki+1 < ki whenever ki > 2 and that 2 is a fixed point of therecurrence. Then, there exists j(m) such that kl = 2 for all l ≥ j; in fact itis easily shown that j(m) ≤ m. We now prove that fki(0) = 0 for all i ∈ N.Suppose that the statement is true for some i ∈ N; then using inequality
18
(2.16) with m1 = 0 and m2 = ki we obtain
|F (ki/2)(x)|2 ≤ (−1)ki/2F (0)F (ki)(0)
for every x ∈ I. Since F (ki)(0) = 0 we conclude that F (ki/2)(x) for all x ∈ I,which implies in particular that F (ki/2)+1(0) = 0. According to the definitionof the ki, we conclude that F (ki+1)(0) = 0. Hence F (ki)(0) = 0 for all i ∈ N.But as observed ki reaches 2 in a finite number of steps. In particular, thisimplies that F ′′(0) = 0 and |F ′(x)| = 0 for every x ∈ I, completing theproof.
If f is meromorphic in C and analytic in z ∈ D(f), we denote by r(z)the radius of convergence of the Taylor series of f about z. Defining l(z) =n
√|f (n)(z)|
n!, of course r(z) = 1/l(z) if l(x) 6= 0 and r(z) =∞ if l(z) = 0.
Lemma 2.3. Let f be a meromorphic function in C. Suppose f is positivedefinite in S ∩ D(f) for some open codifference set S ⊂ C and that ±a +ib, b± iβ ∈ S ∩D(f) for some a, b, β ∈ R. Then
r2(a+ ib) ≥ r(i(b+ β))r(i(b− β)). (2.17)
Proof. For any z where f is analytic, define un(z) = n
√|f (n)(z)|
n!and observe,
by considering the odd and even subsequences of un(z), that
lim supn→∞
un(z) = maxlim supn→∞
u2n(z), lim supn→∞
u2n+1(z). (2.18)
Suppose, in addition, that z ∈ S is a point on the imaginary axis. The ideais to show that
lim supn→∞
u2n+1(z) ≤ lim supn→∞
u2n(z), (2.19)
since this implieslim supn→∞
un(z) = lim supn→∞
u2n(z).
For z = ib, using inequality (2.15) with a = 0, m1 = n and m2 = n + 1 wehave
|f (2n+1)(ib)|2 ≤ f (2n)(ib)f (2n+2)(ib).
19
Then we have(|f (2n+1)(ib)|
(2n+ 1)!
) 22n+1
≤(|f (2n)(ib)|
(2n)!
) 12n
22n+1
(|f (2n+2)(ib)|
(2n+ 2)!
) 12n+2
2n+22n+1
(2n+ 2
2n+ 1
) 12n+1
establishing (2.19) and that
l(ib) = lim supn→∞
2n
√|f (2n)(ib)|
2n!. (2.20)
To conclude the proof, consider now the more generic points ±a+ib, (b±β)i.Direct use of inequality (2.15) with m1 = n and m2 = n+ 1 yields
|f (2n+1)(a+ ib)|2 ≤ f (2n)(i(b+ β))f (2n+2)(i(b− β)).
By a similar calculation to the one above we obtain
lim supn→∞
(|f (2n+1)(a+ ib)|
(2n+ 1)!
) 22n+1
≤
lim supn→∞
(|f (2n)(i(b+ β))|
(2n)!
) 12n
lim supn→∞
(|f (2n+2)(i(b− β))|
(2n+ 2)!
) 12n+2
or, in view of (2.20),(lim supn→∞
u2n+1(a+ ib)
)2
≤ l(i(b+ β))l(i(b− β)).
On the other hand using inequality (2.15) with m1 = m2 = n
|f (2n)(a+ ib)|2 ≤ |f (2n)(i(b+ β))||f (2n)(i(b− β)|,
implying that (lim supn→∞
u2n(a+ ib)
)2
≤ l(i(b+ β))l(i(b− β)).
Therefore, according to (2.18)
l2(a+ ib) ≤ l(i(b+ β))l(i(b− β)).
20
Hence, we haver2(a+ ib) ≥ r(i(b+ β))r(i(b− β)),
finishing the proof.
Theorem 2.14. Let S ⊂ C be an open codifference set containing z = ib,b ∈ R. Suppose f is meromorphic in C and positive definite in S ∩ D(f),where D(f) is its domain. If f has no poles on the imaginary axis, then fis entire.
Proof. If f has no poles on the imaginary axis, then there exists h > 0 suchthat f is positive definite and holomorphic on the square Qh(ib). Hence usingthe results of Lemma 2.3 it is possible to conclude that
r(a+ ib) ≥ r(ib) (2.21)
for every a ∈ (−h, h). If r(ib) < ∞, then f must have a pole z0 = a0 + ib0such that |z − z0| = r(ib) and a0 6= 0 since by hypothesis f has no poles onthe imaginary axis. Choose a ∈ (−h, h) such that |a− a0| ≤ |a0|, and writez = a+ ib. Then
|z − z0| =√|a− a0|2 + |b− b0|2 <
√a20 + |b− b0|2 = |z0 − ib|
implying that r(a+ib) ≤ |z−z0| < |z0−ib0| = r(ib) and contradicting (2.21).Hence r(ib) must be infinite and we conclude that f has no poles.
Theorem 2.15. Suppose S ⊂ C is an open codifference set. Let L(b0) bethe horizontal line defined by L(b0) = z ∈ C : z = a + ib0, for b0 ∈ R,and let f be a meromorphic function in C. Suppose f is positive definitein S ∩ D(f) and that L(b0) ⊂ S ∩ D(f). Then f has no poles on the stripS = z = a + ib ∈ C : a ∈ R and |b− b0| < r(b0). If r(b0) is finite, then atleast one of i(b± r(b0)) is a pole of f .
Proof. From Lemma 2.3 we have that r(ib0) ≤ r(a + ib0) for every a ∈ R.Since r(a + ib0), a ∈ R, is the radius of convergence of the Taylor series off centered at z = a + ib0, the distance from the set of poles of f to the lineL(b0) must be greater or equal than r(b0) and the first assertion follows. Ifr(ib0) is finite we conclude that at least one of i(b+ β) and i(b− β) is a poleof f , finishing the proof.
Corollary 2.3. In the conditions of Theorem 2.15, f extends holomorphi-cally to a maximal strip SM = z ∈ C : −α + b0 < Im(z) < β + b0,
21
where α, β ∈ (0,+∞[, as a positive definite function admitting, for somenon-negative measure µ, the integral representation
f(z) =
∫ +∞
−∞e−(iz−b0)tdµ(t), ∀ z ∈ SM . (2.22)
Moreover, r(b0) = minα, β and f has a pole at b0− iα (resp. b0 + iβ) if α(resp. β) is finite.
