Universidade de Lisboa
Faculdade de Ciˆ
encias
Departamento de Matem´
atica
Complex positive definiteness,
including characteristic and moment
generating functions
Alexandra Symeonides
Disserta¸c˜
ao
Mestrado em Matem´
atica
Universidade de Lisboa
Faculdade de Ciˆ
encias
Departamento de Matem´
atica
Complex positive definiteness,
including characteristic and moment
generating functions
Alexandra Symeonides
Disserta¸c˜
ao
Mestrado em Matem´
atica
Orientador: Professor Doutor Jorge Buescu
2013
Resumo
A partir do in´ıcio do s´eculo passado, as fun¸c˜oes definidas positivas foram ob-jecto de estudos em muitas e diferentes ´areas da matem´atica como teoria da probabilidade, teoria dos operadores, an´alise de Fourier etc. Foi por causa disto que nota¸c˜oes e generaliza¸c˜oes das fun¸c˜oes definidas positivas prove-nientes das diversas ´areas nunca foram reunidas numa ´unica doutrina. O prop´osito desta tese, ´e estudar com maior detalhe fun¸c˜oes definidas positivas de vari´avel complexa em dom´ınios particulares do plano complexo.
No Cap´ıtulo 1, daremos a defini¸c˜ao de fun¸c˜ao definida positiva, algu-mas propriedades b´asicas, o teorema de representa¸c˜ao de Bochner e tamb´em algumas propriedades diferenciais destas fun¸c˜oes. Em particular, vamos con-siderar o caso de fun¸c˜ao definida positiva e anal´ıtica sobre o eixo real e vamos ver, como neste caso, ´e poss´ıvel estender a fun¸c˜ao ao plano complexo, assim generalizando o conceito de fun¸c˜ao definida positiva no caso de fun¸c˜ao de variavel complexa. ´E a partir deste resultado devido a Z. Sasvari, see [2], que J. Buescu e A. C. Paix˜ao deram a primeira defini¸c˜ao de fun¸c˜ao definida positiva de vari´avel complexa, sem requerer nenhuma ulterior regularidade sobre a fun¸c˜ao. Veremos, como muitas das propriedas b´asicas e diferenciais de fun¸c˜oes definidas positivas reais s˜ao v´alidas tamb´em no caso complexo com generaliza¸c˜oes oportunas. Al´em disso, J. Buescu e A. C. Paix˜ao carac-terizaram os conjuntos do plano complexo onde a defini¸c˜ao de fun¸c˜ao definida positiva est´a bem dada, e chamaram a estes conjuntos codifference sets. En-fim, neste Cap´ıtulo 1, vamos apresentar tamb´em o conceito de fun¸c˜ao real co-definida positiva e vamos estudar rela¸c˜oes e analogias desta fun¸c˜ao com as de uma fun¸c˜ao definida positiva cl´assica. Por exemplo, enunciaremos o an´alogo do teorema de Bochner, o teorema de Widder, que garante a ex-istˆencia de uma representa¸c˜ao integral para fun¸c˜oes co-definidas positivas.
No Cap´ıtulo 2, vamos estudar fun¸c˜oes definidas positivas, mas a par-tir de um ponto de vista da teoria da probabilidade. De facto, a nota¸c˜ao probabil´ıstica revela-se particularmente ´util quando se trabalha com repre-senta¸c˜oes integrais de fun¸c˜oes definidas positivas, sejam de vari´avel real ou de vari´avel complexa. Os teoremas de representa¸c˜ao de Bochner e de Widder
para fun¸c˜oes respectivamente definidas e co-definidas positivas explicitam a rela¸c˜ao destas fun¸c˜oes com a bem conhecida ferramenta da teoria da proba-bilidade, ou seja fun¸c˜oes caracter´ısticas, fun¸c˜oes geradoras dos momentos e problema dos momentos. Portanto, iremos estudar estas fun¸c˜oes na ´optica do nosso interesse acerca das fun¸c˜oes definidas positivas, logo n˜ao iremos fornecer uma cl´assica revis˜ao desta ferramenta, que de facto pode ser en-contrada em qualquer manual de teoria da probabilidade. Referimos por exemplo os livros de J. S. Rosenthal [13] e de R. Ash [1].
Enfim, no Cap´ıtulo 3, vamos concentrar-nos sobre fun¸c˜oes definidas pos-itivas de vari´avel complexa em faixas do plano complexo. De facto, veremos como as faixas parecem ser os ´unicos conjuntos onde faz sentido considerar uma fun¸c˜ao definida positiva que possui um m´ınimo de regularidade. Provar isto, foi um dos prop´ositos, indirectos, desta tese. De facto, os resultados desta tese sugerem e n˜ao refutam, mas ainda n˜ao provam, a suposi¸c˜ao prece-dente. Daremos condi¸c˜oes sobre fun¸c˜oes complexas definidas positivas em faixas para garantir a existˆencia e eventualmente a unicidade de uma repre-seta¸c˜ao integral. Observaremos, que a existˆencia ou n˜ao desta representa¸c˜ao depende da regularidade da fun¸c˜ao e que a regularidade da fun¸c˜ao em toda a faixa `e dominada pela regularidade da fun¸c˜ao sobre o intervalo do eixo imagin´ario que intersecta a faixa considerada. Em particular, iremos provar que uma fun¸c˜ao complexa definida positiva numa faixa que seja pelo menos cont´ınua no intervalo do eixo imagin´ario que intersecta a faixa ´e de facto uma fun¸c˜ao anal´ıtica em toda a faixa. Tamb´em, demonstraremos que uma fun¸c˜ao anal´ıtica definida positiva numa faixa do plano complexo possui uma ´
unica representa¸c˜ao integral. Al´em disso, daremos uma generaliza¸c˜ao no caso complexo do problema de extens˜ao para fun¸c˜oes definidas positivas. Veremos como, dada uma fun¸c˜ao definida positiva num codifference set qualquer, nas componentes conexas do codifference set que intersectam o eixo imagin´ario ´
e poss´ıvel, comforme a regularidade da fun¸c˜ao, extender a fun¸c˜ao e a pro-priedade de ser definida positiva, a todas as faixas horizontais que cont´em as componentes conexas do codifference set original. Infelizmente, veremos tamb´em como este conjunto de resultados resolve s´o parcialmente a quest˜ao de estabelecer as faixas como codifference sets por excelˆencia.
