An analytic characterization of the symmetric extension of a Herglotz-Nevanlinna function in several variables
AAn analytic characterization of the symmetric extension of aHerglotz-Nevanlinna function in several variables
Mitja Nedic
Abstract.
In this paper, we derive an analytic characterization of the sym-metric extension of a Herglotz-Nevanlinna function in several variables. Here,the main tools used are the so-called variable non-dependence property andthe symmetry formula satisfied by Herglotz-Nevanlinna and Cauchy-type func-tions. We also provide an extension of the Stieltjes inversion formula forCauchy-type functions.
1. Introduction
On the upper half-plane C + := { z ∈ C | Im[ z ] > } , the class of holomorphicfunctions with non-negative imaginary part plays an important role in many areasof analysis and applications. These functions, called Herglotz-Nevanlinna functions ,appear, to name but a few examples, in the theory of Sturm-Liouville operators andtheir perturbations [
3, 4, 7, 10 ], when studying the classical moment problem [ ], when deriving physical bounds for passive systems [ ] or as approximatingfunctions in certain convex optimization problems [
8, 9 ].A classical integral representation theorem [
6, 16 ] states that any Herglotz-Nevanlinna function h can be written, for z ∈ C + , as(1.1) h ( z ) = a + bz + 1 π (cid:90) R (cid:18) t − z − t t (cid:19) d µ ( t ) , where a ∈ R , b ≥ and µ is a positive Borel measure on R for which (cid:82) R (1 + t ) − d µ ( t ) < ∞ . Although this representation is a prioir established for z ∈ C + , itis well-defined, as an algebraic expression, for any z ∈ C \ R . Hence, for a Herglotz-Nevanlinna function h , we define its symmetric extension h sym as the right-handside of representation (1.1) where we now take z ∈ C \ R . It is now an easyconsequence of the definitions that a holomorphic function f : C \ R → C equals thesymmetric extension of some Herglotz-Nevanlinna function if and only if it holdsthat Im[ f ( z )] ≥ for z ∈ C + and f ( z ) = f ( z ) for all z ∈ C \ R . In this way, weobtain an analytic characterzation of the symmetric extension.When considering, instead, functions in the poly-upper half-plane C + n := ( C + ) n = (cid:8) z ∈ C n (cid:12)(cid:12) ∀ j = 1 , , . . . , n : Im[ z j ] > (cid:9) , Mathematics Subject Classification.
Key words.
Herglotz-Nevanlinna functions, Cauchy-type functions, symmetric extension. a r X i v : . [ m a t h . C V ] F e b MITJA NEDIC the analogous situation becomes more involved. Herglotz-Nevanlinna functionsin several variables, cf.
Definition 2.2, appear e.g. when considering operatormonotone functions [ ] or with representations of multidimensional passive systems[ ]. Their corresponding integral representation is recalled in detail in Theorem 2.3later on and leads, in an analogous way as in the one-variable case, to the definitionof a symmetric extension , which is now a holomorphic function on ( C \ R ) n . Assuch, the main goal of this paper is to give an analytic characterization of symmetricextensions of a Herglotz-Nevanlinna function in several variables, i.e. we wish to beable to determine when a function f : ( C \ R ) n → C is, in fact, equal to the symmetricextension of a Herglotz-Nevanlinna function. This is answered by Theorem 3.3 andCorollary 3.4.The structure of the paper is as follows. After the introduction in Section 1we review the different classes of functions that will appear throughout the paperin Section 2. Section 3 is then devoted to presenting the main result of the paperas well as some important examples. Finally, Section 4 discusses how the Stieltjesinversion formula can be extended to certain functions on ( C \ R ) n .
