Pick Functions Research Articles

Analytic Continuation of Concrete Realizations and the McCarthy Champagne Conjecture

Abstract In this paper, we give formulas that allow one to move between transfer function type realizations of multi-variate Schur, Herglotz, and Pick functions, without adding additional singularities except perhaps poles coming from the conformal transformation itself. In the two-variable commutative case, we use a canonical de Branges–Rovnyak model theory to obtain concrete realizations that analytically continue through the boundary for inner functions that are rational in one of the variables (so-called quasi-rational functions). We then establish a positive solution to McCarthy’s Champagne conjecture for local to global matrix monotonicity in the settings of both two-variable quasi-rational functions and $d$-variable perspective functions.

Complete monotonicity-preserving numerical methods for time fractional ODEs

The time fractional ODEs are equivalent to convolutional Volterra integral equations with completely monotone kernels. We therefore introduce the concept of complete monotonicity-preserving ($\mathcal{CM}$-preserving) numerical methods for fractional ODEs, in which the discrete convolutional kernels inherit the $\mathcal{CM}$ property as the continuous equations. We prove that $\mathcal{CM}$-preserving schemes are at least $A(\pi/2)$ stable and can preserve the monotonicity of solutions to scalar nonlinear autonomous fractional ODEs, both of which are novel. Significantly, by improving a result of Li and Liu (Quart. Appl. Math., 76(1):189-198, 2018), we show that the $\mathcal{L}$1 scheme is $\mathcal{CM}$-preserving, so that the $\mathcal{L}$1 scheme is at least $A(\pi/2)$ stable, which is an improvement on stability analysis for $\mathcal{L}$1 scheme given in Jin, Lazarov and Zhou (IMA J. Numer. Analy. 36:197-221, 2016). The good signs of the coefficients for such class of schemes ensure the discrete fractional comparison principles, and allow us to establish the convergence in a unified framework when applied to time fractional sub-diffusion equations and fractional ODEs. The main tools in the analysis are a characterization of convolution inverses for completely monotone sequences and a characterization of completely monotone sequences using Pick functions due to Liu and Pego (Trans. Amer. Math. Soc. 368(12):8499-8518, 2016). The results for fractional ODEs are extended to $\mathcal{CM}$-preserving numerical methods for Volterra integral equations with general completely monotone kernels. Numerical examples are presented to illustrate the main theoretical results.

Universally starlike and Pick functions

Denote by \({{\cal P}_{\log}}\) the set of all non-constant Pick functions f whose logarithmic derivatives f′/f also belong to the Pick class. Let \({\cal U}({\rm{\Lambda}})\) be the family of functions z · f(z), where \(f \in {{\cal P}_{\log}}\) and f is holomorphic on Λ ≔ ℂ [1, + √). Important examples of functions in \({\cal U}({\rm{\Lambda}})\) are the classical polylogarithms \(L{i_\alpha}(z): = \sum\nolimits_{k = 1}^\infty {{z^k}} /{k^\alpha}\) for α ≥ 0; see [5](2015).In this note we prove that every \(\varphi \in {\cal U}({\rm{\Lambda}})\) is universally starlike, i.e., φ maps every circular domain in Λ containing the origin one-to-one onto a starlike domain. Furthermore, we show that every non-constant function \(f \in {{\cal P}_{\log}}\) belongs to the Hardy space Hp on the upper half-plane for some constant p = p(f) > 1, unless f is proportional to some function (a − z)−θ with a ∊ ℝ and 0 <θ ≤ 1. Finally we derive a necessary and sufficient condition on a real-valued function υ for which there exists \(f \in {{\cal P}_{\log}}\) such that υ (x) = limε↓0 lim f(x + iε) for almost all x ∊ ℝ.

Equality of norms for a class of Bloch and symmetrically weighted Lipschitz spaces of vector valued functions and derivation inequalities for Pick functions

If X and Y are Banach spaces and f:BX→Y is Fréchet differentiable on the open unit ball BX of X, then for every operator monotone function φ:(−1,1)→R, which satisfies φ′′⩾0 on [a,b),(1)supa,b∈BX,a≠b⁡‖f(a)−f(b)‖φ′(‖a‖)‖a−b‖φ′(‖b‖)=supa∈BX⁡‖Df(a)‖φ′(‖a‖). This generalizes Holland–Walsh–Pavlović criterium for the membership in Bloch type spaces for functions defined in the unit ball of a Banach space and taking values in another Banach space. We also established relations of the induced Bloch and Lipschitz spaces with other spaces of vector valued functions.