Palavras-chave Fun¸c˜oes definidas positivas, An´alise complexa, Fun¸c˜oes caracter´ısticas e outras transformadas.
Mathematics Subject Classification (2010) Prim´ario 42A82; Secund´ario 30A10, 60E10.
Abstract
In Chapter 1, we will give the definition of positive definite functions on R and we will present some basic and differential properties of these functions. In particular, we will consider the case of analytic positive definite functions on R in order to construct continuations to the complex plane. In view of this, we will present the definition of complex-variable positive definite function mainly due to J. Buescu and A. Paix˜ao and we will see how several of the differential properties valide in the real case can be generalized in the complex settings. Moreover, is given here the notion of codifference set as the set of the complex plane in which the definition of complex positive definite function is well-given. In Chapter 1, we will also introduce another similar property to positive definiteness, namely co-positive definiteness.
In Chapter 2, we will look at the concept of positive definite function from a probabilistic point of view. In order to do that, we will recall the notion of characteristic function and moment generating function and we will show how, thanks to Bochner’s and Widder’s representation theorems, these objects respectively correspond to positive definite and co-positive definite functions. Furthermore, we will present the so-called moment problem.
In Chapter 3 we will focus on complex-variable positive definite functions on strips of the complex plane. We tried to understand under which condi-tions a complex positive definite function on a strip benefits of an integral representation and eventually when it is unique. We found out that the existence or not of such a representation depends on the regularity of the function; and that the regularity of a complex positive definite function on a strip is completely imposed by the regularity of the function on the interval of 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.
Acknowledgements
I would like to thank my advisor Jorge Buescu and the Professor Antonio Carlos Paix˜ao that to all effects is co-advisor of this thesis. I want to thank them 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 my Master thesis and I simply couldn’t do it without their support.
Thanks to S´ergio for his encouragements. Thanks to all the friends of Rua da Saudade. Grazie a mamma e pap`a, e a Sara, sempre tanto vicini.
Contents
Resumo i
Abstract ii
1 Introduction 1
2 Positive definite functions 5
2.1 Real-variable positive definite functions . . . 5
2.1.1 Co-positive definite functions . . . 8
2.2 Complex-variable positive definite functions . . . 9
2.2.1 Codifference sets . . . 10
2.2.2 Properties of complex positive definite functions . . . . 12
3 Characteristic functions 23 3.1 Moment generating functions . . . 29
3.2 Moment problem . . . 32
3.2.1 Hamburger moment problem . . . 32
4 Complex positive definite functions on strips 37 4.1 Propagation of regularity . . . 37
4.2 Integral representations . . . 40
4.3 The extension problem . . . 45
Chapter 1
Introduction
The concept of positive definiteness appears for the first time in 1907 in a paper of the mathematician Carath´eodory. He was looking for necessary and sufficient conditions on the coefficients of the power series
1 +
∞
X
k=1
(ak+ ibk)zk
analytic on the unit disc in order to have positive real part. Carath´eodory characterized these point, (a1, b1, . . . , an, bn) for n ∈ N, as the points that lie
in the smallest convex set containing the points
2(cos ϕ, sin ϕ, . . . , cos nϕ, sin nϕ), with 0 ≤ ϕ ≤ 2π.
In 1911 Toepliz noticed that Carath´eodory’s condition is equivalent to
n
X
k,l=1
dk−lckc¯l ≥ 0, ∀ n ∈ N, ∀ {ck}nk=1 ∈ C (1.1)
where d0 = 2, dk = ak− ibk, d−k = ¯dk. That is, if and only if dk is a positive
definite sequence. In the same year, thanks to Toepliz’s deduction, Herglotz solved the so-called trigonometric moment problem. Indeed, he stated that a sequence {dn} satisfies (1.1) if and only if there exists a unique non-negative
and finite Borel measure µ such that dn=
Z 2π 0
eintdµ(t), n ∈ Z.
In 1923 Mathias introduced the notion of positive definite function. A function f : R → C is positive definite if
f (−x) = f (x), x ∈ R (1.2) and m X j,k=1 ξjξkf (xj − xk) ≥ 0 (1.3)
for all m ∈ N, {ξk}mk=1 ⊂ C and {xk}mk=1 ⊂ R. That is, if every square
matrix [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 a bounded non-decreasing function µ on R such that f is the Fourier-Stieltjes transform of µ, that is
f (x) = Z +∞
−∞
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 Hilbert spaces have several applications to the theory of integral equations. Actually, positive definite kernels were introduced by Mercer in 1909, that is before positive definite functions. We call k a positive definite kernel if k(x, y) is any complex-valued function on R2 such that
n
X
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 could be used to prove an important theorem on one-parameter groups of unitary operators, namely Stone Theorem; and with the appearance of harmonic analysis on groups in 1940’s the role of positive definite functions in Fourier analysis became apparent.
Perhaps the area of mathematics in which most people are familiar with positive 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
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 L´evy who first recognized that in general it is easier to work with characteristic functions instead of probability distributions. It is not surprising that the central limit problem (the problem of convergence of sums of laws of probability) was in fact solved with the aid of positive definite functions.
It is because of the concept of positive definite function being such a central notion in so many different theories that it never had been unified in a unique doctrine; and that still today there is a big disparity between the notations from distinct mathematical areas.