2. Classes of functions in the poly cut-plane
Throughout this paper, we will primarily consider two classes of holomorphicfunctions on the poly cut-plane ( C \ R ) n , both of which are intricately connectedto a certain kernel function. These objects are defined as follows. K n and Cauchy-type functions. We begin by introducingthe kernel K n : ( C \ R ) n × R n → C as(2.1) K n ( z , t ) := i (cid:32) i ) n n (cid:89) (cid:96) =1 (cid:18) t (cid:96) − z (cid:96) − t (cid:96) + i (cid:19) − i ) n n (cid:89) (cid:96) =1 (cid:18) t (cid:96) − i − t (cid:96) + i (cid:19)(cid:33) . If the vector z is restricted to C + n , then the kernel K n is a complex-constantmultiple of the Schwartz kernel of C + n viewed as a tubular domain over the cone [0 , ∞ ) n [ , Sec. 12.5].When n = 1 , it holds that K ( z, t ) = 1 t − z − t t . As such, the kernel K satisfies, for all z ∈ C \ R and all t ∈ R , the symmetryproperty K ( z, t ) = K ( z, t ) . When n ≥ , the symmetry satisfied by the kernel becomes more involved andrequires the introduction of some additional notation. First, given two numbers z, w ∈ C , an indexing set B ⊆ { , , . . . , n } and an index j ∈ { , , . . . , n } , define ψ jB ( z, w ) := (cid:26) z ; j (cid:54)∈ B,w ; j ∈ B. Second, given an indexing set B ⊆ { , , . . . , n } , define the map Ψ B : C n × C n → C n as Ψ B ( z , w ) := ζ with ζ j := ψ jB ( z j , w j ) . In other words, the map Ψ B functionsas a way of selectively combining two vectors into one where the set B determines HARACTERIZATION OF THE SYMMETRIC EXTENSION 3 which components of z should be replaced by the conjugates of the components of w . It now holds that(2.2) K n ( z , t ) = (cid:88) B ⊆{ ,...,n } B (cid:54) = ∅ ( − | B | +1 K n (Ψ B ( i , z ) , t ) for every z ∈ ( C \ R ) n and every t ∈ R n [ , Prop. 6.1].Using the kernel K n , the largest class of functions that will be considered isthe following. Definition . A function g : ( C \ R ) n → C is called a Cauchy-type function if there exists a positive Borel measure µ on R n satisfying the growth condition(2.3) (cid:90) R n n (cid:89) (cid:96) =1
11 + t (cid:96) d µ ( t ) < ∞ such that g ( z ) = 1 π n (cid:90) R n K n ( z , t )d µ ( t ) for every z ∈ ( C \ R ) n .Note that this definition is different from [ , Def. 3.1] in that it assumes fromthe beginning that a Cauchy-type function is defined on ( C \ R ) n and not only on C + n . Furthermore, it would be possible to define an even larger class of functionsusing the same kernel, but general distributions instead of measures, see [ , Ex.7.7] for an example. However, this extension will not be considered here. Moreover,Definition 2.1 allows, in principle, for two (or more) different measure to yield thesame function g , though we will show that this is not the case later in Section 4.An immediate consequence of the symmetry formula (2.2) is an analogous sym-metry formula for Cauchy-type functions. In particular, it holds, for any Cauchy-type function g , that(2.4) g ( z ) = (cid:88) B ⊆{ ,...,n } B (cid:54) = ∅ ( − | B | +1 g (Ψ B ( i , z )) for every z ∈ ( C \ R ) n and every t ∈ R n [ , Prop. 6.5].The growth of a Cauchy-type function along a coordinate parallel complex linecan be described using non-tangential limits. These are taken in so-called Stoltzdomains and are defined as follows. An upper Stoltz domain with centre ∈ R andangle θ ∈ (0 , π ] is the set { z ∈ C + | θ ≤ arg( z ) ≤ π − θ } and the symbol z ∨−→ ∞ thendenotes the limit | z | → ∞ in any upper Stoltz domain with centre . A lower Stoltzdomain and the symbol z ∧−→ ∞ are defined analogously. Furthermore, we note thatin the literature, slightly different notations are sometimes used to describe theselimits. Two examples of Stlotz domains are visualized in Figure 1 below.For any Cauchy-type function g it now holds, for any z ∈ ( C \ R ) n and any j ∈ { , . . . , n } , that lim z j ∨ −→ ∞ g ( z ) z j = lim z j ∧ −→ ∞ g ( z ) z j = 0 , see [ , Lem. 3.2]. MITJA NEDIC x i y θ x i yθ Figure 1.