Cauchy transforms arising from homomorphic conditional expectations parametrize noncommutative Pick functions

Abstract Nevanlinna showed that Cauchy transforms of probability measures parametrize all functions from the upper half plane into itself satisfying a certain asymptotic condition at infinity. We show that the correspondence fails in general for the unbounded case for somewhat trivial reasons; however, we show that in a setting of “homomorphic” operator valued free probability that Cauchy transforms of homomorphic conditional expectations parametrize noncommutative Pick functions.

The first hitting time of the integers by symmetric Lévy processes

For one-dimensional Brownian motion, the exit time from an interval has finite exponential moments and its probability density is expanded in exponential terms. In this note we establish its counterpart for certain symmetric Lévy processes. Applying the theory of Pick functions, we study properties of the Laplace transform of the first hitting time of the integer lattice as a meromorphic function in detail. Its density is expanded in exponential terms and the poles and the zeros of a Pick function play a crucial role.Intermediate results concerning finite exponential moments are also obtained for a class of nonsymmetric Lévy processes.

An Inductive Julia-Carathéodory Theorem for Pick Functions in Two Variables

AbstractClassically, Nevanlinna showed that functions from the complex upper half plane into itself which satisfy nice asymptotic conditions are parametrized by finite measures on the real line. Furthermore, the higher order asymptotic behaviour at infinity of a map from the complex upper half plane into itself is governed by the existence of moments of its representing measure, which was the key to his solution of the Hamburger moment problem. Agler and McCarthy showed that an analogue of the above correspondence holds between a Pick function f of two variables, an analytic function which maps the product of two upper half planes into the upper half plane, and moment-like quantities arising from an operator theoretic representation for f. We apply their ‘moment’ theory to show that there is a fine hierarchy of levels of regularity at infinity for Pick functions in two variables, given by the Löwner classes and intermediate Löwner classes of order N, which can be exhibited in terms of certain formulae akin to the Julia quotient.

Orthogonal Polynomials Related to $$_{g}$$ g -Fractions with Missing Terms

The purpose of the present paper is to investigate some structural and qualitative aspects of two different perturbations of the parameters of g-fractions. In this context, the concept of gap g-fractions is introduced. While tail sequences of a continued fraction play a significant role in the first perturbation, Schur fractions are used in the second perturbation of the g-parameters that is considered. Illustrations are provided using Gaussian hypergeometric functions. Using a particular gap g-fraction, some members of the class of Pick functions are also identified.

A wedge‐of‐the‐edge theorem: analytic continuation of multivariable Pick functions in and around the boundary

In 1956, quantum physicist N. Bogoliubov discovered the edge-of-the-wedge theorem, a theorem used to analytically continue a function through the boundary of a domain under certain conditions. We discuss an analogous phenomenon, a wedge-of-the-edge theorem, for the boundary values of Pick functions, functions from the poly upper half plane into the half plane. We show that Pick functions which have a continuous real-valued extension to a union of two hypercubes with a certain orientation in R d have good analytic continuation properties. Furthermore, we establish bounds on the behavior of this analytic continuation, which makes normal families arguments accessible on the boundary for Pick functions in several variables. Moreover, we obtain a Hartog's phenomenon type result for locally inner functions.

Constructing measures with identical moments

When the moment problem is indeterminate, the Nevanlinna parametrization establishes a bijection between the class of all measures having a prescribed set of moments and the class of Pick functions. The fact that all measures constructed through the Nevanlinna parametrization have identical moments follows from the theory of orthogonal polynomials and continued fractions. In this paper we explore the opposite direction: we construct a set of measures and we show that they all have identical moments, and then we establish a Nevanlinna-type parametrization for this set of measures. Our construction does not require the theory of orthogonal polynomials and it exposes the analytic structure behind the Nevanlinna parametrization.

Free Pick functions: Representations, asymptotic behavior and matrix monotonicity in several noncommuting variables

R. Nevanlinna showed that elements of the Pick class, the set of complex analytic functions taking the upper half plane into itself, have certain integral representations which reflect their asymptotic behavior at infinity. Later, Löwner connected the Pick class to matrix monotone functions. We generalize the Nevanlinna representation theorems and Löwner's theorem on matrix monotone functions to the free Pick class, the collection of functions that map tuples of matrices with positive imaginary part into the matrices with positive imaginary part which obey the free functional calculus.

Coagulation\u2013Fragmentation Model for Animal Group-Size Statistics

We study coagulation–fragmentation equations inspired by a simple model proposed in fisheries science to explain data for the size distribution of schools of pelagic fish. Although the equations lack detailed balance and admit no H-theorem, we are able to develop a rather complete description of equilibrium profiles and large-time behavior, based on recent developments in complex function theory for Bernstein and Pick functions. In the large-population continuum limit, a scaling-invariant regime is reached in which all equilibria are determined by a single scaling profile. This universal profile exhibits power-law behavior crossing over from exponent -frac{2}{3} for small size to -frac{3}{2} for large size, with an exponential cutoff.