In Chapter 1, after recalling the definition of positive definite functions on R, we will present some basic properties and some differential properties of these functions. In particular, we will consider the case of analytic positive definite functions on R in order to construct continuations to the complex plane, see Sasvari [2], and thus, in order to extend the condition of positive definiteness to analytic functions of the complex variable. In view of this, we will present the a priori definition, that is requiring no further regularity on the function, of complex-variable positive definite function mainly due J. Buescu and A. Paix˜ao, see [10] and we will see how several of the differential properties 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 and A. Paix˜ao, see [10], as the set of the complex plane in which the definition of complex positive definite function is well-given. In Chapter 1, we will also introduce another property very similar to positive definiteness, namely co-positive definiteness, and we will show that even for such functions exists a representation theorem of 1933 due to Widder, see [18].
In Chapter 2, we will look at the concept of positive definite function from a probabilistic point of view, since the notation of the probability theory re-sulted pretty useful when dealing with integral representations of positive definite functions both of real or complex variable. In order to do that, we will recall the notion of characteristic function and moment generating function and we will show how, thanks to Bochner’s and Widder’s repre-sentation theorems, these objects respectively correspond to positive definite and co-positive definite functions. However, we will explore these tools in the perspective of what we are interested in, thus we will not offer a common exposition of characteristic and moment generating functions as can be found in any manual of probability theory, see for example J. S. Rosenthal [13] and R. Ash [1]. Furthermore, we will present the so-called moment problem in order to show its relations with positive definite functions.
Finally, in Chapter 3 we will focus on complex-variable positive definite functions on strips of the complex plane. In fact, our interest is particularly
focused on this kind of codifference sets, since at a first sight they seems to be the only sets in which it makes sense to consider a complex positive definite function. To prove this was one of the purposes, an indirect one, of this thesis. Indeed, we tried to understand under which conditions a complex positive definite function benefits of an integral representation and eventually when it is unique. We found out that the existence or not of such a representation depends on the regularity of the function; and that the regularity of a complex positive definite function on a strip is completely imposed by the regularity of the function on the interval of the imaginary axis contained in the strip. In particular, we will see that a complex positive definite function on a strip, that is at least continuous on the imaginary axis will result holomorphic in the whole strip; and that holomorphy will ensure the existence of a unique integral representation in the strip. Moreover, we will state a generalization of the extension problem for complex positive definite functions. In fact, we will show when and how a positive definite function on an arbitrary codifference set can be extended to strips of the complex plane. However, this results accomplish only in part the problem of establishing the strips as the only set in which make sense to consider positive definite functions.
Chapter 2
Positive definite functions
The purpose of this chapter is to introduce the theory of positive definite functions of real variable and to extend, in case of analyticity, this concept to complex-variable positive definite functions, see Z. Sasv´ari [2]. Moreover, we will present the recent a priori description of positive definite functions of complex variable, that is without requiring further regularity on the func-tions, mainly due to J. Buescu and A. C. Paix˜ao, see [10] and [9].
2.1
Real-variable positive definite functions
Definition 2.1. A function f : R → C is positive definite if
m
X
j,k=1
ξjξkf (xj − xk) ≥ 0 (2.1)
∀ m ∈ N, ∀ {ξk}mk=1 ⊂ C and ∀ {xk}mk=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 follow from the definition considering the cases n = 1, 2 with a suitable choice of the sequences {xk}mk=1 and {ξk}mk=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 the
functions ¯f1, f1(−x), Re(f1), |f1|2 and f1f2 are positive definite. Moreover,
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 is positive definite if and only if it is the Fourier-Stieltjes transform of a finite and non-negative measure µ on R, that is
f (x) = Z +∞
−∞
eitxdµ(t). (2.2) Proof. We will only prove that for a function f to be a Fourier-Stieltjes transform of a finite non-negative measure µ on R is sufficient to be a positive definite function. m X j,k=1 ξjξkf (xj − xk) = Z +∞ −∞ m X j,k=1 ξjξke i(xj−xk)tdµ(t) = Z +∞ −∞ m X j,k=1 ξjξke ixjteixktdµ(t) = Z +∞ −∞ m X 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 neighborhood of the origin is uniformly continuous on R.
Theorem 2.3 (Propagation of regularity). Let f : R → C be a positive definite function of class C2n in some neighborhood of the origin for some positive 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 in Buescu and Paix˜ao [8] and the corresponding literature. Note that propa-gation of regularity only occurs for even-order derivatives, in fact even-order derivatives play a central role in the theory of positive definite functions, as follows from the next results too.
Proposition 2.1. Let f : R → C be a positive definite function of class C2n in some neighborhood of the origin for some positive integer n. Then
f ∈ C2n(R) and for all integers 0 ≤ m ≤ n, the function (−1)mf2m(x) is
positive definite. Proof. See [8].
This result gives rise to a two-parameter family of differential inequalities for positive definite functions which is very useful when dealing for example with integral equations. In the context of positive definite kernel Hilbert spaces, these inequalities may be interpreted as a generalized Cauchy-Schwarz inequality.
Proposition 2.2. Let f : R → C be a positive definite function of class C2n in some neighborhood of the origin for some positive integer n. Then
f ∈ C2n(R) and for all integers m
1, m2 with 0 ≤ m1 ≤ n, 0 ≤ m2 ≤ n and
every 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 every 0 ≤ m ≤ n, the right hand-side of (2.3) is positive because of basic property 1 of positive definite functions, thus the inequality is meaningful.
Theorem 2.4. Let f : R → C be a positive definite function of class C2n in
some neighborhood of the origin for some positive integer n. If f(2m)(0) = 0
for 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 |f0(x)|2 ≤ −f (0)f00(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 stating the 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 open interval J , with α ∈ J ⊂ I, and constants C > 0 and R > 0 such that the derivatives satisfy
|f(k)(x)| ≤ C k!
