Asymptotic behavior of Bergman kernels with logarithmic weight

We describe singularities of weighted Bergman kernels on the unit disc with respect to radial logarithmic weights.     

Weighted reproducing kernels and Toeplitz operators on harmonic Bergman spaces on the real ball

For the standard weighted Bergman spaces on the complex unit ball, the Berezin transform of a bounded continuous function tends to this function pointwise as the weight parameter tends to infinity. We show that this remains valid also in the context of harmonic Bergman spaces on the real unit ball of any dimension. This generalizes the recent result of C. Liu for the unit disc, as well as the original assertion concerning the holomorphic case. Along the way, we also obtain a formula for the corresponding weighted harmonic Bergman kernels.

     

方程△g-aKg=0无正函数解的充分条件

Fischer-Colbrie和Schoen曾在1980年研究过复平面中单位圆盘当赋予某个完备度量时,方程Δg-aKg=0在其上无正函数解的充分条件,并将其结果应用到三维非负数量曲率流形中完备稳定的极小曲面上.这里Δ是Laplace算子,K为高斯曲率,a是常数,g是所讨论的单位圆盘上的函数.本文给出了此方程在该圆盘上无正函数解的一个更弱的充分条件.     

An Integral Formula for Weighted Bergman Reproducing Kernels

An integral formula is obtained for reproducing kernels in weighted Bergman spaces with radial and logarithmically subharmonic weights in the unit disk. We deduce from it that these reproducing kernels have a special structure leading to the contractive divisor property of extermal functions.     

Finite Blaschke product interpolation on the closed unit disc

We show how to construct all finite Blaschke product solutions and the minimal scaled Blaschke product solution to the Nevanlinna-Pick interpolation problem in the open unit disc by solving eigenvalue problems of the interpolation data. Based on a result of Jones and Ruscheweyh we note that there always exists a finite Blaschke product of degree at most n−1 that maps n distinct points in the closed unit disc, of which at least one is on the unit circle, into n arbitrary points in the closed unit disc, provided that the points inside the unit circle form a positive semi-definite Pick matrix of full rank. Finally, we discuss a numerical limiting procedure.     

几个多复变数加权全纯函数空间的边界值与对偶性

该文将几个全纯函数空间的定义从单复变数推广到多复变数, 这些空间中全纯函数的增长性依赖于某个权函数. 作者研究了它们的增长性与边界值的关系以及这些空间相互之间的对偶关系, 所用方法和技巧与单复变数有不少区别. 所得到的结论是单复变数许多已知结果的推广.     

Perturbed kernel approximation on homogeneous manifolds

Current methods for interpolation and approximation within a native space rely heavily on the strict positive-definiteness of the underlying kernels. If the domains of approximation are the unit spheres in euclidean spaces, then zonal kernels (kernels that are invariant under the orthogonal group action) are strongly favored. In the implementation of these methods to handle real world problems, however, some or all of the symmetries and positive-definiteness may be lost in digitalization due to small random errors that occur unpredictably during various stages of the execution. Perturbation analysis is therefore needed to address the stability problem encountered. In this paper we study two kinds of perturbations of positive-definite kernels: small random perturbations and perturbations by Dunkl's intertwining operators [C. Dunkl, Y. Xu, Orthogonal polynomials of several variables, Encyclopedia of Mathematics and Its Applications, vol. 81, Cambridge University Press, Cambridge, 2001]. We show that with some reasonable assumptions, a small random perturbation of a strictly positive-definite kernel can still provide vehicles for interpolation and enjoy the same error estimates. We examine the actions of the Dunkl intertwining operators on zonal (strictly) positive-definite kernels on spheres. We show that the resulted kernels are (strictly) positive-definite on spheres of lower dimensions.     

Norm of Berezin transformon <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> space

It is shown that the Berezin transform B on L ^p (D), where D is the unit disc, has norm . Furthermore, the norms of a family of operators (on L ^p (D)) whose kernels are moduli of Bergman type kernels are also calculated. Partially supported by MNZZS, Grant ON144010     

A property of Hermite-Padé interpolation on the roots of unity

We extend a theorem of Ivanov and Saff to show that for the Hermite-Padé interpolant at the roots of unity to a function meromorphic in the unit disc, its leading coefficients vanish if and only if the corresponding interpolant to a related function vanishes at given points outside the unit disc. The result is then extended to simultaneous Hermite-Padé inter polation to a finite set of functions.     

The V. A. Steklov problem in the theory of orthogonal polynomials

An exact order of growth of Szego polynomial kernels and moduli of polynomials orthogonal in the unit circle is obtained in the zeros of a weight function of special form. Other questions are also examined.     

