Improvement of convergence of an iterative method for finding polynomial factors of analytic functions

In this paper, we consider an iterative method for evaluating the coefficients of a monic factor of an analytic function using complex circular arithmetic. In a previous paper, the authors presented a factoring method that finds a cluster of zeros as a polynomial factor. We analyze the convergence behavior of this method and discuss a technique for improving convergence. Numerical examples illustrate the aspects of the improved method.     

Infinite systems of linear equations for real analytic functions

We study the problem when an infinite system of linear functional equations

has a real analytic solution on for every right-hand side and give a complete characterization of such sequences of analytic functionals . We also show that every open set has a complex neighbourhood such that the positive answer is equivalent to the positive answer for the analogous question with solutions holomorphic on .

     

A counterexample for approximable functions

Let be the unit disk. We show that for some relatively closed set there is a function that can be uniformly approximated on by functions of , but such that cannot be written as , with and uniformly continuous on . This answers a question of Stray.

     

Inheritance of surjectivity for partial differential operators on spaces of real analytic functions

For an open set let A(Ω) be the space of real analytic functions on Ω. Improving our previous results, we prove a new quantitative characterization of the linear partial differential operators P(D) which are surjective on A(Ω). This implies that P(D) is surjective on if P(D) is surjective on A(Ω) for some Ω≠∅. Further inheritance properties for the surjectivity of P(D) on A(Ω) are also obtained.     

Elementary equivalence of some rings of definable functions

We characterize elementary equivalences and inclusions between von Neumann regular real closed rings in terms of their boolean algebras of idempotents, and prove that their theories are always decidable. We then show that, under some hypotheses, the map sending an L-structure R to the L-structure of definable functions from R ⁿ to R preserves elementary inclusions and equivalences and gives a structure with a decidable theory whenever R is decidable. We briefly consider structures of definable functions satisfying an extra condition such as continuity.      

Free interpolation by nonvanishing analytic functions

We are concerned with interpolation problems in where the values prescribed and the function to be found are both zero-free. More precisely, given a sequence in the unit disk, we ask whether there exists a nontrivial minorant (i.e., a sequence of positive numbers bounded by 1 and tending to 0) such that every interpolation problem has a nonvanishing solution whenever for all . The sequences with this property are completely characterized. Namely, we identify them as `` thin" sequences, a class that arose earlier in Wolff's work on free interpolation in VMO.

     

Fredholm composition operators on algebras of analytic functions on Banach spaces

We prove that Fredholm composition operators acting on the uniform algebra H^∞(BE) of bounded analytic functions on the open unit ball of a complex Banach space E with the approximation property are invertible and arise from analytic automorphisms of the ball.     

A Beurling-Carleson set which is a uniqueness set for a given weighted space of analytic functions

Let be a sequence of positive real numbers. We define as the space of functions which are analytic in the unit disc , continuous on and such that

where is the Fourier coefficient of the restriction of to the unit circle . Let be a closed subset of . We say that is a Beurling-Carleson set if

where denotes the distance between and . In 1980, A. Atzmon asked whether there exists a sequence of positive real numbers such that for all and that has the following property: for every Beurling-Carleson set , there exists a non-zero function in that vanishes on . In this note, we give a negative answer to this question.

     

k解析函数的Fourier级数

讨论了复平面上k解析函数的性质,并利用k解析函数的泰勒展开定理研究了k解析函数的Fourier级数,推广了经典的解析函数的Fourier级数理论.     

On analytic functions related with generalized Robertson functions

Analytic functions f are called Robertson functions for which zf^′ is α-spiral-like. This concept is generalized by several authors and a class real, of analytic functions is introduced and studied. It is noted that the functions in are of bounded boundary rotation and consists of Robertson functions.In this paper, we use the class to define a new class of analytic functions which unifies a number of classes previously studied such as the class of close-to-convex functions of higher order. Some interesting properties of this class, including coefficient problems, inclusion results and a sufficient condition for univalency are studied.     

On the spectrum of the Cesàro operator on spaces of analytic functions

This paper concerns the Cesàro operator acting on various spaces of analytic functions on the unit disc. The remarkable fact that this operator is subnormal when acting on the Hardy space H² has lead to extensive studies of its spectral picture on other spaces of this type. We present some of the methods that have been used to obtain information about the spectrum of the Cesàro operator acting on Hardy and Bergman spaces and give a unified approach to these problems which also yields new results in this direction. In particular, we prove that the Cesàro operator is subdecomposable on H¹ and on the standard weighted Bergman spaces , α0.     

Some novel applications of certain higher order ordinary complex differential equations to normalized analytic functions

The aim of this work is first to reveal some nonlinear connections between normalized analytic functions and certain higher order ordinary(complex) differential equations and then to point some of their geometric implications appertaining to normalized analytic functions out.     

Starlike approximations of analytic functions

When an analytic function is not univalent, it is often of interest to approximate it by univalent functions. In this paper we introduce a measure of the non-univalency of a function and we derive a method for constructing the best starlike univalent approximations of analytic functions with respect to it, suitable for both practical problems and numerical implementation.     

Orderings and maximal ideals of rings of analytic functions

We prove that there is a natural injective correspondence between the maximal ideals of the ring of analytic functions on a real analytic set and those of its subring of bounded analytic functions. By describing the maximal ideals in terms of ultrafilters we see that this correspondence is surjective if and only if is compact. This approach is also useful for studying the orderings of the field of meromorphic functions on .

     

Non-reflexivity of the derivation space from Banach algebras of analytic functions

Let be an open connected subset of the plane, and let be a Banach algebra of analytic functions on . We show that the space of bounded derivations from into is not reflexive. We also obtain similar results when for .

     

Multi-point Taylor expansions of analytic functions

Taylor expansions of analytic functions are considered with respect to several points, allowing confluence of any of them. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals. The method is also used for obtaining Laurent expansions in several points as well as Taylor-Laurent expansions.

     

Boundary behavior in Hilbert spaces of vector-valued analytic functions

In this paper we study the boundary behavior of functions in Hilbert spaces of vector-valued analytic functions on the unit disc D. More specifically, we give operator-theoretic conditions on Mz, where Mz denotes the operator of multiplication by the identity function on D, that imply that all functions in the space have non-tangential limits a.e., at least on some subset of the boundary. The main part of the article concerns the extension of a theorem by Aleman, Richter and Sundberg in [A. Aleman, S. Richter, C. Sundberg, Analytic contractions and non-tangential limits, Trans. Amer. Math. Soc. 359 (2007)] to the case of vector-valued functions.