Theorem 2.5. Let f : R → C be a positive definite function of class C∞ in some neighborhood of the origin. Then, if there exist positive constants M and 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 2nq|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 extends holomorphically 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 and z = −ih simultaneously, implying in particular that h = min{α, β}. Proof. See [8].
Remark 2.2. The statement of this theorem is slightly different from others already known in the literature, for example Z. Sasvari [2] using stronger hypothesis, that is including statement 1, concluding that the holomorphic extension of f to the maximal strip S must present singularities in both z = −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 convenient to observe that a different sign in the definition with respect to positive definite functions will lead to a completely different, but analogous, variety of properties for these functions.
Definition 2.2. A function f : R → C is co-positive definite if
m
X
j,k=1
ξjξkf (xj + xk) ≥ 0 (2.6)
∀ m ∈ N, ∀ {ξk}mk=0 ⊂ C and ∀ {xk}mk=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 positive definite functions. However, considering the case n = 1 in (2.6) we conclude that f (x) ≥ 0 for every x ∈ R, thus f has real values. Moreover, even for this kind of function there exists a representation theorem due to Widder (1933).
Theorem 2.6. A function f can be represented in the form f (x) =
Z +∞
−∞
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-variable positive definite functions in the conditions of Theorem 2.5. Indeed, an analytic real-variable positive definite function extends holomorphically to a horizontal strip of the complex plane S = {z ∈ C : −α < Im(z) < β}, with α, β > 0 and maximal with this property. Bochner’s integral representation (2.2) extends holomorphically to S too, so that
f (z) = 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
X
j,k=1
ξjξkf (zj− zk) ≥ 0 (2.9)
∀m ∈ N, ∀{ξk}mk=1 ⊂ C, ∀zj, zk∈ S such that zj− zk ∈ S. In fact, m X j,k=1 ξjξkf (zj− zk) = Z +∞ −∞ m X j,k=1 ξjξke i(zj−zk)tdµ(t) = Z +∞ −∞ m X j,k=1 ξjξkeizjteizktdµ(t) = Z +∞ −∞ m X j=1 ξjeizjt 2 dµ(t) ≥ 0.
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. Paix˜ao [10] give a definition of complex-variable positive definite function that naturally arises from the above observation of Z. Sasvari , but that requires no further assumption on the regularity of the function. From this new definition of complex positive definite function, Buescu and Paix˜ao deduce a list of properties for that kind of function and they figure out on which kind of complex set make sense to consider a complex positive definite function. In the following, I will report the main results (and their proofs) of this paper [10].
Definition 2.3. A function f : C → C is positive definite in the open set S ⊂ C if m
X
j,k=0
ξjξ¯kf (zj− ¯zk) ≥ 0 (2.10)
∀m ∈ N, ∀{ξk}mk=0 ⊂ C, ∀zj, zk∈ S such that zj− ¯zk ∈ S.
Remark that Definition 2.3 does not require any regularity on the function f and that in the complex case Bochner’s representation theorem is not valid. Thus a complex function as in Definition 2.3 does not have an integral representation (2.8). However, we already saw that holomorphic extensions of real analytic positive definite functions have an integral representation (2.8) and provide examples of complex positive definite functions on a complex strip containing the real axis.
Moreover, another matter is now open: which kind of set S is such that for every zj, zk ∈ S, then zj− ¯zk∈ S? On which kind of set S is then possible
to 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. Buescu and A. C. Paix˜ao, [10], introduce the codifference sets.
Definition 2.4. A set S ⊂ C is a codifference set if there exists a set Ω ⊂ C such 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 set difference.
Here are some properties of codifference sets that directly follow from Definition 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 under any translation of the codifference-generating set Ω parallel to the real axis.
2. If z ∈ S, there exist z1, z2 ∈ Ω such that z = z1 − z2, obviously
z2 − z1 = −z ∈ S. Hence any codifference set is symmetric with
respect 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, then
there 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 codifference
set 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 concon-nected.
Figure 2.1: Codifference sets
2.2.2
Properties of complex positive definite functions
We now present some basic properties of complex-variable positive definite functions directly derived from Definition 2.3 by J. Buescu and A. C. Paix˜ao [10]. Observe that most of the following properties are the complex analog of the corresponding properties of real-variable positive definite functions. Proposition 2.3 (Positivity on the imaginary axis). Let f be a complex positive definite function on a codifference set S, f (ib) ≥ 0, ∀ib ∈ S, where b ∈ R.
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, thus f (ib) ≥ 0.
Therefore positive definite functions are always real and non-negative on the imaginary axis.
Proposition 2.4 (Basic properties). Let f be a complex positive definite function 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 = −a2 + i b+β2 such that zi− zj ∈ S for
i, j = 1, 2. From Definition 2.3 with m = 2 follows that
2
X
i,j=1
ξiξ¯jf (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 positive definite functions on strips of the complex plane, that directly follow from the definitions of positive and co-positive definiteness.
Proposition 2.5. Let f be a complex-variable positive definite function on the 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 definite
function 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 open set Ω, that is S =codiff(Ω). Therefore, in order to prove statement 1, let
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 X k,j=1 ξkξjFy(xk− xj) = n X k,j=1 ξkξjf (xk− xj + iy) = n X k,j=1 ξkξjf (xk+ i y 2 − xj + i y 2) = n X k,j=1 ξkξjf (zk− zj) ≥ 0,
and statement 1 is proved. Similarly, to prove statement 2, let zk = iyk and
zj = iyj in Ω such that zk− ¯zj ∈ S. Then n X k,j=1 ξkξjG(yk+ yj) = n X k,j=1 ξkξjf (i(yk+ yj)) = n X k,j=1 ξkξjf (zk− zj) ≥ 0.
Thus G(y) is a co-positive definite function on (a, b), that is statement 2 is proved.