Proof. Let F (x) = f(x+ ib0) for x ∈ R. It is readily seen that F (x) is a real-variable positive definite function and that it is analytic on R, and thereforeadmits a holomorphic extension F(z) to the strip S0 = z ∈ C : −α <Im(z) < β, where α, β are maximal with this property. Then, according toTheorem 1.12.5 in [2], we write
F(z) =
∫ +∞
−∞e−itzdµ(t), −α < Im(z) < β (2.23)
and conclude, by virtue of this formula, that F is positive definite in S0.Furthermore, we also have that −iα (resp. iβ) is a singularity of f if α <∞(resp. β < ∞). Now, since F(z) is a holomorphic extension of F (x) =f(x+ ib), x ∈ R, and f is meromorphic in C, it follows that
F (z) = f(z + ib), for z ∈ S0. (2.24)
Hence from (2.23) we derive that
f(z) =
∫ +∞
−∞e−(iz−b0)tdµ(t) (2.25)
for every z ∈ SM and conclude that f is positive definite on this strip. From(2.24) it now follows that i(b0 − α) (resp. i(b0 + β)) is a pole of f wheneverα <∞ (resp. β <∞). As a direct consequence of Theorem 2.15, r(b0) mustbe the minimun between α, β.
22
Chapter 3
Characteristic functions
Positive definite functions and their various analogs and generalizations havearisen in different parts of mathematics since the beginning of the 20th cen-tury. They occur naturally in Fourier analysis, probability theory, operatortheory, complex-variable function theory, moment problems, integral equa-tions and other areas. Since the concept of positive definite function is sucha fundamental entity in so many distinct mathematical theories, the resultsnever had been collected in one single body doctrine. In what follows, wewill go into more detail on probability theory’s analogs of positive definitefunctions, namely characteristic functions, moment generating functions andmoment problem. In fact, we found the probabilistic point of view extremelyuseful when dealing with integral representations of complex-variable positivedefinite functions, as we will see in the next chapter. However, instead of pre-senting a common description of these tools, as can be found in any manualof probability theory, we will look at characteristic and moment generatingfunctions as good examples of respectively positive definite and co-positivedefinite functions. In order to do this, we will just present properties of thesefunctions that will be useful for the purpose of this thesis. Let us start withsome basic recalls from the probability theory. For a more in-depth analysisand eventual clarifications about what is next we refer to the book of R. Ash[1].
Definition 3.1. Let F be a collection of subsets of a set Ω. Then F is calleda algebra if and only if
1. Ω ∈ F ,
2. if A ∈ F , then Ac ∈ F .
3. if A1, A2, . . . , An ∈ F , then⋃ni=1Ai ∈ F .
23
If 3 is replaced by closure under countable union, that is,
3. if A1, A2, . . . ∈ F , then⋃∞i=1Ai ∈ F .
F is called σ-algebra.
Example 3.1. If Ω is the set R of extended real numbers, and F consist ofall finite disjoint unions of right-semiclosed intervals —(a, b] with −∞ ≤ a <b ≤ +∞—, then F forms an algebra, but not a σ-algebra.
The collection of Borel sets of R, denoted by B(R), is defined as thesmallest σ-algebra containing all the intervals (a, b] with a, b ∈ R. Note thatB(R) is garanteed to exist, and it may be described as the intersection of allσ-algebras containing the intervals (a, b]. Also if a σ-algebra contains all theopen intervals, it must contain all the intervals (a, b], and conversely. In fact
(a, b] =∞⋃n=1
(a, b+
1
n
)and (a, b) =
∞⋃n=1
(a, b− 1
n
]. (3.1)
Thus B(R) is the smallest σ-algebra containing all the open intervals. Simi-larly we can generate the Borel σ-algebra B(R) replacing the intervals (a, b]by other classes of intervals, for example
[a, b),
[a, b],
with −∞ ≤ a < b ≤ +∞.
Definition 3.2. A measure on a σ-algebra F is a non-negative, extendedreal-valued function µ such that whenever A1, A2, . . . form a finite or count-ably infinite collection of disjoint sets in F , we have
µ
(⋃n
An
)=∑n
µ(An). (3.2)
A measure space is a triple (Ω,F , µ) where Ω is a set, F is a σ-algebra ofsubsets of Ω, and µ is a measure on F .
Definition 3.3. A measure µ defined on F is said to be finite if and only ifµ(Ω) is finite.A measure µ on F is said to be σ-finite on F if and only if Ω can be writtenas⋃∞n=1An where An belong to F and µ(An) <∞ for all n.
24
Theorem 3.1 (Caratheodory’s extension theorem). Let µ be a measure onthe algebra F0 of subsets of Ω and assume that µ is σ-finite on F0, so that Ωcan be decomposed as
⋃+∞n=1An where An ∈ F0, and µ(An) < ∞, ∀ n. Then
µ has a unique extension to a measure on the minimal σ-algebra F over F0
Proof. See [1].
Definition 3.4. A Lebesgue-Stieltjes measure on R is a measure µ on B(R)such that µ(I) <∞ for each bounded interval I. A map F : R→ R that isincreasing and right-continuous is a distribution function
We are going to show that µ(a, b] = F (b) − F (a) sets up a one-to-onecorrespondence between Lebesgue-Stieltjes measures and distribution func-tions.
This, in particular, will better explain the statement of Widder’s theorem2.6 where an integral representation with respect to a non-decreasing functionappears, and will allow us to make the notation of this thesis uniform. Weneeded to enlight this correspondence in order to clarify the relation betweenBochner’s and Widder’s integral representations. In fact, this relation willbe useful in the next chapter, when dealing with integral representations inthe complex settings.
Theorem 3.2. Let µ be a Lebesgue-Stieltjes measure on R. Let F : R→ Rdefined up to an additive constant, by F (x) − F (a) = µ(a, b]. (For examplefix F (0) arbitrarily and set F (x) − F (0) = µ(0, x], x > 0; F (0) − F (x) =µ(x, 0], x < 0). Then F is a distribution function.
Proof. See [1].
Theorem 3.3. Let F be a distribution function on R and let µ(a, b] = F (b)−F (a), a < b. There is a unique extension of µ to a Lebesgue-Stieltjes measureon R.
Proof. This is an application of Caratheodory’s extension theorem, see [1].
Furthermore, µ is always σ-finite and is finite whenever F is bounded.
Example 3.2. For F (x) = x we have µ(a, b] = b− a, for a < b, such µ is theLebesgue measure on B(R).
For the complete theory and the missing results and proofs we refer to R.Ash [1].