An upper Stoltz domain with centre and angle θ (left) and a lower Stoltz domain with centre and angle θ (right). These functions are defined as follows, cf. [
12, 13, 18, 19 ]. Definition . A holomorphic function h : C + n → C is called a Herglotz-Nevanlinna function if it is holomorphic with non-negative imaginary part.In contrast to the definition of Cauchy-type function, the above definition isanalytic in nature, i.e. it describes the function class in terms of conditions on thefunction itself. In order to be able to relate it to the kernel K n , we introduce, givenambient numbers z ∈ C \ R and t ∈ R , the expressions N − ( z, t ) := 12 i (cid:18) t − z − t − i (cid:19) ,N ( z, t ) := 12 i (cid:18) t − i − t j + i (cid:19) ,N ( z, t ) := 12 i (cid:18) t + i − t − z (cid:19) . Note that N is independent of z ∈ C \ R and N ( z, t ) ∈ R while N − ( z, t ) = N ( z, t ) for all z ∈ C \ R and t ∈ R . Using these expression, one may give an integralrepresentation formula for Herglotz-Nevanlinna functions involving the kernel K n [ , Thm. 4.1]. Theorem . A function h : C + n → C is a Herglotz-Nevanlinna function ifand only if h can be written as (2.5) h ( z ) = a + n (cid:88) j =1 b j z j + 1 π n (cid:90) R n K n ( z , t )d µ ( t ) , HARACTERIZATION OF THE SYMMETRIC EXTENSION 5 where a ∈ R , b ∈ [0 , ∞ ) n , the kernel K n is as before and µ is a positive Borelmeasure on R n satisfying the growth condition (2.3) and the Nevanlinna condition (2.6) (cid:88) ρ ∈{− , , } n − ∈ ρ ∧ ∈ ρ (cid:90) R n n (cid:89) j =1 N ρ j ( z j , t j )d µ ( t ) = 0 for all z ∈ C + n . Furthermore, for a given function h , the triple of representingparameters ( a, b , µ ) is unique. The integral representation in formula (2.5) is well-defined for any z ∈ ( C \ R ) n ,which may be used to extend any Herglotz-Nevanlinna function h from C + n to ( C \ R ) n . This extension is called the symmetric extension of the function h and isdenoted as h sym . The symmetric extension of a Herglotz-Nevanlinna function h isdifferent from its possible analytic extension as soon as µ (cid:54) = 0 [ , Prop. 6.10] andsatisfies the following variable-dependence property [ , Prop. 6.9]. Proposition . Let n ≥ and let h sym be the symmetric extension of aHerglotz-Nevanlinna function h in n variables for which b = . Let z ∈ ( C \ R ) n besuch that z j ∈ C − for some index j ∈ { , , . . . , n } . Then, the value h sym ( z ) doesnot depend on the components of z that lie in C + . Furthermore, if h is a Herglotz-Nevanlinna function for which b = , then itssymmetric extension h sym will satisfy the symmetry formula(2.7) h sym ( z ) = (cid:88) B ⊆{ ,...,n } B (cid:54) = ∅ ( − | B | +1 h sym (Ψ B ( i , z )) , where z ∈ ( C \ R ) n [ , Prop. 6.7]. When n = 1 , it is not necessary to assume that b = 0 for formula (2.7) to hold. However, when n > , this is required.The representing vector b describes the growth of the function h along coordi-nate parallel complex lines in C + n . More precisely, we recall from [ , Cor. 4.6(iv)]that, for any j ∈ { , . . . , n } , we have(2.8) b j = lim z j ∨ −→ ∞ h ( z ) z j . In particular, the above limit is independent of the entries of the vector z at thenon- j -th positions. This result carries over to the symmetric extension, for whichit holds, for any j ∈ { , . . . , n } , that b j = lim z j ∨ −→ ∞ h sym ( z ) z j = lim z j ∧ −→ ∞ h sym ( z ) z j . Every Herglotz-Nevanlinna function that is represented by a data-triple of theform (0 , , µ ) in the sense of Theorem 2.3 is also a Cauchy-type function. The con-verse, i.e. that every Cauchy-type function equals a Herglotz-Nevanlinna functionrepresented by a data-triple of the form (0 , , µ ) , is true only when n = 1 . This isdue to the fact that when n = 1 , the Nevanlinna condition (2.6) becomes emptilyfulfilled by every positive Borel measures µ satisfying the growth condition (2.3). MITJA NEDIC