Complex positive definite functions are controlled by their behaviour on the imaginary axis as real positive definite function are controlled by their behaviour 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 some real 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
2 if 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 that
f (ic) = 0. The statement is true for n = 1 by hypothesis. For each n we set un = b − β, un+1= b, a = 0 and b + β = 2un+1− un, then using statement 2
of 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 and
2, observe that since by hypothesis f (ib) 6= 0 for every b ∈ I, then because of Property 2.3, f (ib) > 0 for all b ∈ I. Thus g = log(f ) is well-defined on iI. Taking a = 0, b1 = b + β and b2 = b − β in statement 2 of Proposition
2.4 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 proved by Dugu´e [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 positive definite 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 the existence of β such that b ± β ∈ S, while statement 2 of Proposition 2.4 asserts that |f (a + ib)|2 ≤ f (i(b − β))f (i(b + β)). By virtue of Lemma 2.2 the hypothesis 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-variable positive definite functions, J. Buescu and A. C. Paix˜ao, [10], state a collec-tion of differential properties for complex positive definite funccollec-tions. How-ever, this time the use of positive definite kernels in two complex variable is mandatory since without further assumption of regularity a complex posi-tive definite function does not possess of an integral representation. Posiposi-tive definite functions are related with positive definite kernels in two complex variables in the following way. Suppose f is positive definite in S ⊂ C and that V = {(z, u) ∈ C2 : z − ¯u ∈ S}. Define k : V → C such that
k(z, u) := f (z − ¯u). Let Ω ⊂ C such that Ω2 ⊂ V , that is such that codiff(Ω) ⊂ S. Therefore
n
X
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) in Ω2, thus k is a holomorphic positive definite kernel in Ω. Under this further
assumption of regularity much more can be said. The following results are proved in [7] for holomorphic positive definite kernels of several complex variable.
Theorem 2.8. Let Ω ⊂ C be an open set and k : Ω2 → C be a holomorphic positive 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 holomorphic
positive 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 holomorphic
positive definite kernel on Ω. Then for all m1, m2 ∈ N and for all z, u ∈ Ω
we 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 complex positive 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 that f : S → C is positive definite and holomorphic in S. Then (−1)mf(2m)(z) is
a positive definite function in S for every m ∈ N. Proof. We want to show that
n
X
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 ∈ C
for i = 1, . . . , n such that codiff(zi) ∈ S, that is zij := zi − ¯zj ∈ S for all
i, j = 1, . . . , n. Consider Ω as the union of n squares Qr(z), that is Ω =
S
i=1,...,nQr/2(zi), and choose r such that U :=codiff(Ω) =
S
i=1,...,nQr(zij) is
contained in S. Then k(z, u) := f (z − ¯u) is a positive definite kernel in Ω and because of Theorem 2.8
∂2m
∂ ¯um∂zmk(z, u) = (−1)
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 that f : S → C is positive definite and holomorphic in S. Suppose that S contains the points ±a + ib and (b ± β)i for a, b, β ∈ R. Then, for every non-negative integers 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) = z11and (b−β) = z22. Choose r > 0 such that U =Si,j=1,2Q2r(zij) ⊂
S. Consider the points z1 = a2+ i b+β2 and z2 = −a2+ i b−β2 such that zij =
zi − zj ∈ S for i, j = 1, 2. Defining Ω = Qr(z1) ∩ Qr(z2), U =codiff(Ω) ⊂ S
and then z − ¯u ∈ U ⊂ S for all z, u ∈ Ω. Therefore k(z, u) := f (z − ¯u) is a positive 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 the
chain rule.
Let’s see now what it means for a meromorphic function to be positive definite. In particular, the interest of J. Buescu and A. C. Paix˜ao in [10] is to understand if, for example, under the assumption of being positive definite the poles of a meromorphic function can be easily found.
Theorem 2.12. Let Ω ⊂ C be an open set such that S =codiff(Ω). Suppose f is meromorphic in S and positive definite in S ∩ D(f ), where D(f ) is the domain of f . Then f is holomorphic in S.
Proof. Observe that, since Ω is open, S is open. Let z = a + ib ∈ S, then as proved in property 2 of codifference sets, −z ∈ S. Moreover, from property 3 it follows that z11 := z1− z1 and z22 := z2− z2 lie in S whenever we write
z = z1− z2 for some z1, z2 ∈ Ω and β = Im(z1− z2). Since S is an open set
and the singularities of f are isolated, we may choose z1, z2, and therefore
β, such that z11, z22 are points where f is analytic. Let zn := an+ ib be
a sequence converging to z, that is, such that limn→+∞an = a. Since f is
meromorphic the set of singularities of f has no accumulation points, there exists p ∈ N such that f is analytic in both zn and −zn for all n ≥ p. For
each 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 that
limn→+∞|f (zn)| = +∞, contradicting the previous inequality. Therefore z
cannot be a pole of f . Since z is an arbitrary point of S, then f is holomorphic in S.
Corollary 2.2. Suppose f is meromorphic in C and positive definite in its domain. 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 S
with infinite r, α1, α2, it follows immediately from Theorem 2.12 that f is
entire.
Theorem 2.13. Suppose S is an open codifference set and let f : S → C be 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 constant on 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 some positive ε. Moreover, F (x) is positive definite as proved in Proposition 2.5 and such that F(k)(x) = f(n)(x + ib) for every x ∈ I and any non-negative integer 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
|F0(x)|2 ≤ −F (0)F00(0) ∀x ∈ I. (2.16) If F(m)(0) = 0 for m = 0 or m = 1 the thesis is trivially true. Consequently we will consider that m > 1. The idea of the proof is to show that F(2m)(0) =
0 implies F00(0) = 0 for m > 1 since under that hypothesis it is possible to conclude from (2.16) that F0 vanishes identically on I and, consequently, that f0(x + ib) = 0 for every x ∈ I. Since f is holomorphic on S, analytic continuation of f ensures that f0(z) = 0 on the open connected component of S containing ib, implying that f is constant on this set and proving the statement. To prove the implication, suppose m > 1 and define by recurrence a sequence of even numbers, with k1 = 2m and
ki+1 =
ki
2 if ki/2 is even ki
2 + 1 if ki/2 is odd.
