E_m函数类中Nevanlinna-Pick插值与广义Stieltjes矩量问题

令E\-m=(-∞,∞)＼∪[DD(]m[]j=1[DD)](α\-j,β\-j).函数类[WTHT]N[WTBX](E\-m)表示在上半复平面解析且虚部非负,在诸(α\-j,β\-j)(j=1,…,m)内解析且为实值的函数全体.该文用Hankel 向量方法建立[WTHT]N[WTBX](E\-m)函数类 中含有限(或无限可数)插值点的Nevanlinna Pick 问题与集合E\-m上 相关的非标准截断(或全)广义Stieltjes 矩量问题解集之间的一一对应.用类似于Riesz的办法建立E\-m上非标准截断广义Stieltjes矩量问题的可解性准则,从而获得了[WTHT]N[WTBX](E\-m)函数类中Nevanlinna Pick问题的可解性准则.     

Linear Fractional Transformations of Nevanlinna Functions Associated with a Nonnegative Operator

In the present paper a subclass of scalar Nevanlinna functions is studied, which coincides with the class of Weyl functions associated to a nonnegative symmetric operator of defect one in a Hilbert space. This class consists of all Nevanlinna functions that are holomorphic on (?∞, 0) and all those Nevanlinna functions that have one negative pole a and are injective on ${(-\infty, a)\,\cup\, (a, 0)}$ . These functions are characterized via integral representations and special attention is paid to linear fractional transformations which arise in extension and spectral problems of symmetric and selfadjoint operators.     

Boundary-value problems for x-analytical functions with weighted boundary conditions

We consider boundary-value problems for x-analytical functions of a complex variable z = x + iy in a number of domains. Limit values with the weight (ln x)^–1 are given for the real part of the x-analytical function on the sections of the boundary that follow the imaginary axis, and simple limits are given for the real part of the x-analytical functions on the part of the boundary outside the imaginary axis. The apparatus of integral representations of x-analytical functions is applied to obtain a solution of the problem in quadratures.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 74, pp. 3–11, 1992;     

Local spectral theory for normal operators in Krein spaces

Sign type spectra are an important tool in the investigation of spectral properties of selfadjoint operators in Krein spaces. It is our aim to show that also sign type spectra for normal operators in Krein spaces provide insight in the spectral nature of the operator: If the real part and the imaginary part of a normal operator in a Krein space have real spectra only and if the growth of the resolvent of the imaginary part (close to the real axis) is of finite order, then the normal operator possesses a local spectral function defined for Borel subsets of the spectrum which belong to positive (negative) type spectrum. Moreover, the restriction of the normal operator to the spectral subspace corresponding to such a Borel subset is a normal operator in some Hilbert space. In particular, if the spectrum consists entirely out of positive and negative type spectrum, then the operator is similar to a normal operator in some Hilbert space. We use this result to show the existence of operator roots of a class of quadratic operator polynomials with normal coefficients.     

Some Extensions of Loewner's Theory of Monotone Operator Functions

Several extensions of Loewner's theory of monotone operator functions are given. These include a theorem on boundary interpolation for matrix-valued functions in the generalized Nevanlinna class. The theory of monotone operator functions is generalized from scalar-to matrix-valued functions of an operator argument. A notion of κ-monotonicity is introduced and characterized in terms of classical Nevanlinna functions with removable singularities on a real interval. Corresponding results for Stieltjes functions are presented.     

A Note on Purely Imaginary Roots of Generalized Kac–Moody Algebras

Abstract

In this paper, we generalize the concept of purely imaginary roots of Kac–Moody algebras to generalized Kac–Moody algebras. Also we give a complete classification of those generalized Kac–Moody algebras with the purely imaginary property. We also define a new class of indefinite non-hyperbolic generalized Kac–Moody algebras called extended hyperbolic generalized Kac–Moody algebras and find that it does not always possess the purely imaginary property whereas the extended hyperbolic Kac–Moody algebras possess the purely imaginary property.     

Holonomic functions in a polycircle with nonnegative imaginary part

A well-known theorem of Nevanlinna on the representation of nonnegative measure of a function holomorphic in a circle and having nonnegative imaginary part is extended to functions of many complex variables, holomorphic in a polycircle and having there a nonnegative imaginary part.Translated from Matematicheskie Zametki, Vol. 15, No. 1, pp. 55–61, January, 1974.     