3. Symmetry and variable non-dependence
We begin by recalling that the symmetric extension of a Herglotz-Nevanlinnafunction h in one variable is uniquely determined by its values in C + . Indeed, when n = 1 , the symmetry formula (2.7) takes the form h sym ( z ) = h sym ( z ) , providing away to recover the values of the function in C − using only the values of the functionin C + .For functions of several variables, the appropriate analogue involves the follow-ing definition. Definition . A function f : ( C \ R ) n → C is said to satisfy the variablenon-dependence property if for every vector z ∈ ( C \ R ) n such that z j ∈ C − forsome index j ∈ { , , . . . , n } the value f ( z ) does not depend on the components of z that lie in C + .By Proposition 2.4, the symmetric extension of a Herglotz-Nevanlinna functionsatisfies the variable non-dependence property 3.1 if b = . In particular, thesymmetric extension of any Herglotz-Nevanlinna function that is also a Cauchy-typefunction will always satisfy the variable non-dependence property 3.1. However, ageneral Cauchy-type function need not satisfy it, as shown by the function f inExample 3.5 later on.We may now describe the precise circumstances under which we can recoverthe values of a function defined on ( C \ R ) n purely in terms of its values in C + n . Proposition . Let f : ( C \ R ) n → C be a holomorphic function satisfyingthe symmetry formula (2.7) and the variable non-dependence property 3.1. Then,the values of the function f on ( C \ R ) n are uniquely determined by its values in C + n . Proof.
Using the symmetry formula (2.7), let us investigate the values ofthe function f in a connected component of ( C \ R ) n where at least one of thecoordinates has a negative sign of the imaginary part, i.e. we are investigating aconnected component X ⊆ ( C \ R ) n where exist at lest one index j ∈ { , . . . , n } suchthat the j -th coordinate lies in C − . For any such chosen connected component X ,let B (cid:48) ⊆ { , . . . , n } be the set of those indices for which the corresponding variableslie in C − . In particular, ≤ | B (cid:48) | ≤ n . For z ∈ X , it holds, by the symmetryformula (2.7), that f ( z ) = (cid:88) B ⊆{ ,...,n } B (cid:54) = ∅ ( − | B | +1 f (Ψ B ( i , z ))= (cid:88) B ⊆{ ,...,n } B (cid:54) = ∅∧ B ⊆ B (cid:48) ( − | B | +1 f (Ψ B ( i , z )) + (cid:88) B ⊆{ ,...,n } B (cid:54)⊆ B (cid:48) ( − | B | +1 f (Ψ B ( i , z )) . Due to the definition of the set B (cid:48) , it holds that Ψ B ( i , z ) ∈ C + n for any z ∈ X andany indexing set B ⊆ B (cid:48) . Furthermore, by the variable non-dependence property3.1, it holds that Ψ B ( i , z ) = Ψ B \ B (cid:48) ( i , z ) HARACTERIZATION OF THE SYMMETRIC EXTENSION 7 for any z ∈ X and any indexing set B where B (cid:54)⊆ B (cid:48) . Hence, f ( z ) = (cid:88) B ⊆{ ,...,n } B (cid:54) = ∅∧ B ⊆ B (cid:48) ( − | B | +1 f (Ψ B ( i , z )) + (cid:88) B ⊆{ ,...,n } B (cid:54)⊆ B (cid:48) ( − | B | +1 f (Ψ B \ B (cid:48) ( i , z )) . We now claim that the second sum is always equal to zero. Indeed, if | B (cid:48) | = n ,there is nothing left to prove. Otherwise, we may assume that | B (cid:48) | < n , where weclaim that there is a way to "pair up" the indexing sets in the second sum in sucha way that the two sets in each pair only differ by one element in B (cid:48) . We constructthis pairing in the following way. Let j be the smallest index in B (cid:48) . Then, exactlyhalf of the sets B ⊆ { , . . . , n } that are not subsets of B (cid:48) contain the index j andexactly half of them do not contain the index j . This follows from the generalobservation that exactly half of the subsets of a given set contain a specific elementof the set. An indexing set B is then paired with the indexing set B ∪ { j } . Inthis case, ( B ∪ { j } ) \ B (cid:48) = B \ B (cid:48) and ( − | B | +1 f (Ψ B \ B (cid:48) ( i , z )) + ( − | B ∪{ j }| +1 f (Ψ ( B ∪{ j } ) \ B (cid:48) ( i , z ))= ( − | B | +1 f (Ψ B \ B (cid:48) ( i , z )) − ( − | B | +1 f (Ψ B \ B (cid:48) ( i , z )) = 0 , yielding the desired result. (cid:3) Using Proposition 3.2, we may now given an analytic characterization of thesymmetric extension of a Herglotz-Nevanlinna function.
Theorem . Let f : ( C \ R ) n → C be a holomorphic function such that lim z j ∨ −→ ∞ f ( z ) z j = lim z j ∧ −→ ∞ f ( z ) z j = 0 for all indices j ∈ { , . . . , n } . Then f = h sym for some Herglotz-Nevanlinna func-tion h if and only if (i) it holds that Im[ f ( z )] ≥ for all z ∈ C + n , (ii) the function f satisfies the symmetry formula (2.7) , (iii) the function f satisfies the variable non-dependence property 3.1. Proof. If f = h sym for some Herglotz-Nevanlinna function h , then this func-tion must have b = due to the assumption on the growth of f . Then, properties(i) – (iii) are satisfied by the previously known results discussed in Section 2.2.Conversely, if we are given the function f satisfies the properties (i) – (iii), weconstruct a Herglotz-Nevanlinna function out the function f by setting h := f | C + n . This function h may then be symmetrically extended to ( C \ R ) n . However, f and h sym are now two holomorphic functions on ( C \ R ) n satisfying the symmetryformula (2.7) and the variable non-dependence property 3.1 which, furthermore,agree on C + n . Therefore, by Proposition 3.2, they agree everywhere on ( C \ R ) n ,as desired. (cid:3) The assumption on the growth of the function f may be slightly weakened, but,to compensate, conditions (ii) and (iii) need to be slightly modified. MITJA NEDIC
Corollary . Let f : ( C \ R ) n → C be a holomorphic function such that lim z j ∨ −→ ∞ f ( z ) z j = lim z j ∧ −→ ∞ f ( z ) z j = d j ≥ for all indices j ∈ { , . . . , n } . In particular, for a fixed j ∈ { , . . . , n } , the abovelimits are assumed to be independent of the values of the vector z ∈ ( C \ R ) n at thenon- j -th positions. Then f = h sym for some Herglotz-Nevanlinna function h if andonly if (i) it holds that Im[ f ( z )] ≥ for all z ∈ C + n , (ii’) the function z (cid:55)→ f ( z ) − (cid:80) nj =1 d j z j satisfies the symmetry formula (2.7) , (iii’) the function z (cid:55)→ f ( z ) − (cid:80) nj =1 d j z j satisfies the variable non-dependenceproperty 3.1. The three conditions on the function f in Theorem 3.3 are independent ofeach-other. To verify this, consider the following functions on ( C \ R ) n . Example . Table 1 presents eight explicit functions defined on ( C \ R ) n and Table 2 summarizes which conditions of Theorem 3.3 are fulfilled by whichfunction. Note