Notice that ki+1 < ki whenever ki > 2 and that 2 is a fixed point of the
recurrence. Then, there exists j(m) such that kl = 2 for all l ≥ j; in fact it
is easily shown that j(m) ≤ m. We now prove that fki(0) = 0 for all i ∈ N.
(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 definition
of 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, this
implies that F00(0) = 0 and |F0(x)| = 0 for every x ∈ I, completing the proof.
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
q
|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 positive definite 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
q
|f(n)(z)|
n! and observe,
by considering the odd and even subsequences of un(z), that
lim sup n→∞ un(z) = max{lim sup n→∞ u2n(z), lim sup n→∞ u2n+1(z)}. (2.18)
Suppose, in addition, that z ∈ S is a point on the imaginary axis. The idea is to show that lim sup n→∞ u2n+1(z) ≤ lim sup n→∞ u2n(z), (2.19)
since this implies
lim sup
n→∞
un(z) = lim sup n→∞
u2n(z).
For z = ib, using inequality (2.15) with a = 0, m1 = n and m2 = n + 1 we
have
Then we have |f(2n+1)(ib)| (2n + 1)! 2n+12 ≤ |f(2n)(ib)| (2n)! 2n1 2 2n+1 |f(2n+2)(ib)| (2n + 2)! 2n+21 2n+2 2n+1 2n + 2 2n + 1 2n+11 establishing (2.19) and that
l(ib) = lim sup
n→∞
2n
r
|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 sup n→∞ |f(2n+1)(a + ib)| (2n + 1)! 2n+12 ≤ lim sup n→∞ |f(2n)(i(b + β))| (2n)! 2n1 lim sup n→∞ |f(2n+2)(i(b − β))| (2n + 2)! 2n+21 or, in view of (2.20), lim sup n→∞ 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 sup n→∞ u2n(a + ib) 2 ≤ l(i(b + β))l(i(b − β)). Therefore, according to (2.18)
Hence, we have
r2(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 f is entire.
Proof. If f has no poles on the imaginary axis, then there exists h > 0 such that f is positive definite and holomorphic on the square Qh(ib). Hence using
the 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 + ib0
such that |z − z0| = r(ib) and a0 6= 0 since by hypothesis f has no poles on
the imaginary axis. Choose a ∈ (−h, h) such that |a − a0| ≤ |a0|, and write
z = a + ib. Then
|z − z0| =p|a − a0|2+ |b − b0|2 <
q a2
0+ |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) be
the 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 definite in S ∩ D(f ) and that L(b0) ⊂ S ∩ D(f ). Then f has no poles on the strip
S = {z = a + ib ∈ C : a ∈ R and |b − b0| < r(b0)}. If r(b0) is finite, then at
least 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 of
f centered at z = a + ib0, the distance from the set of poles of f to the line
L(b0) must be greater or equal than r(b0) and the first assertion follows. If
r(ib0) is finite we conclude that at least one of i(b + β) and i(b − β) is a pole
of 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},
where α, β ∈ (0, +∞[, as a positive definite function admitting, for some non-negative measure µ, the integral representation
f (z) = Z +∞
−∞
e−(iz−b0)tdµ(t), ∀ z ∈ S
M. (2.22)
Moreover, r(b0) = min{α, 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 therefore admits 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 to Theorem 1.12.5 in [2], we write
F (z) = 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) =
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 whenever
α < ∞ (resp. β < ∞). As a direct consequence of Theorem 2.15, r(b0) must
Chapter 3
Characteristic functions
Positive definite functions and their various analogs and generalizations have arisen in different parts of mathematics since the beginning of the 20th cen-tury. They occur naturally in Fourier analysis, probability theory, operator theory, complex-variable function theory, moment problems, integral equa-tions and other areas. Since the concept of positive definite function is such a fundamental entity in so many distinct mathematical theories, the results never had been collected in one single body doctrine. In what follows, we will go into more detail on probability theory’s analogs of positive definite functions, namely characteristic functions, moment generating functions and moment problem. In fact, we found the probabilistic point of view extremely useful when dealing with integral representations of complex-variable positive definite 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 manual of probability theory, we will look at characteristic and moment generating functions as good examples of respectively positive definite and co-positive definite functions. In order to do this, we will just present properties of these functions that will be useful for the purpose of this thesis. Let us start with some basic recalls from the probability theory. For a more in-depth analysis and 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 called a algebra if and only if
1. Ω ∈ F ,
2. if A ∈ F , then Ac∈ F . 3. if A1, A2, . . . , An ∈ F , then
Sn
If 3 is replaced by closure under countable union, that is, 3. if A1, A2, . . . ∈ F , thenS
∞
i=1Ai ∈ F .
F is called σ-algebra.
Example 3.1. If Ω is the set R of extended real numbers, and F consist of all 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 the smallest σ-algebra containing all the intervals (a, b] with a, b ∈ R. Note that B(R) is garanteed to exist, and it may be described as the intersection of all σ-algebras containing the intervals (a, b]. Also if a σ-algebra contains all the open 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, extended real-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 ! =X n µ(An). (3.2)
A measure space is a triple (Ω, F , µ) where Ω is a set, F is a σ-algebra of subsets 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 if µ(Ω) is finite.
A measure µ on F is said to be σ-finite on F if and only if Ω can be written as S∞
Theorem 3.1 (Carath´eodory’s extension theorem). Let µ be a measure on the algebra F0 of subsets of Ω and assume that µ is σ-finite on F0, so that Ω
can be decomposed as S+∞
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 is increasing 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-one correspondence between Lebesgue-Stieltjes measures and distribution func-tions.