On generating functions of Hausdorff moment sequences

The class of generating functions for completely monotone sequences (moments of finite positive measures on [ 0 , 1 ] [0,1] ) has an elegant characterization as the class of Pick functions analytic and positive on ( − ∞ , 1 ) (-\infty ,1) . We establish this and another such characterization and develop a variety of consequences. In particular, we characterize generating functions for moments of convex and concave probability distribution functions on [ 0 , 1 ] [0,1] . Also we provide a simple analytic proof that for any real p p and r r with p &gt; 0 p&gt;0 , the Fuss-Catalan or Raney numbers r p n + r ( p n + r n ) \frac {r}{pn+r}\binom {pn+r}{n} , n = 0 , 1 , … n=0,1,\ldots , are the moments of a probability distribution on some interval [ 0 , τ ] [0,\tau ] if and only if p ≥ 1 p\ge 1 and p ≥ r ≥ 0 p\ge r\ge 0 . The same statement holds for the binomial coefficients ( p n + r − 1 n ) \binom {pn+r-1}n , n = 0 , 1 , … n=0,1,\ldots \, .

Symmetric polynomials in information theory: Entropy and subentropy

Entropy and other fundamental quantities of information theory are customarily expressed and manipulated as functions of probabilities. Here we study the entropy H and subentropy Q as functions of the elementary symmetric polynomials in the probabilities and reveal a series of remarkable properties. Derivatives of all orders are shown to satisfy a complete monotonicity property. H and Q themselves become multivariate Bernstein functions and we derive the density functions of their Levy-Khintchine representations. We also show that H and Q are Pick functions in each symmetric polynomial variable separately. Furthermore, we see that H and the intrinsically quantum informational quantity Q become surprisingly closely related in functional form, suggesting a special significance for the symmetric polynomials in quantum information theory. Using the symmetric polynomials, we also derive a series of further properties of H and Q.

Hankel vector moment sequences and the non-tangential regularity at infinity of two variable Pick functions

A Pick function of d variables is a holomorphic map from Πd to Π, where Π is the upper halfplane. Some Pick functions of one variable have an asymptotic expansion at infinity, a power series ∑∞ n=1 ρnz −n with real numbers ρn that gives an asymptotic expansion on non-tangential approach regions to infinity. In 1921 H. Hamburger characterized which sequences {ρn} can occur. We give an extension of Hamburger’s results to Pick functions of two variables.

A one-parameter family of Pick functions defined by the Gamma function and related to the volume of the unit ball in $n$-space

We show that can be considered as a Pick function when a > 1, i.e. extends to a holomorphic function mapping the upper half-plane into itself. We also consider the function and show that In f(x+1) is a Stieltjes function and that f(x + 1) is completely monotonic on ]0, ∞[. In particular, f(n) = Ω 1/(n ln n) n , n ≥ 2, is a Hausdorff moment sequence. Here Ω n is the volume of the unit ball in Euclidean n-space.

Completely monotone sequences and universally prestarlike functions

We introduce universally convex, starlike and prestarlike functions in the slit domain ℂ [1, ∞), and show that there exists a very close link to completely monotone sequences and Pick functions.

Interpolation classes, operator and matrix monotone functions

Connections between interpolation spaces, Pick functions, and matrix monotone functions are investigated. Characterizations, inclusion results, open problems, and conjectures on these function classes and their interrelations are presented.

On functions that preserve M-matrices and inverse M-matrices

In earlier works, authors such as Varga, Micchelli and Willoughby, Ando, and Fiedler and Schneider have studied and characterized functions which preserve the M-matrices or some subclasses of the M-matrices, such as the Stieltjes matrices. Here we characterize functions which either preserve the inverse M-matrices or map the inverse M-matrices to the M-matrices. In one of our results we employ the theory of Pick functions to show that if A and B are inverse M-matrices such that B −1 ≤ A −1, then (B+tI)−1 ≤ (A+tI)−1, for all t ≥ 0.

Canonical products of small order and related Pick functions

We show that various functions related to the logarithms of the canonical products P ρ ( z ) = ∏ n = 1 ∞ ( 1 + z / n ρ ) , ρ > 1 and Q ( z ) = ∏ n = 0 ∞ ( 1 + zq n ) , q ∈ ( 0 , 1 ) are Pick functions. As a consequence we find an integral expansion of a function involving the logarithm of Jacksons q-gamma function.