also that all eight functions satisfy the assumption on the growthof the function from Theorem 3.3. The functions are constructed as follows.The functions f is defined to equal: a negative imaginary constant on C + × C + ,breaking condition (i); a function depending only on the second variable on C − × C + ,breaking condition (iii); and identically zero in the remaining connected componentsof ( C \ R ) n , ensuring that condition (ii) is not satisfied. The function f is obtainedform f by changing the definition on C + × C + to a positive imaginary constant,thereby satisfying condition (i), but still neither (ii) nor (iii).The function f is the Cauchy-type function given by a measure µ on R defined on Borel subsets U ⊆ R as µ ( U ) := π (cid:90) R χ U ( t, t )d t, where χ denotes the characteristic function of a set. This measure obviously satisfiesthe growth condition (2.3) and it does not satisfy the Nevanlinna condition (2.6)as it supported on the diagonal in R - an impossibility for Nevanlinna measuresas shown in [ , Ex. 3.14]. This function does not satisfy condition (i) as, forexample, f (4 i , i ) = − i . As a Cauchy-type function, it is guaranteed to satisfycondition (ii). It also clearly does not satisfy condition (iii) as the values in e.g. C + × C − depend explicitly on both variables. Note now that while the function f takes values with negative imaginary part in C + × C + , its imaginary part isbounded form below. Indeed, the functions z (cid:55)→ − i + z and z (cid:55)→ − i + z areHerglotz-Nevanlinna functions of one variable, implying that Im[ f ( z , z )] ≥ − for all ( z , z ) ∈ C + × C + . Hence, the function f is obtained by adding to thefunction f the symmetric extension of the Herglotz-Nevanlinna function ( z , z ) (cid:55)→ i (represented by the measure λ R ). This new function now satisfies condition(i) in addition to (ii), while clearly still not satisfying condition (iii). Note that thefunction f | C + × C + ( z , z ) = 9 i − i + z − i + z HARACTERIZATION OF THE SYMMETRIC EXTENSION 9 as a Herglotz-Nevanlinna function is not represented by the measure µ + 5 λ R inthe sense of Theorem 2.3, but rather by the measure λ R + ( τ (cid:55)→ (1 + τ ) − ) λ R ⊗ λ R + λ R ⊗ ( τ (cid:55)→ (1 + τ ) − ) λ R . The function f is defined as zero on all the connected components of ( C \ R ) other than C + × C + to ensure that it satisfies condition (iii), while setting thefunction equal to a negative imaginary constant in C + × C + ensures that it satisfiesneither condition (i) nor (ii). Changing this definition to a positive imaginaryconstant in C + × C + gives the function f which satisfies condition (i) and (iii),but not (ii).The function f is simply taken as the symmetric extension of a Herglotz-Nevanlinna function, thereby satisfying all three properties automatically. Finally,the function f is chosen as f := − f , satisfying conditions (ii) and (iii), but not(i). ♦ C + × C + C − × C + C + × C − C − × C − f − i z f i z f − i − i + z − i + z − i + z − z − i + z − i − i + z + z − z − i f − i f i − i + z − i + z − i + z − z − i + z − i − i + z + z − z − i f i f − i i i i f i − i − i − i Table 1.
Examples of eight functions defined on ( C \ R ) .(i) (ii) (iii) f × × × f (cid:88) × × f × (cid:88) × f × × (cid:88) f (cid:88) (cid:88) × f (cid:88) × (cid:88) f × (cid:88) (cid:88) f (cid:88) (cid:88) (cid:88) Table 2.
The relation of the eight functions from Table 1 to thethree conditions from Theorem 3.3.
