Algebroid functions, Wirsing's theorem and their relations

In this paper, we first point out a relationship between the Second Main Theorem for algebriod functions in Nevanlinna theory and Wirsing's theorem in Diophantine approximation. This motivates a unified proof for both theorems. The second part of this paper deals with “moving targets” problem for holomorphic maps to Riemann surfaces. Its counterpart in Diophantine approximation follows from a recent work of Thomas J. Tucker. In this paper, we point out Tucker's result in the special case of the approximation by rational points could be obtained by doing a “translation” and applying the corresponding result with fixed target. However, we could not completely recover Tucker's result concerning the approximation by algebraic points. In the last part of this paper, cases in higher dimensions are studied. Some partial results in higher dimensions are obtained and some conjectures are raised. Received August 26, 1997; in final form June 30, 1998     

A simplicial approach to factorization systems and Kurosh-Amitsur radicals

Regarding categories as simplicial sets via the nerve functor, we extend the notion of a factorization system from morphisms in a category, to 1-simplexes in an arbitrary simplicial set. Applied to what we call the simplicial set of short exact sequences, it gives the notion of Kurosh-Amitsur radical. That is, we present a unified approach to factorization systems and radicals.     

Comparison of the Bergman and Szegö kernels

The quotient of the Szegö and Bergman kernels for a smooth bounded pseudoconvex domains in Cn is bounded from above by a constant multiple of for any p>n, where δ is the distance to the boundary. For a class of domains that includes those of D?Angelo finite type and those with plurisubharmonic defining functions, the quotient is also bounded from below by a constant multiple of for any p<−1. Moreover, for convex domains, the quotient is bounded from above and below by constant multiples of δ.     

Slice regular functions on real alternative algebras

In this paper we develop a theory of slice regular functions on a real alternative algebra A. Our approach is based on a well-known Fueter's construction. Two recent function theories can be included in our general theory: the one of slice regular functions of a quaternionic or octonionic variable and the theory of slice monogenic functions of a Clifford variable. Our approach permits to extend the range of these function theories and to obtain new results. In particular, we get a strong form of the fundamental theorem of algebra for an ample class of polynomials with coefficients in A and we prove a Cauchy integral formula for slice functions of class C¹.     

Extension of bounded holomorphic functions¶in convex domains

The E. Amar and G. Henkin theorem on the bounded extendability of bounded holomorphic functions from certain closed complex submanifolds of strictly pseudoconvex domains to the whole domain is generalized to the case of finite type convex domains and their intersections with affine linear hyperplanes. Suitable integral operators of Berndtsson–Andersson type are constructed and estimated for this purpose. Received: 7 July 2000     

Some finite-dimensional backward-shift-invariant subspaces in the ball and a related interpolation problem

We solve Gleason's problem in the reproducing kernel Hilbert space with repoducing kernel . We define and study some finite-dimensional resolvent-invariant subspaces that generalize the finite-dimensional de Branges-Rovnyak spaces to the setting of the ball.This research was supported by a grant from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel, and by the Israeli Academy of Sciences.     

Singular Berezin Transforms

We give examples of pseudoconvex Reinhardt domains where the Berezin transform has integral kernel with singularities and, hence, fails to be a smoothing map. On the other hand, we show that this can never happen for a plane domain – in fact, then the Bergman kernel is always either identically zero or strictly positive everywhere on the diagonal – and also prove that, in contrast to the example by Wiegerinck from 1984, on any pseudoconvex Reinhardt domain the Bergman space can be finite-dimensional only if it reduces to the constant zero. Received: February 02, 2007. Accepted: May 28, 2007.     

Permutation polynomials over finite fields from a powerful lemma

Using a lemma proved by Akbary, Ghioca, and Wang, we derive several theorems on permutation polynomials over finite fields. These theorems give not only a unified treatment of some earlier constructions of permutation polynomials, but also new specific permutation polynomials over Fq. A number of earlier theorems and constructions of permutation polynomials are generalized. The results presented in this paper demonstrate the power of this lemma when it is employed together with other techniques.     