25
Recall that Widder’s theorem 2.6 states that a continuous function f isco-positive definite if and only if there exists a non-decreasing function α(t)such that
f(x) =
∫ +∞
−∞e−xtdα(t). (3.3)
Since α(t) is non-decreasing the set of discontinuity points of α(t) is at mostcountable.In fact, let D the set of discontinuity points of α. For every t0 ∈ D, α(t+0 ) >α(t−0 ), where
α(t+0 ) = limt→t+0
α(t) and α(t−0 ) = limt→t+0
α(t)
and the above limits exist for every t0 ∈ D by monotonicity of α(t). Thus,for every interval (α(t−0 ), α(t+0 )) we can choose qt0 ∈ Q such that α(t−0 ) <qt0 < α(t+0 ). Since α(t) is non-decreasing if t0 6= s0 ∈ D then qt0 6= qs0 , thust0 7−→ qt0 is a one-to-one map from D to Q, and since Q is countable, so isD.Then we can define β(t) : R → R such that t ∈ R : β(t) 6= α(t) is atmost countable, and such that β(t) is non-decreasing and right-continuous,namely a distribution function. Hence, by Theorem 3.3, we can define aLebesgue-Stieltjes measure µ from β(t) such that µ is non-negative, σ-finiteon B(R) and eventually finite whenever β(t) is bounded.Remark that the measure generated from α(t) is equivalent to the measuregenerated from β(t) whenever µ(t == 0 for every t ∈ R. In light ofthis construction, Widder’s theorem 2.6 can be stated as follows. If f is acontinuous co-positive definite function on (a, b) with a, b ∈ R, then thereexists a non-negative and σ-finite measure µ such that
f(x) =
∫ +∞
−∞e−xtdµ(t), x ∈ (a, b). (3.4)
The most significant difference between Widder’s integral representationfor co-positive definite functions and the one of Bochner for positive definitefunctions is in the measure with respect to the integrals are made. In fact,unlike Bochner’s theorem, Widder’s statement not guarantee that the mea-sure is in general finite. Consequently, Bochner’s representation must alwaysconverge in a neighborhood of the origin, while Widder’s representation doesnot necessarily do so. Let us now finally give the definition of characteristicfunction.
Definition 3.5. Let µ be a probability measure on R. The characteristic
26
function of µ is the mapping from R to C given by
h(x) =
∫ ∞−∞
eitxdµ(t), x ∈ R. (3.5)
Thus h is the Fourier transform of µ. If F is a distribution functioncorresponding to µ, we shall also write h(t) =
∫∞−∞ e
itxdF (t), and call hthe characteristic function of F (or of X, if X is a random variable withdistribution function F ). A characteristic function as in (3.5) is of coursedefined for all x ∈ R, whenever t is a real number. In particular, if µ is aprobability measure, h(0) = 1.
Remark 3.1. By virtue of Bochner’s representation theorem, a characteris-tic function is always a real-variable positive definite function and even theconverse is true up to a normalization factor.
According to the Remark above and to the positive definite functions’basic properties, characteristic functions verifies the followings.
Theorem 3.4. Let h be the characteristic function of the bounded distribu-tion F . Then
1. |h(x)| ≤ h(0) for all x,
2. h is continuous on R,
3. h(−x) = h(x),
4. h(x) is real-valued if and only if F is symmetric; that is,∫BdF (t) =∫
−B dF (t) for all Borel sets B, where −B = −x : x ∈ B.
5. If∫R |t|
rdF (t) <∞ for some positive integer r, then the rth derivativeof h exists and is continuous on R, and
h(r)(x) =
∫R(it)reixtdF (t) (3.6)
Proof. See R. Ash [1], Theorem 7.1.5.
Next we present some properties of analytic characteristic functions. Theyare mainly due to Sasvari [2] and are basically results of analytic continuationto the complex plane.
Theorem 3.5. If f is an analytic characteristic function then there existαf , βf ∈ (0,∞] such that f extends to a function which is holomorphic inthe strip z ∈ C : −αf < Im(z) < βf and such that αf and βf are maximalwith this property.
27
Proof. See Sasvari [2], Theorem 1.12.2.
Theorem 3.6. Let f be an analytic characteristic function and let µ be thecorresponding probability measure. Then
f(z) =
∫ ∞−∞
eitzdµ(t), −αf < Im(z) < βf . (3.7)
If αf <∞ (βf <∞) then −iαf (iβf , respectively) is a singularity of f .
Proof. See Sasvari [2], Theorem 1.12.5.
Remark that, since an analytic characteristic function as in the conditionsof Theorem 3.5 can be holomorphically extended to a function as in (3.7)then, according to what we observed in Chapter 1, it is a complex-variablepositive definite function in the strip z ∈ C : −αf < Im(z) < βf. Andthus, all the properties presented in Chapter 1 for complex positive definitefunctions are valid here.
Proposition 3.1. A necessary condition for a function that is analytic insome neighborhood of the origin to be a characteristic function is that in eitherhalf-plane the singularity nearest to the real axis is located on the imaginaryaxis.
Proof. See Lukacs [12].
Proposition 3.2. An analytic characteristic function h(z) has no zeros onthe segment of the imaginary axis inside the strip of analyticity. Moreover,the zeros and the singular points of h(z) are located symmetrically with respectto the imaginary axis.
Proof. See Lukacs [12].
Theorem 3.7 (Levy-Raikov). Let h be an analytic characteristic function,and assume that h = h1h2, where h1 and h2 are both characteristic functions.Then the factors h1 and h2 are also analytic functions, and their represen-tations as Fourier integrals converge at least in the strip of convergence ofh.
Proof. See Theorem II b. Dugue [5] and Lukacs [12].
28
3.1 Moment generating functions
The moment generating function has been widely used by statisticians, andespecially by the English writers, in place of the closely-related characteristicfunction. In fact, from both functions it is possible to extract informationson the corresponding probability measure or distribution function. Beforewe give the definition of moment generating function another notion mustbe recalled, namely the one of moments of a probability measure.
Definition 3.6. Let µ be a probability measure on R, for any n ∈ N, themoment Mn of µ is defined as
Mn =
∫ +∞
−∞xndµ(x). (3.8)
We should note that if n is odd, in order for Mn to be defined we musthave
∫ +∞−∞ |x|
ndµ(x) < ∞. Given a probability distribution µ, either all themoments may exist, or they exist only for 0 ≤ n ≤ n0 for some n0. It couldbe that n0 = 0 as happens for example is for the Cauchy distribution 1
π(1+x2).
The characteristic function of a given probability measure is strictly re-lated to the corresponding moments. In fact, from equation (3.6) of Theorem3.4 it easily follows that
Mn = (−i)nh(n)(0) (3.9)
whenever the n-th derivative of h exists at zero. That is, if all the derivativesof the characteristic function exist at the origin, then all the moments of themeasure exist.
Definition 3.7. Let µ be a probability measure. The function
G(t) =
∫ +∞
−∞etxdµ(x), t ∈ R (3.10)
in which the integral is assumed to converge for t in some neighborhood ofthe origin, is called moment generating function of µ.