ALGEBROIDAL FUNCTION AND ITS DERIVED FUNCTION IN UNIT CIRCULAR DISC

In this article,the authors define the derived function of an algeboidal function in the unit disc,prove it is an algabriodal function,and study the order of algebroidal function and that of its derived function in unit circular disc.     

Root statistics of random polynomials with bounded Mahler measure

The Mahler measure of a polynomial is a measure of complexity formed by taking the modulus of the leading coefficient times the modulus of the product of its roots outside the unit circle. The roots of a real degree N polynomial chosen uniformly from the set of polynomials of Mahler measure at most 1 yield a Pfaffian point process on the complex plane. When N is large, with probability tending to 1, the roots tend to the unit circle, and we investigate the asymptotics of the scaled kernel in a neighborhood of a point on the unit circle. When this point is away from the real axis (on which there is a positive probability of finding a root) the scaled process degenerates to a determinantal point process with the same local statistics (i.e.   scalar kernel) as the limiting process formed from the roots of complex polynomials chosen uniformly from the set of polynomials of Mahler measure at most 1. Three new matrix kernels appear in a neighborhood of ±1 which encode information about the correlations between real roots, between complex roots and between real and complex roots. Away from the unit circle, the kernels converge to new limiting kernels, which imply among other things that the expected number of roots in any open subset of C$\mathbb{C}$ disjoint from the unit circle converges to a positive number. We also give ensembles with identical statistics drawn from two-dimensional electrostatics with potential theoretic weights, and normal matrices chosen with regard to their topological entropy as actions on Euclidean space.     

A system of boundary integral equations for orthotropic shells of positive curvature with slits and holes

We propose a method of constructing a system of boundary integral equations for the problem of the stress state of an orthotropic shell with slits and holes. Using the theory of distributions and the two-dimensional Fourier transform, we reduce the problem to a system of boundary integral equations. In the solution obtained the kernels of the system of integral equations do not contain the direction cosines of the unit outward normal vector explicitly. There are no extra-integral terms. The matrix of the kernels is symmetric. The kernels are regular or have a logarithmic singularity. Two figures. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 59–69.     

Ramadanov conjecture and line bundles over compact Hermitian symmetric spaces

We compute the Szegö kernels of the unit circle bundles of homogeneous negative line bundles over a compact Hermitian symmetric space. We prove that their logarithmic terms vanish in all cases and, further, that the circle bundles are not diffeomorphic to the unit sphere in ${\mathbb C^n}We compute the Szeg? kernels of the unit circle bundles of homogeneous negative line bundles over a compact Hermitian symmetric space. We prove that their logarithmic terms vanish in all cases and, further, that the circle bundles are not diffeomorphic to the unit sphere in \mathbb Cⁿ{\mathbb C^n} for Grassmannian manifolds of higher ranks. In particular, they provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds for which the logarithmic term in the Fefferman expansion of the Szeg? kernel vanishes but whose boundary is not diffeomorphic to the sphere (in fact, it is not even locally spherical). The analogous results for the Bergman kernel are also obtained.     

Becker type univalence conditions for harmonic mappings

We obtain Becker type univalence conditions for locally univalent harmonic mappings defined in one of the following domains: the unit disc, a halfplane, the exterior of the unit disc and prove a generalization of John’s univalence condition.     

Stability of recursive digital filters

If B(z) is an absolutely convergent power series on the unit disk, the requirements for the reciprocal of B to also be absolutely convergent on the unit disc are well known. These requirements are that B does not vanish on the unit disc. This paper gives an alternate characterization of the corresponding theorem in two-variables along with a new proof.     

Reproducing kernels for harmonic Bergman spaces of the unit ball

We study reproducing kernels for harmonic Bergman spaces of the unit ball inR ⁿ . We establish some new properties for the reproducing kernels and give some applications of these properties.     

The growth of solutions of differential equations with coefficients of small growth in the disc

In this paper, we classify the functions of small growth in the unit disc to different degree, and investigate the growth of solutions for certain linear differential equations with coefficients of small growth in the unit disc.     

Clifford–Fourier transform on hyperbolic space

In this paper, we introduce a new generalization of the Helgason–Fourier transform using the angular Dirac operator on both the hyperboloid and unit ball models. The explicit integral kernels of even dimension are derived. Furthermore, we establish the formal generating function of the even dimensional kernels. In the computations, fractional integration plays a key unifying role. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimal methods for specifying information in the solution of integral equations with analytic kernels

We determine the exact order of the minimum radius of information in the logarithmic scale for Fredholm integral equations of the second kind with periodic analytic kernels and free terms. We show that the information complexity of the solution of Fredholm equations with analytic kernels is greater in order than the complexity of the approximation of analytic functions. This distinguishes the analytic case from the case of finite smoothness.