4. The Stieltjes inversion formula for Cauchy-type functions
For Herglotz-Nevanlinna functions, the Stieltjes inversion formula describes howto reconstruct the representing measure µ of a Herglotz-Nevanlinna function h from the values of the imaginary part of the function in C + n . More precisely, it holdsthat (cid:90) R n ϕ ( t )d µ ( t ) = lim y → + (cid:90) R n ϕ ( x )Im[ h ( x + i y )]d x for all C -functions ϕ : R n → R for which there exists a constant D ≥ such that | ϕ ( x ) | ≤ D (cid:81) nj =1 (1 + x j ) − for all x ∈ R n , see e.g. [ ] or [ , Lem. 4.1] for thecase n = 1 and [ , Cor. 4.6(viii)] for the general case.As noted in Section 2.2, Cauchy-type functions are a subclass of Herglotz-Nevanlinna functions when n = 1 and, hence, one only need the values of (theimaginary part of) a Cauchy-type function in C + to reconstruct its measure. How-ever, in Example 3.5, we have seen two different positive Borel measures on R for which the corresponding Cauchy-type functions agree on C +2 , but not on theremaining connected components of ( C \ R ) .The crucial role in the proof of the Stieltjes inversion formula is held by thePoisson kernel of C + n , which, we recall, is defined for z ∈ C + n and t ∈ R n as P n ( z , t ) := n (cid:89) j =1 Im[ z j ] | t j − z j | . Note that P n ( z , t ) > for every z ∈ C + n and t ∈ R n . The imaginary part ofthe kernel K n is equal to the Poisson kernel P n plus a remainder term which canbe expressed in terms of the N j -factors [ , Prop. 3.3] and the integral of theremainder with respect to any Nevanlinna measure is zero.The following lemma now shows how one can recover the value of the Poisonkernel P n at some point z ∈ C + n (and t ∈ R n ) using the values of kernel K n formall of the connected components of the poly cut-plane ( C \ R ) n . Lemma . Let n ∈ N , z ∈ C + n and t ∈ R n . Then, it holds that i P n ( z , t ) = (cid:88) B ⊆{ ,...,n } ( − | B | K n (Ψ B ( z , z ) , t ) , where Ψ B is the selective conjugation map from Section 2.1. Proof.
The proof is done by induction on the dimension n . If n = 1 , then (cid:88) B ⊆{ } ( − | B | K (Ψ B ( z, z ) , t ) = K (Ψ ∅ ( z, z ) , t ) + ( − K (Ψ { } ( z, z ) , t )= K ( z, t ) − K ( z, t ) = 2 i Im[ K ( z, t )] = 2 i P ( z, t ) , as desired.Assume now that the statement of the lemma holds for all n = 1 , , . . . , N − for some N ∈ N . For n = N , take z ∈ C + N and t ∈ R N and let z (cid:48) and t (cid:48) denotethe same vectors with the last component removed, i.e. z (cid:48) := ( z , . . . , z N − ) and t (cid:48) := ( t , . . . , t N − ) . Furthermore, denote A ( z, t ) := 12 i (cid:18) t − z − t + i (cid:19) . Then, we then calculate that (cid:88) B ⊆{ ,...,N } ( − | B | K N (Ψ B ( z , z ) , t ) HARACTERIZATION OF THE SYMMETRIC EXTENSION 11 = (cid:88) B ⊆{ ,...,N } N (cid:54)∈ B ( − | B | K N (Ψ B ( z , z ) , t ) + (cid:88) B ⊆{ ,...,N } N ∈ B ( − | B | K N (Ψ B ( z , z ) , t )= (cid:88) B (cid:48) ⊆{ ,...,N − } ( − | B (cid:48) | (cid:20) i (cid:18) N − (cid:89) j =1 A ( ψ jB (cid:48) ( z j , z j ) , t j ) · A ( z N , t N ) − N (cid:89) j =1 A ( i , t j ) (cid:19)(cid:21) + (cid:88) B (cid:48) ⊆{ ,...,N − } ( − | B (cid:48) | +1 (cid:20) i (cid:18) N − (cid:89) j =1 A ( ψ jB (cid:48) ( z j , z j ) , t j ) · A ( z N , t N ) − N (cid:89) j =1 A ( i , t j ) (cid:19)(cid:21) = 2 i A ( z N , t N ) (cid:88) B (cid:48) ⊆{ ,...,N − } ( − | B (cid:48) | K N − (Ψ B (cid:48) ( z (cid:48) , z (cid:48) ) , t (cid:48) )+ i N − (cid:89) j =1 A ( i , t j ) · ( A ( z N , t N ) − A ( i , t N )) · =0 (cid:122) (cid:125)(cid:124) (cid:123)(cid:88) B (cid:48) ⊆{ ,...,N − } ( − | B (cid:48) | − i A ( z N , t N ) (cid:88) B (cid:48) ⊆{ ,...,N − } ( − | B (cid:48) | K N − (Ψ B (cid:48) ( z (cid:48) , z (cid:48) ) , t (cid:48) ) − i N − (cid:89) j =1 A ( i , t j ) · ( A ( z N , t N ) − A ( i , t N )) · =0 (cid:122) (cid:125)(cid:124) (cid:123)(cid:88) B (cid:48) ⊆{ ,...,N − } ( − | B (cid:48) | = 2 i A ( z N , t N ) P N − ( z (cid:48) , t (cid:48) ) − i A ( z N , t N ) P N − ( z (cid:48) , t (cid:48) ) = 2 i P N ( z , t ) , finishing the proof. (cid:3) The Stieltjes inversion for Cauchy-type functions is, thus, the following.
Theorem . Let g be a Cauchy-type function given by a measure µ . Then,it holds that (4.1) (cid:90) R n ϕ ( t )d µ ( t )= lim y → + i (cid:90) R n ϕ ( x ) (cid:20) (cid:88) B ⊆{ ,...,n } ( − | B | g (Ψ B ( x + i y , x + i y )) (cid:21) d x for all C -functions ϕ : R n → R for which there exists a constant D ≥ such that | ϕ ( x ) | ≤ D (cid:81) nj =1 (1 + x j ) − for all x ∈ R n . Proof.
By the definition of Cauchy-type functions and Lemma 4.1, it holdsthat (cid:88) B ⊆{ ,...,n } ( − | B | g (Ψ B ( x + i y , x + i y ))= 1 π n (cid:90) R n (cid:20) (cid:88) B ⊆{ ,...,n } ( − | B | K n (Ψ B ( x + i y , x + i y ) , t ) (cid:21) d µ ( t )= 2 i π n (cid:90) R n P n ( x + i y , t )d µ ( t ) . Hence, lim y → + i (cid:90) R n ϕ ( x ) (cid:20) (cid:88) B ⊆{ ,...,n } ( − | B | g (Ψ B ( x + i y , x + i y )) (cid:21) d x = lim y → + π n (cid:90) R n ϕ ( x ) (cid:18) (cid:90) R n P n ( x + i y , t )d µ ( t ) (cid:19) d x = lim y → + π n (cid:90) R n (cid:18) (cid:90) R n ϕ ( x ) P n ( x + i y , t )d x (cid:19) d µ ( t ) , where the assumptions on the function ϕ and condition (2.3) for µ justify the use ofFubini’s theorem to change the order of integration. The same assumptions permitfor Lebesgue’s dominated convergence to be used, allowing us to take the limit as y → + before integrating with respect to the measure µ . Noting that, by e.g. [ ,pg. 111], lim y → + (cid:90) R n ϕ ( x ) P n ( x + i y , t )d x = π n ϕ ( t ) finishes the proof. (cid:3) As an immediate corollary of the previous theorem, we may now establish thatthe correspondence between a Cauchy-type function and its defining measure µ is,indeed, a bijection. Corollary . Let µ , µ be two positive Borel measures on R n satisfyingthe growth condition (2.3) . Then, (cid:90) R n K n ( z , t )d µ ( t ) = (cid:90) R n K n ( z , t )d µ ( t ) for all z ∈ ( C \ R ) n if and only if µ ≡ µ . Acknowledgments
The author would like to thank Dale Frymark for many enthusiastic discussionson the subject.
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Mitja Nedic, Department of Mathematics and Statistics, University of Helsinki,PO Box 68, FI-00014 Helsinki, Finland, orc-id: 0000-0001-7867-5874
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