Remark that, if a probability measure has a moment generating functionthat converges in a non-trivial interval, then the domain of the correspondigcharacteristic function can be extended to the complex plane by
h(−it) = G(t). (3.11)
However, remark that the characteristic function of a distribution alwaysexists, while the moment generating function may not.
29
Let us now recall a standard result, namely Leibniz’s integral rule fordifferentiation under a Lebesgue-Stieltjes integral sign. We cite it here inorder to calculate the derivatives of a moment generating function.
Proposition 3.3 (Leibniz’s rule). Let I be an open subset of R and (Ω,F , µ)a measure space. Suppose f : Ω× I → R satisfies:
1. f(x,t) is a µ-integrable function of x for every t ∈ I.
2. For almost all x ∈ Ω, ∂f(x,t)∂t
exist for all t ∈ I.
3. There exists an integrable function g : Ω→ R such that |∂f(x,t)∂t| ≤ g(x)
for all t ∈ I.
Then for all t ∈ I
d
dt
∫I
f(x, t)dµ(t) =
∫I
∂f(x, t)
∂tdµ(t) (3.12)
Thus, a moment generating function (3.10) converging in an interval, say(−a, b) for some a, b ∈ R+, is such that
dn
dtnG(t) =
dn
dtn
∫ +∞
−∞etxdµ(x) =
∫ +∞
−∞
∂n
∂tnetxdµ(x)
for every n ∈ N and t ∈ (−a, b), because of Leibniz’s rule. That is, a momentgenerating function is infinitely differentiable in the interval of convergence.Actually, even more is true, a moment generating function is analytic in theinterval of convergence.
Proposition 3.4. Let µ be a probability measure and let G(t) be the corre-sponding moment generating function such that G(t) <∞ for every |t| < t0,for some t0 > 0. Then
∫ +∞−∞ |x|
ndµ(x) <∞ for n ≥ 0 and G(t) is analytic in|t| < t0 with
G(t) =+∞∑n=0
∫ +∞−∞ xndµ(x)tn
n!. (3.13)
In particular the derivative of order k at zero is given by
G(k)(0) =
∫ +∞
−∞xndµ(x).
Proof. See e.g. [13].
30
Remark 3.2. Proposition 3.4 says that the n-th derivative of G(t) at 0 equalsthe n-th moment of the probability measure µ (thus explaining the termi-nology “moment generating function”). For example, G(0) = 1, G′(0) =∫ +∞−∞ xdµ(x) , G(0) =
∫ +∞−∞ x2dµ(x) , etc.
Remark 3.3. By virtue of Widder’s representation (theorem 2.6) we have thata moment generating function g is of course a co-positive definite function.Conversely, we know that for a co-positive definite function there exists anon-decreasing function F —and thus a non-negative, σ-finite Borel measureµ on R— such that
g(t) =
∫ +∞
−∞etxdF (x) =
∫ +∞
−∞etxdµ(x) (3.14)
for some t ∈ (a, b). Therefore, whenever g(t) converges at zero, and thus theinterval of convergence (a, b) contains the origin, g(t) is a moment generatingfunction. On the other hand, if (a, b) does not contain the origin g(t) is nota moment generating function in the classical sense, but still conserves someof its properties for example being analytic in the interval of convergence.We will prove this fact in the next chapter.
Even more is true for a moment generating function. In fact, accordingto the next statement of Dugue such a function can be analytically continuedto the complex plane.
Proposition 3.5. Consider the moment generating function G(x) corre-sponding to the probability measure µ and let (−a, b) with a, b > 0 be itsinterval of convergence. Then G(z) is analytic in the strip −a < Re(z) < b.In fact,
G(z) =
∫ +∞
−∞eitzdµ(t) (3.15)
is absolutely convergent for −a < Re(z) < b.
Proof. See Dugue [5] and Lukacs [12].
Remark 3.4. The result of Proposition 3.5 remains valid considering a co-positive definite function instead of a moment generating function. Thatis, the result is still true even when the measure µ is just σ-finite. Fromanother point of view, it is well-known that a two-sided Laplace transform isanalytic in its region of absolute convergence. We will return to this in thenext Chapter.
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3.2 Moment problem
The moment problem arises in mathematics as result of trying to invert themapping that takes a measure µ to the sequences of moments Mn. Indeed, itcan be summarized as follows: “Given a sequence Mn, under which conditionsdoes there exist a measure µ on R such that all the moments of µ existand are equivalent to Mn for every positive integer n?”. In the literaturewe distinguish between three different moment problems depending on thesupport of the measure µ, namely
the Hamburger moment problem, if the support of the measure µ is R;
the Stieltjes moment problem, if the support of the measure µ is (0,∞];
the Hausdorff moment problem, if the support of the measure µ is abounded interval, which without loss of generality may be taken as[0, 1].
Obviously, Hamburger, Stieltjes and Hausdorff are the names of the math-ematicians that solved the corresponding moment problems. In the nextsection we will focus on the Hamburger moment problem since it is closelyrelated to the already-known positiveness and co-positiveness conditions.
3.2.1 Hamburger moment problem
Definition 3.8. A sequence Mn is a Hamburger moment sequence if thereexists a positive Borel measure µ on the real line R such that
Mn =
∫ +∞
−∞tndµ(t). (3.16)
We say that a Hamburger moment sequence is determined if the posi-tive Borel measure according to Definition 3.8 exists and is unique. Thereexist further conditions that may be imposed on the moments to guaranteeuniqueness, as for example Carleman’s and Krein’s conditions, but we willnot go into details on it. For in-depth analysis we refer to [16]. Here, we willjust present Carleman’s condition since it is the most general one.
Theorem 3.8 (Carleman’s condition). A sufficient condition for the Ham-burger moment problem to be determined is that
∞∑n=1
M− 1
2n2n = +∞. (3.17)
32
More generally, it is sufficient that
∞∑n=1
γ−12n = +∞, (3.18)
whereγ2n = inf
r≥n(M
12n2n ). (3.19)
Proof. See [16] pag. 19.
As already said, Hamburger provided a complete characterisation for aHamburger moment sequence.
Proposition 3.6 (Hamburger). A sequence Mn is a Hamburger momentsequence if and only if
m∑j,k=0
ξj ξkMj+k ≥ 0 (3.20)
∀m ∈ N, ∀ξkmk=0 ⊂ C.
Therefore, a Hamburger moment sequence must verify a kind of co-positive condition. In light of this, the next result will not be so impressive.
Proposition 3.7 (Hamburger). If f(x) is analytic in a < x < b, and
n∑i,j=0
f (i+j)(c)ξiξj ≥ 0 (3.21)
for a fixed c ∈ (a, b), then
f(x) =
∫ +∞
−∞e−xtdµ(t) (3.22)
where µ(t) is a non-decreasing function, and the integral converges in (a, b).