This, in particular, will better explain the statement of Widder’s theorem 2.6 where an integral representation with respect to a non-decreasing function appears, and will allow us to make the notation of this thesis uniform. We needed to enlight this correspondence in order to clarify the relation between Bochner’s and Widder’s integral representations. In fact, this relation will be useful in the next chapter, when dealing with integral representations in the complex settings.
Theorem 3.2. Let µ be a Lebesgue-Stieltjes measure on R. Let F : R → R defined up to an additive constant, by F (x) − F (a) = µ(a, b]. (For example fix 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 measure on R.
Proof. This is an application of Carath´eodory’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 the Lebesgue measure on B(R).
For the complete theory and the missing results and proofs we refer to R. Ash [1].
Recall that Widder’s theorem 2.6 states that a continuous function f is co-positive definite if and only if there exists a non-decreasing function α(t) such that
f (x) = Z +∞
−∞
e−xtdα(t). (3.3) Since α(t) is non-decreasing the set of discontinuity points of α(t) is at most countable.
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) = lim t→t+0 α(t) and α(t−0) = lim t→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, thus
t0 7−→ qt0 is a one-to-one map from D to Q, and since Q is countable, so is
D.
Then we can define β(t) : R → R such that {t ∈ R : β(t) 6= α(t)} is at most countable, and such that β(t) is non-decreasing and right-continuous, namely a distribution function. Hence, by Theorem 3.3, we can define a Lebesgue-Stieltjes measure µ from β(t) such that µ is non-negative, σ-finite on B(R) and eventually finite whenever β(t) is bounded.
Remark that the measure generated from α(t) is equivalent to the measure generated from β(t) whenever µ({t} == 0 for every t ∈ R. In light of this construction, Widder’s theorem 2.6 can be stated as follows. If f is a continuous co-positive definite function on (a, b) with a, b ∈ R, then there exists a non-negative and σ-finite measure µ such that
f (x) = Z +∞
−∞
e−xtdµ(t), x ∈ (a, b). (3.4) The most significant difference between Widder’s integral representation for co-positive definite functions and the one of Bochner for positive definite functions 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 always converge in a neighborhood of the origin, while Widder’s representation does not necessarily do so. Let us now finally give the definition of characteristic function.
function of µ is the mapping from R to C given by h(x) =
Z ∞
−∞
eitxdµ(t), x ∈ R. (3.5) Thus h is the Fourier transform of µ. If F is a distribution function corresponding to µ, we shall also write h(t) = R∞
−∞e
itxdF (t), and call h
the characteristic function of F (or of X, if X is a random variable with distribution function F ). A characteristic function as in (3.5) is of course defined for all x ∈ R, whenever t is a real number. In particular, if µ is a probability 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 the converse 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, R
BdF (t) =
R
−BdF (t) for all Borel sets B, where −B = {−x : x ∈ B}.
5. If R
R|t|
rdF (t) < ∞ for some positive integer r, then the rth derivative
of h exists and is continuous on R, and h(r)(x) =
Z
R
(it)reixtdF (t) (3.6) Proof. See R. Ash [1], Theorem 7.1.5.
Next we present some properties of analytic characteristic functions. They are mainly due to Sasvari [2] and are basically results of analytic continuation to the complex plane.
Theorem 3.5. If f is an analytic characteristic function then there exist αf, βf ∈ (0, ∞] such that f extends to a function which is holomorphic in
the strip {z ∈ C : −αf < Im(z) < βf} and such that αf and βf are maximal
Proof. See Sasvari [2], Theorem 1.12.2.
Theorem 3.6. Let f be an analytic characteristic function and let µ be the corresponding probability measure. Then
f (z) = 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 conditions of 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-variable positive definite function in the strip {z ∈ C : −αf < Im(z) < βf}. And
thus, all the properties presented in Chapter 1 for complex positive definite functions are valid here.
Proposition 3.1. A necessary condition for a function that is analytic in some neighborhood of the origin to be a characteristic function is that in either half-plane the singularity nearest to the real axis is located on the imaginary axis.
Proof. See Lukacs [12].
Proposition 3.2. An analytic characteristic function h(z) has no zeros on the segment of the imaginary axis inside the strip of analyticity. Moreover, the zeros and the singular points of h(z) are located symmetrically with respect to the imaginary axis.
Proof. See Lukacs [12].
Theorem 3.7 (L´evy-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 of h.
3.1
Moment generating functions
The moment generating function has been widely used by statisticians, and especially by the English writers, in place of the closely-related characteristic function. In fact, from both functions it is possible to extract informations on the corresponding probability measure or distribution function. Before we give the definition of moment generating function another notion must be 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, the moment Mn of µ is defined as
Mn=
Z +∞
−∞
xndµ(x). (3.8) We should note that if n is odd, in order for Mn to be defined we must
have R+∞
−∞ |x|
ndµ(x) < ∞. Given a probability distribution µ, either all the
moments may exist, or they exist only for 0 ≤ n ≤ n0 for some n0. It could
be that n0 = 0 as happens for example is for the Cauchy distribution π(1+x1 2).
The characteristic function of a given probability measure is strictly re-lated to the corresponding moments. In fact, from equation (3.6) of Theorem 3.4 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 derivatives of the characteristic function exist at the origin, then all the moments of the measure exist.
Definition 3.7. Let µ be a probability measure. The function G(t) =
Z +∞
−∞
etxdµ(x), t ∈ R (3.10) in which the integral is assumed to converge for t in some neighborhood of the origin, is called moment generating function of µ.
Remark that, if a probability measure has a moment generating function that converges in a non-trivial interval, then the domain of the correspondig characteristic function can be extended to the complex plane by
h(−it) = G(t). (3.11) However, remark that the characteristic function of a distribution always exists, while the moment generating function may not.
