Remarks on restricted Nevanlinna transforms

The Nevanlinna transform K(z), of a measure and a real constant, plays an important role in the complex analysis and more recently in the free probability theory (boolean convolution). It is shown that its restriction k(it) (the restricted Nevanlinna transform) to the imaginary axis can be expressed as the Laplace transform of the Fourier transform (characteristic function) of the corresponding measure. Finally, a relation between the Voiculescu and the boolean convolution is indicated.

for some finite measures m and ρ and constants a. To get the measure m from G usually one uses the following inversion formula cf. Akhiezer (1965), p. 125 or Lang (1975), p. 380, Bondesson (1992). Thus G m uniquely determines m. However, for the inversion one needs to know the Cauchy transform in strips {x + iy : x ∈ R, 0 < y < ǫ} for some ǫ > 0. In Jurek (2006) it was shown, among others, that the values of G m (it), t = 0 are sufficient to identify m; also cf. Corollary 2 below. Of course, as holomorphic functions both G m and K a ,ρ are determined by their values on sets having condensation point. In this note we will consider G m and K a, ρ only on the imaginary axis without the origin. In particular, we show that the measure ρ can be retrieved from K a, ρ (it), t = 0, using the classical (standard) Fourier and Laplace transforms; (Theorem 1). Using yet another functional (self-energy functional) we introduce the boolean convolution. Finally, we will indicate rather unexpected relations between the Voiculescu ⊕ and the boolean ⊎ operations on probability measures; (Proposition 1). This paper is an exemplification of a general idea that many transforms in the complex analysis and, in particular, in the area of the free probability, are indeed some functionals of the standard Laplace and Fourier transforms when suitably restricted to the imaginary line.
1. Notations, the results and an example. For a real constant a and a finite Borel measure on the real line ρ, we define the restricted Nevanlinna transform by and similarly the restricted Cauchy transform as follows Let us recall also that the Fourier transform (the characteristic function)μ of a measure µ is given bŷ and the Laplace transform of a function h : (0, ∞) → C or of a measure m is given where λ is a such that those integral exist; cf. Gradshteyn and Ryzhik (1994), Chapter 17, for examples of those transforms and their inverses.
Here are the main results, in particular, the formula how to obtain the measure ρ knowing only restricted Nevanlinna transforms. ( Below, ℜz, ℑz, z denote the real part, the imaginary part and the conjugate of a complex z ∈ C, respectively.) Theorem 1. (An inversion formula.) For the restricted Nevalinna functional k a, ρ we have that: a = ℜk a, ρ (i), ρ(R) = −ℑk a, ρ (i); and the identity holds for w > 0. In particular, the constant a and the measure ρ are uniquely determined by the functional k a, ρ .
Since part of the right-hand side formula can be viewed as Laplace transform of some exponential functions we get Corollary 1. For the restricted Nevanlinna functional k a, ρ and w > 1 we have In particular, if a = 0 and ν is a probability measure then for k 0,ν we get Corollary 2. For a finite measure ρ and its restricted Cauchy transform g ρ we have L[ρ; w] = i g ρ (iw), w = 0.
In the following example we show explicitly that shifted reciprocals of restricted Cauchy transforms of discrete measures are, indeed, restricted Nevalinna transforms; ( see the formula (5) below).
Example. For a set b = {b 1 , b 2 , ..., b m } of distinct real numbers let us define a discrete probability measure µ b := 1 m m j=1 δ b j and the canonical polynomial where Note that the procedure described in the Example can be iterated. Namely, in the second step we may start with the probability measure concentrated on the roots ξ j , j = 1, 2, ..., m − 1, and so on.
Recall that the self-energy functional E µ , of the probability measure µ, is defined as follows And similarly as above to e µ (it) := E µ (it), t = 0, we will refer to as a restricted self-energy functional.
Here is how to express a and ρ in terms of µ using only the restricted functionals.
Corollary 3. For a probability measure µ let If e µ (it) = k a, ρ (it), for t = 0, then the constants a and ρ(R) are given by formulae and the Fourier transformρ satisfies the equation Since for any probability measures µ, ν there exists a unique probability measure γ such that we call it the boolean convolution and denote by γ = µ ⊎ ν; for more details cf. Speicher -Woroudi (1997). Remark 1. Boolean convolution has the property that all probability measures are ⊎-infinitely divisible. That feature has also the max -convolution because for each distribution function F, F 1/n (the n-th root) is also distribution function and taking independent identically distributed (as F 1/n ) r.v. X n,1 , X n,2 , ..., X n,n then max{X n,1 , ..., X n,n } has distribution function F.
For a probability measure µ, let where D is so called Stolz angle in which the inverse F −1 µ exists; cf. Bercovici-Voiculescu (1993) and references therein . Since for any probability measures µ, ν there exists a unique probability measure γ such that we call it the Voiculescu convolution and denote by γ = µ ⊕ ν. A relation between ⊕-infinite divisibility and some random integrals with respect to classical Lévy processes is given in Jurek (2007), Corollary 6.
(a) For probability measures µ and ν there exists unique measure µ ⊎ ν such that for t ∈ R; cf. Theorem 2 and Remark 1.1.1 in Jurek (2006) for other forms of the above formula and some comments.
(b) For measures µ 1 and µ 2 there exist unique measures ν 1 , ν 2 and µ 1 ⊕µ 2 such that for their restricted Cauchy transforms we have for all t = 0. (Using Corollary 2 we may express the above identity in terms of classical Laplace and Fourier transforms.)