Thus, Hamburger established a direct relation between co-positive def-inite functions and Hamburger moment sequences. In fact, he stated thatwhenever a function f is analytic on an interval (a, b), if for every fixed pointc ∈ (a, b) the sequences of the derivatives of f in c are Hamburger momentsequences —because of Hamburger’s characterisation 3.6—, then f must bea co-positive definite function in (a, b) for Widder’s representation theorem2.6.
33
On the other hand, the Hamburger moment problem is related to positivedefinite functions too. The following results are basically due to Devinatzand can be found in [3].
Suppose that f(x) is an infinitely differentiable positive definite function.That is
f(x) =
∫ +∞
−∞eixtdµ(t) (3.23)
where µ(t) is a finite and non-negative Borel measure on R. Since f(x) isinfinitely differentiable, obviously
f (n)(x) =
∫ +∞
−∞intneixtdµ(t). (3.24)
Therefore, the sequence (−i)nf (n)(0)∞n=0 is a Hamburger moment sequence.Moreover, if ξk∞n=0 is an arbitrary complex sequence and m is any non-negative integer, then∣∣∣∣∣
n∑k=0
ξk(−i)kfk+m(x)
∣∣∣∣∣2
=
∣∣∣∣∣∫ +∞
−∞tmeixt
n∑k=0
ξktkdµ(t)
∣∣∣∣∣2
≤∫ +∞
−∞t2mdµ(t)
∫ +∞
−∞
∣∣∣∣∣n∑k=0
ξktk
∣∣∣∣∣2
dµ(t)
= Mm
n∑r=0
n∑s=0
ξrξs(−i)r+sf (r+s)(0).
where Mm = (−i)2mf (2m)(0).Conversely, adding to these two necessary conditions a third condition,
namely that (−i)nf (n)(0)∞n=0 is a determined Hamburger moment sequence,then these three conditions are sufficient for an infinitely differentiable func-tion on R to have the representation (3.23) and thus to be a positive definitefunction. In fact, even more is true. In fact, Devinatz proved that if f(x) isinfinitely differentiable just on some open interval containing the origin andsatisfies the above conditions, then it has the representation (3.23), that isit can be extended to a positive definite function on R. Moreover, since theHamburger moment sequence is by hypothesis determined, then the exten-sion is clearly unique. We will return to the problem of extension for positivedefinite functions in the next chapter.
Theorem 3.9 (Devinatz). Let f(x) be an infinitely differentiable functiondefined on the open interval (−a, b) where a, b > 0. If
34
1. (−i)nf (n)(0)∞n=0 is a determined Hamburger moment sequence and
2. for every non-negative integer m there exists an Mn > 0 such that forevery x ∈ (−a, b) and every finite complex sequence ξknk=0∣∣∣∣∣
n∑k=0
ξk(−i)kfk+m(x)
∣∣∣∣∣2
≤Mm
n∑r=0
n∑s=0
ξrξs(−i)r+sf (r+s)(0) (3.25)
then there exists a bounded non-negative measure µ(t) such that
f(x) =
∫ +∞
−∞eixtdµ(t). (3.26)
Proof. See Devinatz [3].
Remark 3.5. Remark that the first two necessary conditions, namely that fis infinitely differentiable at zero and that (−i)nf (n)(0)∞n=0 is a Hamburgermoment sequence, are not sufficient if (−i)nf (n)(0)∞n=0 is not a determinedHamburger moment sequence. Statement 2 is a technical condition neededfor the proof. In fact, Theorem 3.9 admits proof in the more abstract settingof the associated reproducing kernel Hilbert space. We would not discuss thissubject since it would lead far from the purposes of this thesis.
However, Devinatz’s result can be applied to the case of a complex posi-tive definite function on a strip of the complex plane. Suppose then, that S isan open strip of the complex plane. Then there exists an open set Ω such thatS =codiff(Ω). Assume that S = z ∈ C : −a < Im(z) < b with a, b > 0and that f(z) is a complex-variable positive definite function on S. We knowfrom Property 2.5 that Fy(x) = f(x + iy) is a real-variable positive definitefunction for every fixed y ∈ (−a, b) and that g(y) = f(iy) is a real-variableco-positive definite function on (−a, b). On the other hand, the functions Fyare always analytic at zero whenever f(z) is at least continuous on i(−a, b),that is whenever g(y) is continuous on (a, b). In fact, under this assumptiong(y) is a moment generating function, see Remark 3.3, and is thus analyticon (−a, b) by virtue of Proposition 3.4. Therefore, under this suitable con-
dition and according to Theorem 3.9 of Devinatz, (−i)nF (n)y (0)∞n=0 is a
Hamburger moment sequence such that
Myn = (−i)nF (n)
y (0) =
∫ +∞
−∞tndµy(t) (3.27)
35
where Myn indicates the n-th moment corresponding to the Hamburger mo-
ment sequence obtained for fixed y ∈ (−a, b).From this observation and from the inequalities of Proposition 2.4 with
a = 0 and β = y1−y22
for complex-variable positive definite functions, wemay establish a family of inequalities between different Hamburger momentsequences of different orders, namely∣∣∣∣M y1+y2
2m1+m2
∣∣∣∣2 ≤My12m1
My22m2
(3.28)
for every m1,m2 positive integers and for every y1, y2 ∈ (−a, b).We already knew a family of inequalities for a given moment sequence.
It is a result due to Dugue, see [5].
Proposition 3.8. Suppose that all the absolute moments, Mx, of order x ofa certain probability measure exist. Then
Mm1y1+m2y2 ≤ (My1)m1(My2)
m2 (3.29)
for every positive integers m1,m2 such that m1 +m2 = 1.
Proof. See [5].
Remark 3.6. Observe how the inequalities in (3.28) are different from theinequalities already found by Dugue. In fact, in (3.28) we have a family ofinequalities on different moment sequences of different orders, while in (3.29)we just have a family of inequalities on different orders of the same momentsequence.
36
Chapter 4
Complex positive definitefunctions on strips
The aim of this chapter is to establish conditions under which integral repre-sentations for complex-variable positive definite functions exist and to figureout when they are unique. In particular, integral representations for positivedefinite functions of the complex variable ensure a high regularity for thefunctions. Indeed, we will show that continuity on the imaginary axis —thatis, according to Widder’s theorem 2.6, an integral representation on the imag-inary axis— implies analyticity on the corresponding connected componentof the codifference set on which a given function is positive definite. Anotherpurpose is to explore, in the complex settings, the problem of the extensionfor positive definite functions according to its regularity.
4.1 Propagation of regularity
Here, we will present a generalization of Property 2.3 (propagation of regu-larity) for positive definite functions of the complex variable. First of all, letus note some properties of co-positive definite functions that will be usefulfor the next.