Some properties of root functions of ordinary second-order differential operators in the absence of the carleman condition

On a finite interval G of the real line, we consider the root functions of an ordinary second-order differential operator without any boundary conditions for the case in which the imaginary part of the spectral parameter is unbounded.We refine the estimates for the C-and L _p -norms of a root function and its first derivative on a compact set contained in the interior of G for the case in which the Carleman condition fails.A sufficient condition is obtained for the root functions of an ordinary second-order differential operator to have the Bessel property, assuming that the Carleman condition fails. We show that, under certain conditions, this problem can be reduced to analyzing the Bessel property of systems of exponentials.     

Composition operators that improve integrability on weighted Bergman spaces

Composition operators between weighted Bergman spaces with a smaller exponent in the target space are studied. An integrability condition on a generalized Nevanlinna counting function of the inducing map is shown to characterize both compactness and boundedness of such an operator. Composition operators mapping into the Hardy spaces are included by making particular choices for the weights.

     

A Distortion Theorem for Functions Convex in One Direction

We present a characterization of functions convex in the positive direction of the real axis with the bounded imaginary part, which includes a sharp distortion theorem in terms of Poincaré hyperbolic metric and the Bloch norm of those functions.     

Root multiplicities of hyperbolic Kac–Moody algebras and Fourier coefficients of modular forms

In this paper we consider the hyperbolic Kac–Moody algebra $\mathcal {F}$ associated with the generalized Cartan matrix . Its connection to Siegel modular forms of genus 2 was first studied by A. Feingold and I. Frenkel. The denominator function of $\mathcal{F}$ is not an automorphic form. However, Gritsenko and Nikulin extended $\mathcal{F}$ to a generalized Kac–Moody algebra whose denominator function is a Siegel modular form. Using the Borcherds denominator identity, the denominator function can be written as an infinite product. The exponents that appear in the product are given by Fourier coefficients of a weak Jacobi form. P. Niemann also constructed a generalized Kac–Moody algebra which contains $\mathcal {F}$ and whose denominator function is related to a product of Dedekind η-functions. In particular, root multiplicities of the generalized Kac–Moody algebra are determined by Fourier coefficients of a modular form. As the main results of this paper, we compute asymptotic formulas for these Fourier coefficients using the method of Hardy–Ramanujan–Rademacher, and obtain an asymptotic bound for root multiplicities of the algebra $\mathcal{F}$ . Our method can be applied to other hyperbolic Kac–Moody algebras and to other modular forms as demonstrated in the later part of the paper.     

Boundary Relations, Unitary Colligations, and Functional Models

Recently a new notion, the so-called boundary relation, has been introduced involving an analytic object, the so-called Weyl family. Weyl families and boundary relations establish a link between the class of Nevanlinna families and unitary relations acting from one Kreĭn space, a basic (state) space, to another Kreĭn space, a parameter space where the Nevanlinna family or Weyl family is acting. Nevanlinna families are a natural generalization of the class of operator-valued Nevanlinna functions and they are closely connected with the class of operator-valued Schur functions. This paper establishes the connection between boundary relations and their Weyl families on the one hand, and unitary colligations and their transfer functions on the other hand. From this connection there are various advances which will benefit the investigations on both sides, including operator theoretic as well as analytic aspects. As one of the main consequences a functional model for Nevanlinna families is obtained from a variant of the functional model of L. de Branges and J. Rovnyak for Schur functions. Here the model space is a reproducing kernel Hilbert space in which multiplication by the independent variable defines a closed simple symmetric operator. This operator gives rise to a boundary relation such that the given Nevanlinna family is realized as the corresponding Weyl family. Received: January 21, 2008., Revised: March 31, 2008.     

Integration of holomorphic Lie algebroids

We prove that a holomorphic Lie algebroid is integrable if and only if its underlying real Lie algebroid is integrable. Thus the integrability criteria of Crainic–Fernandes (Theorem 4.1 in Crainic, Fernandes in Ann Math 2:157, 2003) do also apply in the holomorphic context without any modification. As a consequence we prove that a holomorphic Poisson manifold is integrable if and only if its real part or imaginary part is integrable as a real Poisson manifold.     