Auxiliary results and proofs. Note that
and therefore we may consider those function only on the positive half-line.
Further, since (1) and (2) we infer that On the other hand, in Jurek (2006) on p. 189, it was noticed that This with (11) and (12) give .
and substituting once again s w = r > 0, one arrives at which gives the formula from Theorem 1. Finally, since the left-hand side is the Laplace transform of the Fourier transformρ of the measure ρ, therefore it is uniquely determined by k a, ρ . This completes the proof.
Proof of Corollary 1. Simple note that which with Theorem 1 give the corollary.
Proof of Corollary 2. Using the definitions (3) and (4) we have which completes the proof.
Here is an auxiliary lemma that might be of an independent interest. It is a key argument for the example.
Proof. (a) Since P ′ (z) = m j=1 m k =j,k=1 (z − b k ) we get the first part of (a). Differentiating both sides of the identity P ′ (z) = P (z) m j=1 1 z−b j we get the second part of (a). (b) Assume that P and P ′ have a common root. Without loss of generality, let say that which contradicts the assumption that all b j are distinct.
Suppose that ξ 1 and and its complex conjugateξ 1 are two complex roots of P ′ (z) = 0. Then from (a) we have Since P (ξ 1 ) = 0 and P (ξ 1 ) = 0 therefore and hence ℑξ 1 = 0 = ℑξ 2 = ℑξ 3 = ... = ℑξ m−1 , that is, all roots of P ′ (z) = 0 are real. Let us note that is a polynomial of degree m-1, (for another polynomial of degree m − 2. Consequently, W m (z) is a rational function (a ratio of two polynomials of degree m-1). Since ξ 1 , ..., ξ m−1 are zeros of P ′ (z) = 0 , i.e., simple poles of W m (z), then invoking the theorem on the decomposition of rational function into a sum of simple fractions Puttingb := (b 1 + ... + b m )/m and multiplying both sides by z − ξ k , from the above we have and then letting z → ξ k we get explicitly that (c) Since P (x) is a polynomial of m-th degree for x ∈ R and P (b k ) = P (b k+1 ) = 0 (for b j ∈ R) then, by the Mean Value Theorem, there exists exactly one ξ j (in that interval) such that P ′ (ξ k ) = 0. If P (ξ k ) > 0 then P must be concave on that interval and therefore P ′′ (ξ k ) < 0. Consequently, α j > 0. In the opposite case we have convex function that also leads to positivity of the parameter α k . This completes the proof of Lemma 1.
Proof of the Example. From Lemma 1 we have that the measure ρ b is finite and positive. Furthermore, for a b given by (6) and using (13) (in Lemma 1) we get Substituting in the above it for z, one gets equality (5) in the Example. Proof of Corollary 3. From (2) and we get immediately the expression (8) for −g µ (i). From (11), e µ (i) = a − iρ(R) and hence we infer equalities in (9).
In view of the assumption, k a, ρ in Corollary 1 may be replaced by e µ then, using (7) and (9), one gets Consequently from Corollary 1 we get the required identity.