Let f be a complex positive definite function on a codifference set suchthat G(y) = f(iy) is a real-variable co-positive definite function because ofProposition 2.5 and such that G(y) is a continuous function. By Proposition3.4 the function G(y) = f(iy) is analytic in its interval of absolute conver-gence, say I, whenever I contains the origin. Remark that G(y) = f(iy) isanalytic in I even when I does not contain the origin. In fact, G(y) is such
37
that
G(y) =
∫ +∞
−∞e−ytdµ(t) (4.1)
and absolutely convergent in (a, b), with 0 < a < b without loss of generality.Then, for every y0 ∈ (a, b) we have
G(y) =
∫ +∞
−∞limk→∞
k∑n=1
(−t)ne−y0t
n!(y − y0)ndµ(t). (4.2)
Since
limk→∞
∣∣∣∣∣k∑
n=1
(−t)ne−y0t
n!(y − y0)n
∣∣∣∣∣ = |e−yt| = e−yt (4.3)
for some y0 ∈ (a, b); and since by hypothesis
G(y) =
∫ +∞
−∞e−ytdµ(t) <∞, y ∈ (a, b), (4.4)
by the dominated convergence theorem, we can take the limit out of theintegral sign in (4.2), obtaining
G(y) = limk→∞
k∑n=1
∫ +∞−∞ (−t)ne−y0tdµ(t)
n!(y − y0)n. (4.5)
If the integrals in (4.5) are all convergent for y0 ∈ (a, b), they correspond tothe derivatives of G in y0 for Leibniz’s rule 3.3, and we can conclude thatG(y) is analytic in (a, b). In order to do this consider α and β such thaty0 < α < b and a < β < y0. We have that∫ +∞
−∞|(−t)ne−y0t|dµ(t) =
∫ 0
−∞|tne−(y0−α)t|e−αtdµ(t)+ (4.6)∫ +∞
0
|tne−(y0−β)t|e−βtdµ(t). (4.7)
The two integrals are convergent because the integrands are the product ofan integrable function and a bounded function respectively on [−∞, 0] and[0,+∞].
Thus, G(y) = f(iy) is analytic on I and by analytic continuation it canbe extended to the whole strip of the complex plane S = z ∈ C : Im(z) ∈ I
38
to a holomorphic function f(z), such that
f(z) =
∫ +∞
−∞eiztdµ(t), z ∈ S (4.8)
that is, such that f(z) is a complex-variable positive definite function on S.As we already saw in Proposition 3.5, this fact was already known by
Dugue [5] in the case where the interval of absolute convergence of (4.14)contains the origin and thus in the case that µ is a probability measure andG(y) a moment generating function. This is in fact a well-known result inprobability theory.
The following statement is a strong consequence of the previous remarksin the theory of positive definite functions of the complex variable.
Theorem 4.1. Let f be a complex-variable positive definite function on theopen strip S = z ∈ C : a < Im(z) < b with a, b ∈ R. If f is continuous oni(a, b) = S ∩ Im(z), then f is analytic in S.
Proof. Since f(z) is positive definite on S, from Proposition 2.5 we knowthat G(y) = f(iy) is a co-positive definite function on the open interval ofthe imaginary axis (a, b) and is continuous here by hypothesis. Thus, byWidder’s theorem 2.6 there exists a non-negative and σ-finite measure on Rsuch that
G(y) =
∫ +∞
−∞e−ytdµ(t) (4.9)
and such that G(y) is analytic in (a, b) by virtue of the previous observation.Obviously, for every fixed y ∈ (a, b) there exists a neighborhood of y, eventu-ally very small, in which G —and thus f— is analytic. Therefore, the real-variable positive definite functions, obtained by setting Fy(x) = f(x + iy),are all analytic in a neighborhood of the origin and by virtue of “propagationof regularity” theorem 2.3, they are analytic on the whole lines iy of the com-plex plane for some y ∈ (a, b). We still cannot conclude that these analyticreal positive definite functions on the lines link continuously one to the other,and thus that f(z) is analytic on S. However, we may easily overcome thisproblem. In fact, from the previous digression, we know that G(y) has ananalytic extension to S, namely
g(z) =
∫ +∞
−∞eiztdµ(t) (4.10)
Obviously, by virtue of the identity theorem for functions of the complexvariable, g(z) ≡ f(x + iy) for every y ∈ (a, b) and x ∈ R. Therefore, f(z) is
39
analytic in S and the assertion is proved.
Remark 4.1. Of course, the statement of Theorem 4.1 is still valid when deal-ing with complex positive definite functions on a codifference set S differentfrom a strip. In this case, if f is continuous on every connected componentof S ∩ Im(z), then f will be holomorphic on every connected component ofS containing S ∩ Im(z).
We just proved that a complex-variable positive definite function on astrip is either discontinuous or holomorphic. In light of this, in the nextsection we will study when a complex positive definite function benefits ofan integral representation.
4.2 Integral representations
We begin by stating two well-known notions from measure theory.
Definition 4.1. Let µ, ν be two measures on the same measurable space,then µ is absolutely continuous with respect to ν if µ(A) = 0 for every mea-surable set A such that ν(A) = 0. In this case we write µ ν. Absolutelycontinuity is a reflexive and transitive, but not symmetric relation. Twomeasures are equivalent if µ ν and ν µ.
Theorem 4.2 (Radon-Nikodym). Let µ and ν be two σ-finite and non-negative measures on the same σ-algebra, with µ ν, then there exists anon-negative and σ-measurable function f such that
µ(A) =
∫A
fdν (4.11)
for every set A in the σ-algebra.
Proof. See R. Ash [1].
We say that f is said the Radon-Nikodym derivative and denote it bydµ/dν.
The next theorem shows that in case f is a holomorphic positive definitefunction on a strip of the complex plane, we can construct a commutativediagram where everything is well-behaved, in order to guarantee the existenceand uniqueness of an integral representation for f .
Theorem 4.3. Suppose f is a holomorphic complex function defined on thehorizontal strip S = z ∈ C, a < Im(z) < b, a, b ∈ R. Define Fy(x) =f(x + iy) for x ∈ R and any y ∈ (a, b); and G(y) = f(iy) for y ∈ (a, b).Then the following statements are equivalent:
40
1.a) f is a complex-variable positive definite function on S.
1.b) There exists a unique, non-negative and σ-finite measure µ on R suchthat
f(z) =
∫ +∞
−∞eitzdµ(t), ∀z ∈ S. (4.12)
2.a) Fy(x) is a positive definite function of the real variable x for some y ∈(a, b).
2.b) There exists a unique, non-negative finite measure µy on R such that
Fy(x) =
∫ +∞
−∞eitxdµy(t), ∀x ∈ R. (4.13)
for some y ∈ (a, b).
3.a) G(y) is a co-positive definite function of the real variable y on the in-terval (a, b).