Inverse Spectral Problem for Analytic {{(\mathbb{Z}/2 \mathbb{Z})^{n}}} -Symmetric Domains in {{\mathbb{R}^{n}}}

We prove that bounded real analytic domains in ${\mathbb{R}^{n}}$ , with the symmetries of an ellipsoid and with one axis length fixed, are determined by their Dirichlet or Neumann eigenvalues among other bounded real analytic domains with the same symmetries and axis length. Some non-degeneracy conditions are also imposed on the class of domains. It follows that bounded, convex analytic domains are determined by their spectra among other such domains. This seems to be the first positive result for the well-known Kac problem, “Can one hear the shape of a drum?”, in higher dimensions.     

On a remarkable formula of Ramanujan

A simple proof of Ramanujan’s formula for the Fourier transform of |Γ (a + it)|² is given where a is fixed and has positive real part and t is real. The result is extended to other values of a by solving an inhomogeneous ODE, and we use it to calculate the jump across the imaginary axis.     

Schur Algorithm in The Class {\mathcal{SI}} of J-contractive Functions Intertwining Solutions of Linear Differential Equations

In the PhD thesis of the second author under the supervision of the third author was defined the class ${\mathcal{SI}}$ of J-contractive functions, depending on a parameter and arising as transfer functions of overdetermined conservative 2D systems invariant in one direction. In this paper we extend and solve in the class ${\mathcal{SI}}$ , a number of problems originally set for the class ${\mathcal{S}}$ of functions contractive in the open right-half plane, and unitary on the imaginary line with respect to some preassigned signature matrix J. The problems we consider include the Schur algorithm and the Nevanlinna–Pick interpolation problem. The arguments rely on a correspondence between elements in a given subclass of ${\mathcal{SI}}$ and elements in ${\mathcal{S}}$ . Another important tool in the arguments is a new result pertaining to the classical tangential Schur algorithm.     

Hamilton算子的一类n次数值域的对称性

以Hamilton算子的数值域为基础,研究了一类算子的二次数值域关于实轴,虚轴的对称性.此外从α-J-自伴算子的n次数值域关于过原点直线对称出发,得到了有界Hamilton算子的一类n次数值域关于虚轴的对称性.     

Sharp Estimates for the Coefficients of the Inverse Functions of the Nevanlinna Univalent Functions of the Classes N1 and N2

In the paper the extremum of a typical functional in the class N₂ is found. In particular, it is shown its application for determination of the extremums of the coefficients of the inverse functions of the Nevanlinna univalent functions of the class N₂. A conjecture for these coefficients is stated.     

An uniqueness theorem for characteristic functions

Suppose that f is the characteristic function of a probability measure on the real line \(\mathbb R\). In this paper, we deal with the following problem posed by N.G. Ushakov: Is it true that f is never determined by its imaginary part \(\mathfrak {I}f\)? In other words, is it true that for any characteristic function f there exists a characteristic function g such that \(\mathfrak {I}f\equiv \mathfrak {I}g\) but \( f\not \equiv g\)? We study this question in the more general case of the characteristic function defined on an arbitrary locally compact abelian group. A characterization of what characteristic functions are uniquely determined by their imaginary parts are given. As a consequence of this characterization, we obtain that several frequently used characteristic functions on the classical locally compact abelian groups are uniquely determined by their imaginary parts.     

Two classes of multisecant methods for nonlinear acceleration

Many applications in science and engineering lead to models that require solving large‐scale fixed point problems, or equivalently, systems of nonlinear equations. Several successful techniques for handling such problems are based on quasi‐Newton methods that implicitly update the approximate Jacobian or inverse Jacobian to satisfy a certain secant condition. We present two classes of multisecant methods which allow to take into account a variable number of secant equations at each iteration. The first is the Broyden‐like class, of which Broyden's family is a subclass, and Anderson mixing is a particular member. The second class is that of the nonlinear Eirola–Nevanlinna‐type methods. This work was motivated by a problem in electronic structure calculations, whereby a fixed point iteration, known as the self‐consistent field (SCF) iteration, is accelerated by various strategies termed ‘mixing’. Copyright © 2008 John Wiley & Sons, Ltd.