Let us now recall a standard result, namely Leibniz’s integral rule for differentiation under a Lebesgue-Stieltjes integral sign. We cite it here in order 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 Z I f (x, t)dµ(t) = Z I ∂f (x, t) ∂t dµ(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 Z +∞ −∞ etxdµ(x) = Z +∞ −∞ ∂n ∂tne txdµ(x)
for every n ∈ N and t ∈ (−a, b), because of Leibniz’s rule. That is, a moment generating function is infinitely differentiable in the interval of convergence. Actually, even more is true, a moment generating function is analytic in the interval 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
R+∞
−∞ |x|
ndµ(x) < ∞ for n ≥ 0 and G(t) is analytic in
|t| < t0 with G(t) = +∞ X n=0 R+∞ −∞ x ndµ(x)tn n! . (3.13) In particular the derivative of order k at zero is given by
G(k)(0) = Z +∞
−∞
xndµ(x). Proof. See e.g. [13].
Remark 3.2. Proposition 3.4 says that the n-th derivative of G(t) at 0 equals the n-th moment of the probability measure µ (thus explaining the termi-nology “moment generating function”). For example, G(0) = 1, G0(0) = R+∞
−∞ xdµ(x) , G(0) =
R+∞
−∞ x
2dµ(x) , etc.
Remark 3.3. By virtue of Widder’s representation (theorem 2.6) we have that a moment generating function g is of course a co-positive definite function. Conversely, we know that for a co-positive definite function there exists a non-decreasing function F —and thus a non-negative, σ-finite Borel measure µ on R— such that g(t) = Z +∞ −∞ etxdF (x) = Z +∞ −∞ etxdµ(x) (3.14) for some t ∈ (a, b). Therefore, whenever g(t) converges at zero, and thus the interval of convergence (a, b) contains the origin, g(t) is a moment generating function. On the other hand, if (a, b) does not contain the origin g(t) is not a moment generating function in the classical sense, but still conserves some of 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, according to the next statement of Dugu´e such a function can be analytically continued to 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 its interval of convergence. Then G(z) is analytic in the strip −a < Re(z) < b. In fact,
G(z) = Z +∞
−∞
eitzdµ(t) (3.15) is absolutely convergent for −a < Re(z) < b.
Proof. See Dugu´e [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. That is, the result is still true even when the measure µ is just σ-finite. From another point of view, it is well-known that a two-sided Laplace transform is analytic in its region of absolute convergence. We will return to this in the next Chapter.
3.2
Moment problem
The moment problem arises in mathematics as result of trying to invert the mapping that takes a measure µ to the sequences of moments Mn. Indeed, it
can be summarized as follows: “Given a sequence Mn, under which conditions
does there exist a measure µ on R such that all the moments of µ exist and are equivalent to Mn for every positive integer n?”. In the literature
we distinguish between three different moment problems depending on the support 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 a bounded 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 next section we will focus on the Hamburger moment problem since it is closely related 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 there
exists a positive Borel measure µ on the real line R such that Mn =
Z +∞
−∞
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. There exist further conditions that may be imposed on the moments to guarantee uniqueness, as for example Carleman’s and Krein’s conditions, but we will not go into details on it. For in-depth analysis we refer to [16]. Here, we will just 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
∞ X n=1 M− 1 2n 2n = +∞. (3.17)
More generally, it is sufficient that ∞ X n=1 γ2n−1 = +∞, (3.18) where γ2n = inf r≥n(M 1 2n 2n). (3.19)
Proof. See [16] pag. 19.
As already said, Hamburger provided a complete characterisation for a Hamburger moment sequence.
Proposition 3.6 (Hamburger). A sequence Mn is a Hamburger moment
sequence if and only if
m
X
j,k=0
ξjξ¯kMj+k ≥ 0 (3.20)
∀m ∈ N, ∀{ξk}mk=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
X
i,j=0
f(i+j)(c)ξiξj ≥ 0 (3.21)
for a fixed c ∈ (a, b), then
f (x) = Z +∞
−∞
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 that whenever a function f is analytic on an interval (a, b), if for every fixed point c ∈ (a, b) the sequences of the derivatives of f in c are Hamburger moment sequences —because of Hamburger’s characterisation 3.6—, then f must be a co-positive definite function in (a, b) for Widder’s representation theorem 2.6.
On the other hand, the Hamburger moment problem is related to positive definite functions too. The following results are basically due to Devinatz and can be found in [3].
Suppose that f (x) is an infinitely differentiable positive definite function. That is
f (x) = Z +∞
−∞
eixtdµ(t) (3.23) where µ(t) is a finite and non-negative Borel measure on R. Since f (x) is infinitely differentiable, obviously
f(n)(x) = Z +∞
−∞
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 X k=0 ξk(−i)kfk+m(x) 2 = Z +∞ −∞ tmeixt n X k=0 ξktkdµ(t) 2 ≤ Z +∞ −∞ t2mdµ(t) Z +∞ −∞ n X k=0 ξktk 2 dµ(t) = Mm n X r=0 n X s=0 ξrξ¯s(−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=0is a determined Hamburger moment sequence,
then these three conditions are sufficient for an infinitely differentiable func-tion on R to have the representafunc-tion (3.23) and thus to be a positive definite function. In fact, even more is true. In fact, Devinatz proved that if f (x) is infinitely differentiable just on some open interval containing the origin and satisfies the above conditions, then it has the representation (3.23), that is it can be extended to a positive definite function on R. Moreover, since the Hamburger moment sequence is by hypothesis determined, then the exten-sion is clearly unique. We will return to the problem of extenexten-sion for positive definite functions in the next chapter.
Theorem 3.9 (Devinatz). Let f (x) be an infinitely differentiable function defined on the open interval (−a, b) where a, b > 0. If