3.b) There exists a unique, non-negative and σ-finite measure µI on R suchthat
G(y) =
∫ +∞
−∞e−ytdµI(t), ∀y ∈ (a, b). (4.14)
Proof. Since f is holomorphic on S, we have in particular that Fy(x) is realanalytic on R and that G(y) is continuous on (a, b). The equivalence of 2.aand 2.b follows from Bochner’s representation theorem 2.2 and the equiva-lence of 3.a and 3.b is consequence of Widder’s theorem 2.6. That 1.a implies2.a and 3.a —and therefore 2.b and 3.b— is an immediate consequence of thedefinitions of Fy and G and the conditions of positiveness or co-positivenessassumed for f , Fy and G, see Proposition 2.5.
Direct calculation, see the beginning of Section 2.2, can be used to showthat 1.b implies 1.a.
We now show that 2.a implies 1.a. According to the hypothesis, Fy(x) =f(x + iy) is positive definite and analytic on R for some y ∈ (a, b) and wehave from 2.b that
Fy(x) =
∫ +∞
−∞eitxdµy(t), ∀x ∈ R. (4.15)
41
for some non-negative and finite measure µy on R. Then, according to theo-rems 3.5 and 3.6, there exists an extension Fy(z) of Fy(x) which is holomor-phic on the strip SM = z ∈ C : −α < Im(z) < β, where α, β ∈ (0,∞]are maximal with this property, and presents singularities at both points−iα and iβ, whenever they are finite. This extension admits the integralrepresentation
Fy(z) =
∫ +∞
−∞eitzdµy(t), ∀z ∈ SM (4.16)
and its uniqueness implies that Fy(z) = f(z + iy) whenever both sides ofthe identity are defined. Since f is holomorphic on S and −iα and iβ aresingularities of Fy, we conclude that y − α < a < b < y + β. Therefore wemay write
f(z) =
∫ +∞
−∞ei(z−iy)tdµy(t) (4.17)
=
∫ +∞
−∞eizteytdµy(t), ∀z ∈ S. (4.18)
Defining µ(t) bydµ(t) = eytdµy(t) (4.19)
we finally obtain
f(z) =
∫ +∞
−∞eitzdµ(t), ∀z ∈ S. (4.20)
for the non-negative and σ-finite measure µ(t). According to the Radon-Nikodym theorem, µ is absolutely continuous with respect to µy and eyt isthe corresponding Radon-Nikodym derivative. Moreover, from (4.19) followsthat µ and µy are equivalent measures, in the sense of Definition 4.1, forevery y ∈ (a, b). We finish the proof by showing that 3.a implies 1.b. Since,according to the hypothesis, G(y) = f(iy) is continuous and co-positivedefinite on (a, b) we have from 3.b that
G(y) =
∫ +∞
−∞e−ytdµI(t), ∀y ∈ (a, b) (4.21)
for some non-negative and σ-finite measure µI . Then, by virtue of whatwe observed in the beginning of Section 4.1, Gy may be holomorphically
42
extended to the set S = z ∈ C : a < Im(z) < b by the function
g(z) =
∫ +∞
−∞e−ztdµI(t). (4.22)
Since G(y) = f(iy), the uniqueness of this extension implies that g(z) =f(iz), ∀z ∈ S. Therefore, we must have
f(z) =
∫ +∞
−∞eiztdµI(t), ∀z ∈ S. (4.23)
Defining µ(t) = µI(t) we finally obtain
f(z) =
∫ +∞
−∞eiztdµ(t), ∀z ∈ S. (4.24)
Remark 4.2. For every y ∈ (a, b) the measures µy in the conditions of Theo-rem 4.3 are equivalent between them. In fact, from statement 2 of Proposition2.4 with a = 0 and b = y, we have that
|f(iy)|2 ≤ f(i(y + β))f(i(y − β))
for y, β ∈ R such that iy, i(y ± β) ∈ (a, b). Using that for every y ∈ (a, b),by Bochner’s theorem f(iy) = Fy(0) =
∫ +∞−∞ dµy(t) for some finite and non-
negative Borel measure, µy, on R, we have∣∣∣∣∫ +∞
−∞dµy(t)
∣∣∣∣2 ≤ ∫ +∞
−∞dµy−β(t)
∫ +∞
−∞dµy+β(t) (4.25)
for every y, β ∈ R such that iy, i(y ± β) ∈ (a, b). Then, whenever µy+β orµy−β vanish on some Borel set B, µy(B) must be zero too, implying thatµy µy±β. Since y, β are arbitrary points, the assertion is proved.
Remark 4.3. If two measures are equivalent according to Definition 4.1, when-ever one of the two is finite the other one need not be finite too. For example,Lebesgue measure λ on R and the Normal distribution γ are mutually abso-lutely continuous and
γ(B) =1√2π
∫ +∞
−∞e−t
2/2dλ(t)
for every Borel set B. However, it is well-known that Lebesgue measure is
43
σ-finite and not finite, while the Normal distribution is finite since it is aprobability measure.
This is in general what occurs in Theorem 4.3 for the measures µ andµy for some y ∈ (a, b), where the µy are all finite measures by virtue ofBochner’s theorem and µ is initially just σ-finite. Only when the interval(a, b) contains the origin can we state that µ and µy are finite for everyy ∈ (a, b), and consequently that the maximal strip of the complex plane,S, on which we extend the positive definite function contains the whole realline.
We just proved that a complex-variable positive definite function on astrip is either discontinuous and does not admit an integral representation oris holomorphic and has a unique integral representation. This is the state-ment of the next Corollary.
Corollary 4.1. Let f be a complex-variable positive definite function on theopen strip S = z ∈ C : a < Im(z) < b with a, b ∈ R. If f is continuous oni(a, b) = S ∩ Im(z), then f has a unique integral representation on S,
f(z) =
∫ +∞
−∞eiztdµ(t), ∀z ∈ S (4.26)
for the non-negative and σ-finite Borel measure µ.
Proof. From Theorem 4.1 the assumptions of Theorem 4.3 are verified, thusproving the result.
Example 4.1. Here some examples of what happens when, given an analyticpositive definite function on the real axis —that is, an analytic characteristicfunction— we extend it to a maximal strip of the complex plane accordingto Theorem 4.3.
1. The real-variable positive definite function 11+x2
is the characteristic
function of the probability distribution e−|t|
2, that is
1
1 + x2=
∫ +∞
−∞eixt
e−|t|
2dt.
By analyticity it is possible to extend this function holomorphically tothe non-trivial strip S = z ∈ C : −i < Im(z) < i so that on theimaginary axis we have
1
1− y2=
∫ +∞
−∞e−yt
e−|t|
2dt.
44
That is, 11−y2 is the moment generating function of the probability
distribution e−|t|
2and is co-positive definite.
2. The real-variable positive definite function cos(x) is the characteristicfunction of the probability distribution δ(t− 1) + δ(t+ 1), that is
cos(x) =
∫ +∞
−∞eixt[δ(t− 1) + δ(t+ 1)]dt.
