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   

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.     

Entropy Computation on the Unit Disc of a Meromorphic Map

We propose a new definition of entropy based on both topological and metric entropy for the meromorphic maps. The entropy is then computed on the unit disc of a meromorphic map, which is called the extended Blaschke function, and is a nonlinear extension of the normalized Lorentz transformation. We nd that the de ned entropy is computable and observe several interested results, such as maximal entropy, entropy overshoot due to topological transition, entropy reduction to zero, and scaling invariance in conjunction with parameter space.     

On local non‐compactness in recursive mathematics

A metric space is said to be locally non‐compact if every neighborhood contains a sequence that is eventually bounded away from every element of the space, hence contains no accumulation point. We show within recursive mathematics that a nonvoid complete metric space is locally non‐compact iff it is without isolated points. The result has an interesting consequence in computable analysis: If a complete metric space has a computable witness that it is without isolated points, then every neighborhood contains a computable sequence that is eventually computably bounded away from every computable element of the space. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

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.     

Weakly Computable Real Numbers

A real number x is recursively approximable if it is a limit of a computable sequence of rational numbers. If, moreover, the sequence is increasing (decreasing or simply monotonic), then x is called left computable (right computable or semi-computable). x is called weakly computable if it is a difference of two left computable real numbers. We show that a real number is weakly computable if and only if there is a computable sequence (x_s)_s of rational numbers which converges to x weakly effectively, namely the sum of jumps of the sequence is bounded. It is also shown that the class of weakly computable real numbers extends properly the class of semi-computable real numbers and the class of recursively approximable real numbers extends properly the class of weakly computable real numbers.     

Computable dimension for ordered fields

The computable dimension of a structure counts the number of computable copies up to computable isomorphism. In this paper, we consider the possible computable dimensions for various classes of computable ordered fields. We show that computable ordered fields with finite transcendence degree are computably stable, and thus have computable dimension 1. We then build computable ordered fields of infinite transcendence degree which have infinite computable dimension, but also such fields which are computably categorical. Finally, we show that 1 is the only possible finite computable dimension for any computable archimedean field.     

Some constructions on the class of groups of computable automorphisms

We show that the computable inverse limit of a computable family of computable groups, the computable wreath product of a group of computable automorphisms and a computable group, as well as the commutant and center of every computable group can be realized as groups of computable automorphisms of suitable computable models.     

Interpolating sequences and the Shilov boundary of H∞(Δ)

Let H^∞(Δ) denote the Banach algebra of bounded analytic functions on the open unit disc, let $\text{M}$ denote its maximal ideal space, and let ? denote its Shilov boundary. D. J. Newman has shown that a homomorphism ? in $\text{M}$ will be in ? if and only if ? is unimodular on all Blaschke products. We answer a question of K. Hoffman by showing that ? will be in ? if and only if ? is unimodular on every Blaschke product whose zero set is an interpolating sequence. Our method is based on a construction due to L. Carleson, originally developed for the proof of the Corona theorem.     

Expanding a Finite Blaschke Product

It is shown that a finite Blaschke product with finite poles, has a nonzero residue. The proofs for the two types of Blaschke products are essentially different.     

The computable dimension of ordered abelian groups

Let G be a computable ordered abelian group. We show that the computable dimension of G is either 1 or ω, that G is computably categorical if and only if it has finite rank, and that if G has only finitely many Archimedean classes, then G has a computable presentation which admits a computable basis.     

Extremal solutions of Nevanlinna-Pick problems and certain classes of inner functions

Consider a scaled Nevanlinna-Pick interpolation problem and let ∏ be the Blaschke product whose zeros are the nodes of the problem. It is proved that if ∏ belongs to a certain class of inner functions, then the extremal solutions of the problem or most of them are in the same class. Three different classical classes are considered: inner functions whose derivative is in a certain Hardy space, exponential Blaschke products and the well-known class of α-Blaschke products, for 0 < α < 1.     

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.     

Cluster sets of interpolating Blaschke products

We construct interpolating Balschke products whose radial cluster sets at a given point of the unit circle can be prescribed to be one of the following: the closed unit disk; an arbitrary closed arc on the unit circle; an arbitrary interval of the form [x, y], wherexy ≠ 0 and −1≤1x≤y≤1. We also show that there does not exist an interpolating Blaschke product having [0,y] or [x, 0] as a radial cluster set. On the other hand, there do exist finite products of interpolating Blaschke products that have [0, 1] as a radial cluster set. Research supported by the RIP-program Oberwolfach, 2002/2003.     

Peaceman–Rachford splitting for a class of nonconvex optimization problems

We study the applicability of the Peaceman–Rachford (PR) splitting method for solving nonconvex optimization problems. When applied to minimizing the sum of a strongly convex Lipschitz differentiable function and a proper closed function, we show that if the strongly convex function has a large enough strong convexity modulus and the step-size parameter is chosen below a threshold that is computable, then any cluster point of the sequence generated, if exists, will give a stationary point of the optimization problem. We also give sufficient conditions guaranteeing boundedness of the sequence generated. We then discuss one way to split the objective so that the proposed method can be suitably applied to solving optimization problems with a coercive objective that is the sum of a (not necessarily strongly) convex Lipschitz differentiable function and a proper closed function; this setting covers a large class of nonconvex feasibility problems and constrained least squares problems. Finally, we illustrate the proposed algorithm numerically.     

On Blaschke products with derivatives in Bergman spaces with normal weights

We generalize a well-known sufficient condition for interpolating sequences for the Hilbert Bergman spaces to other Bergman spaces with normal weights (as defined by Shields and Williams) and obtain new results regarding the membership of the derivative of a Blaschke product or a general inner function in such spaces. We also apply duality techniques to obtain further results of this type and obtain new results about interpolating Blaschke products.     

Derivatives of Computable Functions

As is well known the derivative of a computable and C¹ function may not be computable. For a computable and C∞ function f, the sequence {f⁽ⁿ⁾} of its derivatives may fail to be computable as a sequence, even though its derivative of any order is computable. In this paper we present a necessary and sufficient condition for the sequence {f⁽ⁿ⁾} of derivatives of a computable and C^∞ function f to be computable. We also give a sharp regularity condition on an initial computable function f which insures the computability of its derivative f′.     

Toeplitz算子在相似意义下的强不可约性

该文研究了Hardy空间上带两个Blaschke因子的解析Toeplitz算子在相似意义下的强不可约性, 并讨论了Blaschke因子的缠绕数与由Blaschke因子诱导的解析Toeplitz算子强不可约性之间的关系.     

Multiplication operators on the Bergman space via analytic continuation

In this paper, using the group-like property of local inverses of a finite Blaschke product ?, we will show that the largest C^?-algebra in the commutant of the multiplication operator M? by ? on the Bergman space is finite dimensional, and its dimension equals the number of connected components of the Riemann surface of ?−1°? over the unit disk. If the order of the Blaschke product ? is less than or equal to eight, then every C^?-algebra contained in the commutant of M? is abelian and hence the number of minimal reducing subspaces of M? equals the number of connected components of the Riemann surface of ?−1°? over the unit disk.