Value Distribution of Interpolating Blaschke Products

A Blaschke product B with zero-sequence (a_n) is called almostinterpolating if the inequality lim inf_n(1 – |a_n|²)|B'(a_n)|     

Interpolating Blaschke Products: Stolz and Tangential Approach Regions

A result of D.J. Newman asserts that a uniformly separated sequence contained in a Stolz angle is a finite union of exponential sequences. We extend this by obtaining several equivalent characterizations of such sequences. If the zeros of a Blaschke product B lie in a Stolz angle, then for all and it has recently been shown that this result cannot be improved. Also, Newman's result can be used to prove that if B is an interpolating Blaschke product whose zeros lie in a Stolz angle, then 相似文献   

A New Class of Inner Functions Uniformly Approximable by Interpolating Blaschke Products

We study the class of inner functions Q{\Theta} whose zero set Z(Q){Z(\Theta)} stays hyperbolically close to [`(Z_\mathbbD(Q))]{\overline{Z_\mathbb{D}(\Theta)}} on the corona of H ^∞ and show that these functions are uniformly approximable by interpolating Blaschke products.     

Interpolating Blaschke products solving Pick-Nevanlinna problems

We study when a Pick-Nevanlinna problem with more than one solution can be solved by an interpolating Blaschke product. Supported in part by the grant PB85-0374 of the CYCIT, Ministerio de Educación y Ciencia, Spain.     

Critical Values of Finite Blaschke Products

Doklady Mathematics - For finite Blaschke products B of degree $$n geqslant 2,$$ $$B(0) = 0,$$ $$B'(0) ne 0$$ a sharp upper bound for the least critical values and a sharp lower bound for the...     

Two Theorems on Factorization of Inequalities

The factorization of inequalities for infinite numerical series is a new research field. A monograph giving a systematic treatment of this subject is due to BENNETT. Here two theorems of this type are proved. Our new results extend previous theorems to the so-called Mulholland functions.     

Boundary Interpolation by Finite Blaschke Products

Given 2n distinct points z₁, z′₁, z₂, z′₂, ..., z_n, z′_n (in this order) on the unit circle, and n points w₁, ..., w_n on the unit circle, we show how to construct a Blaschke product B of degree n such that B(z_j) = w_j for all j and, in addition, B(z′_j) = B(z′_k) for all j and k. Modifying this example yields a Blaschke product of degree n - 1 that interpolates the z_j's to the w_j's. We present two methods for constructing our Blaschke products: one reminiscent of Lagrange's interpolation method and the second reminiscent of Newton's method. We show that locating the zeros of our Blaschke product is related to another fascinating problem in complex analysis: the Sendov Conjecture. We use this fact to obtain estimates on the location of the zeros of the Blaschke product.     

Factorization Theorems on Symmetric Spaces of Noncompact Type

We prove the analogs of the Khinchin factorization theorems for K-invariant probability measures on symmetric spaces X=G/K with G semisimple noncompact. We use the Kendall theory of delphic semigroups and some properties of the spherical Fourier transform and spherical functions on X.     

Blaschke, Privaloff, Reade and Saks Theorems for Diffusion Equations on Lie Groups

We prove some asymptotic characterizations for the subsolutions to a class of diffusion equations on homogeneous Lie groups. These results are the diffusion counterpart of the classical Blaschke, Privaloff, Reade and Saks Theorems for harmonic functions.     

Factorization Theorems for Multiplication Operators on Banach Function Spaces

Let X Y and Z be Banach function spaces over a measure space \({(\Omega, \Sigma, \mu)}\) . Consider the spaces of multiplication operators \({X^{Y'}}\) from X into the Köthe dual Y′ of Y, and the spaces X ^Z and \({Z^{Y'}}\) defined in the same way. In this paper we introduce the notion of factorization norm as a norm on the product space \({X^Z \cdot Z^{Y'} \subseteq X^{Y'}}\) that is defined from some particular factorization scheme related to Z. In this framework, a strong factorization theorem for multiplication operators is an equality between product spaces with different factorization norms. Lozanovskii, Reisner and Maurey–Rosenthal theorems are considered in our arguments to provide examples and tools for assuring some requirements. We analyze the class \({d_{p,Z}^*}\) of factorization norms, proving some factorization theorems for them when p-convexity/p-concavity type properties of the spaces involved are assumed. Some applications in the setting of the product spaces are given.     

A Note on Pluripolar Hulls of Graphs of Blaschke Products

In the paper we study the problem of characterization of pluripolar hulls of a class of functions containing Blaschke products. Mathematics Subject Classifications (2000) 32U30, 31A15, 30D50.Research partially supported by the KBN grant No. 5 P03A 033 21 and the Rectors Scholarship Fund at the Jagiellonian University.     

图的伴随多项式的两个因式分解定理及其应用

设G是m阶连通图,P_m是m个顶点的路.令S_km+1^G(i)表示把kG的每一个分支的第i(1≤i≤m)个顶点依次与星图S_k+1的k个1度顶点重迭后得到的图;令G_i1^S*(q,km)表示q阶图G的顶点V_i1与S_km+1^p(1)的k度顶点重迭后得到的图     

Spectra of Toeplitz Operators and Compositions of Muckenhoupt Weights with Blaschke Products

We discuss the optimality of a sufficient condition from [12] for a point to belong to the essential spectrum of a Toeplitz operator with a bounded measurable coefficient. Our approach is based on a new sufficient condition for a composition of a Muckenhoupt weight with a Blaschke product to belong to the same Muckenhoupt class. The first author was partially supported by CONACYT project U46936-F, Mexico.     

Quotient Algebras of Toeplitz-Composition C^{*}-Algebras for Finite Blaschke Products

Let $R$ be a finite Blaschke product. We study the $C^*$ -algebra $\mathcal TC _R$ generated by both the composition operator $C_R$ and the Toeplitz operator $T_z$ on the Hardy space. We show that the simplicity of the quotient algebra $\mathcal OC _R$ by the ideal of the compact operators can be characterized by the dynamics near the Denjoy–Wolff point of $R$ if the degree of $R$ is at least two. Moreover we prove that the degree of finite Blaschke products is a complete isomorphism invariant for the class of $\mathcal OC _R$ such that $R$ is a finite Blaschke product of degree at least two and the Julia set of $R$ is the unit circle, using the Kirchberg–Phillips classification theorem.     

Skew Products and Ergodic Theorems for Group Actions

New ergodic theorems for the action of a free semigroup on a probabilistic space by measure-preserving maps are obtained. The method applied consists of associating with the original semigroup action a skew product over the shift on the space of infinite one-sided sequences of generators of the semigroup and then integrating the BirkhoffKhinchin ergodic theorems along the base of the skew product. Bibliography: 17 titles.     

On Finite Blaschke Products whose Restrictions to the unit circle are Exact Endomorphisms

An easily checked sufficient condition is given for the restrictionof a finite Blaschke product to the unit circle to be an exactendomorphism. A formula for the entropy of such restrictionswith respect to the unique finite invariant measure equivalentto Lebesgue measure is given and it is shown that if such arestriction has maximal entropy then it is conformally equivalentto the product of a rotation and a power.     

Maximal Blaschke products

We consider the classical problem of maximizing the derivative at a fixed point over the set of all bounded analytic functions in the unit disk with prescribed critical points. We show that the extremal function is essentially unique and always an indestructible Blaschke product. This result extends the Nehari–Schwarz Lemma and leads to a new class of Blaschke products called maximal Blaschke products. We establish a number of properties of maximal Blaschke products, which indicate that maximal Blaschke products constitute an appropriate infinite generalization of the class of finite Blaschke products.