The holomorphic extension of the analytic function cos(x), cos(z), isan entire function; this is the case S ≡ C. On the imaginary axis
cos(iy) = cosh(y) =
∫ +∞
−∞e−yt[δ(t− 1) + δ(t+ 1)]dt
is the moment generating function corresponding to the probabilitydistribution δ(t− 1) + δ(t+ 1), which is co-positive definite.
3. As we know from probability theory, it is always possible to define thecharacteristic function of a probability distribution. However, the mo-ment generating function may not exist, since the moments of a prob-ability distribution do not necessary all exist. In this case, the deriva-tives at zero of the characteristic function do not exist all. Consider forexample the well-known Cauchy distribution 1
π(1+t2). Its characteristic
functione−|x|
2π
of course does not posses all the derivatives at zero. With respect toTheorem 4.3 this is the trivial case S ≡ R.
4.3 The extension problem
Let us now present the well-known extension problem for positive definitefunctions, first explicitly posed by M.G. Krein in 1940. If we assume thata complex-valued function f defined on (−2a, 2a), a > 0, is called positivedefinite if the inequality
m∑j,k=1
ξjξkf(xj − xk) ≥ 0 (4.27)
45
holds, whenever ξj ∈ C and xj ∈ (−a, a), the extension problem can bestated as: Can every positive definite function f in (−a, a) be extended to apositive definite function on R?
Theorem 4.4 (M. G. Krein’s extension theorem–1940). Any continuous pos-itive definite function f on (−c, c) can be extended to a positive definite func-tion on R.
Remark 4.4. By Bochner’s theorem 2.2 any such f is a Fourier-Stieltjes trans-form. That is, the extension admits an integral representation.
Krein also showed that the extension needs not be unique, and, by usingmethods reminescent of those used in the classical moment problems, gaveseveral criteria for uniqueness of the extension. A year later A. P. Artjomenkopointed out that the continuity assumption can be dropped, at the cost ofgiving up at the existence of an integral representation by Bochner’s theorem.
Theorem 4.5 (A. P. Artjomenko-1941). Any positive definite function f on(−c, c) can be extended to a positive definite function on R.
For a deeper digression on the extension problem for positive definitefunctions we refer to [15, 17] and their corresponding references.
The next result is analogous to the previous theorems, but it is stated inthe complex setting.
Theorem 4.6. Let f be a complex positive definite function in an open codif-ference set S =codiff(Ω). On every connected component, iI, of S∩Im(z) onwhich f is continuous, there exists a unique Borel non-negative and σ-finitemeasure µI on R such that
f(z) =
∫ +∞
−∞eiztdµI(t), (4.28)
for all z = x+ iy ∈ C such that y ∈ I and x ∈ R. That is, f can be extendedto strips of the complex plane SI = z ∈ C : Im(z) ∈ I where it is positivedefinite and holomorphic.
Proof. Suppose that f is a complex-variable positive definite in the opencodifference set S and that f is continuous on the connected component iI ofS∩Im(z). From Theorem 4.1 f is holomorphic on the connected componenton S containing iI, say it SI . Therefore, on every SI , the functions defined byFy(x) = f(x+ iy) are analytic positive definite functions of the real variablefor some y ∈ I. Thus, by Krein’s extension theorem 4.4 they can be extendedto the horizontal lines of the complex plane L(y) = z = x+ iy ∈ C : x ∈ R
46
as analytic and positive definite functions. However, we still cannot concludethat these functions defined on the lines link continuously one to the other,and thus that f(z) is an analytic positive definite function on the whole stripSI = z ∈ C : Im(z) ∈ I of the complex plane. On the other hand, as wedid in the proof of Theorem 4.1 and saw in the beginning of Section 4.1, thefunction G(y) defined by G(y) = f(iy) has an analytic extension, g(z), tothe strip SI , namely
g(z) =
∫ +∞
−∞eiztdµI(t) z ∈ SI . (4.29)
Therefore, by virtue of the identity theorem for functions of the complexvariable, we have that g(z) ≡ f(z) on SI .
If we drop the hypothesis of continuity in Theorem 4.6 the result is ratherweak. In fact, we have to give up on integral representations, analyticity anduniqueness of the extensions. However, the condition of positive definitenessstill can be extended to the whole strip SI , but just by lines.
Theorem 4.7. Let f be a complex positive definite function on an opencodifference set S =codiff(Ω). For every connected component, iI, of S ∩Im(z), the real-variable functions Fy(x) = f(x + iy) are positive definite onthe horizontal lines L(y) = z = x+ iy ∈ C : x ∈ R, for every fixed y ∈ I.
Proof. On the connected component of S containing iI = S∩Im(z), Fy(x) =f(x + iy) is a real-variable positive definite function for some y ∈ I, thenfor Artjomenko’s extension theorem 4.5 Fy(x) can be extended to a positivedefinite function on the line L(y) = z = x + iy ∈ C : x ∈ R. Since y isarbitrary in I we complete the proof.
Remark 4.5. A priori, the real-variable functions Fy(x) do not continuouslylink one to the other, do not benefit of integral representations and theextensions are not unique. That is, without further assumption of regularity,the extension of a complex positive definite function from codifference set tostrips is valid only by lines and is not univocally determined.
These results and remarks still do not prove, but strongly suggest, thatin the complex setting it may only make sense to consider positive definitefunctions on strips of the complex plane, and that the regularity of thesefunctions establish the existence and eventually the uniqueness of an integralrepresentation. These results are not sufficient to prove our conjecture. Infact, as we saw in Section 2.2.1 there exist codifference sets, as for exampleS2 in Figure 2.1, that is not connected and some of its component do not
47
intersect the imaginary axis, that is we cannot apply Theorem 4.6 and noteven Theorem 4.7. In the future, may be interesting to understand if evenin these cases we can state something similar to the previous Theorem 4.3,4.1, 4.6, 4.7. Another open question remains: the characterisation of thepoles of a complex-variable positive definite function. From a first analysis,it seems that poles can only be located on the imaginary axis and that theorder of a pole may indicate the positive definite nature of the function on aneighboring strip.
Example 4.2. As an example we refer to the function f(z) = 1cosh(πz
2). This
function is positive definite in any horizontal strip of the form
sn = S(∞, 4n− 1, 4n+ 1) = z ∈ C : (4n− 1)i < z < (4n+ 1)i, n ∈ Z.
In fact, f admits the integral representation
1
cosh(πz2
)=
∫ +∞
−∞eizt
1
cosh tdt (4.30)
on the strip s0 = S(∞,−1, 1) = z ∈ C : |Im(z)| < 1, which implies thatit is positive definite on s0. However, since f is periodic of period 4i mustbe positive definite in every strip sn, even though the integral representation(4.30) is divergent outside s0.
48
Chapter 5
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