A characterization of products of balls by their isotropy groups

In this paper we will characterize products of balls – especially the ball and the polydisc – in by properties of the isotropy group of a single point. It will be shown that such a characterization is possible in the class of Siegel domains of the second kind, a class that extends the class of bounded homogeneous domains, and that such a characterization is no longer possible in the class of bounded domains with noncompact automorphism groups. The main result is that a Siegel domain of the second kind is biholomorphically equivalent to a product of balls, iff there is a point such that the isotropy group of p contains a torus of dimension n. As an application it will be proved that the only domains biholomorphically equivalent to a Siegel domain of the second kind and to a Reinhardt domain are exactly the domains biholomorphically equivalent to a product of b alls. Received: 27 February 1998 / In final form: 6 August 1998     

Commutant lifting and interpolation: The time-varying case

We show how the commutant lifting theorem for nest algebras due to Paulsen and Power can be used to give a unified framework for the treatment of a variety of interpolation problems for nest algebras which have been considered recently in the literature. Applications include the treatment of robust control for time-varying systems.Partially supported by NSF grant DMS-9500912     

Regular closure operators

In an E,M-categoryX for sinks, we identify necessary conditions for Galois connections from the power collection of the class of (composable pairs) of morphisms inM to factor through the lattice of all closure operators onM, and to factor through certain sublattices. This leads to the notion ofregular closure operator. As one byproduct of these results we not only arrive (in a novel way) at the Pumplün-Röhrl polarity between collections of morphisms and collections of objects in such a category, but obtain many factorizations of that polarity as well. (One of these factorizations constituted the main result of an earlier paper by the same authors). Another byproduct is the clarification of the Salbany construction (by means of relative dominions) of the largest idempotent closure operator that has a specified class ofX-objects as separated objects. The same relation that is used in Salbany's relative dominion construction induces classical regular closure operators as described above. Many other types of closure operators can be obtained by this technique; particular instances of this are the idempotent and modal closure operators that in a Grothendieck topos correspond to the Grothendieck topologies.Dedicated to Professor Dieter Pumplün, on his 60th birthdayResearch partially supported by the Faculty of Arts and Sciences, University of Puerto Rico, Mayagüez Campus during a sabbatical visit at Kansas State University.     

A general approach to counterexamples in numerical analysis

Summary The purpose of this paper is to indicate a unified approach to quantitative negative results in numerical analysis. This is done via a rather general theorem which in fact subsumes our previous quantitative uniform boundedness principles. The proof is based upon a gliding hump method. The general theorem is exemplarily applied to discuss the sharpness of various direct and inverse approximation results, known for the compound trapezoidal rule and for the approximate solution of the heat equation. The treatment outlines a program which may also be worked out for other procedures.Supported by Deutsche Forschungsgemeinschaft Grant No. Ne 171/5-1     

Rigidity for regular functions over Hamilton and Cayley numbers and a boundary Schwarz' Lemma

A new theory of regular functions over the skew field of Hamilton numbers (quaternions) and in the division algebra of Cayley numbers (octonions) has been recently introduced by Gentili and Struppa (Adv. Math. 216 (2007) 279–301). For these functions, among several basic results, the analogue of the classical Schwarz' Lemma has been already obtained. In this paper, following an interesting approach adopted by Burns and Krantz in the holomorphic setting, we prove some boundary versions of the Schwarz' Lemma and Cartan's Uniqueness Theorem for regular functions. We are also able to extend to the case of regular functions most of the related “rigidity” results known for holomorphic functions.     

Kergin Interpolants of Holomorphic Functions

Let D be a C-convex domain in C ⁿ . Let , and d = 0,1,2, ..., be an array of points in a compact set . Let f be holomorphic on and let K _d (f) denote the Kergin interpolating polynomial to f at A _d0 ,... , A _dd . We give conditions on the array and D such that . The conditions are, in an appropriate sense, optimal. This result generalizes classical one variable results on the convergence of Lagrange—Hermite interpolants of analytic functions. Date received: October 21, 1995. Date revised: May 1, 1996.     

Numerical solution of integral equations,fast algorithms and Krein-Sobolev equation

Summary This paper contains a unified rigorous approach for the treatment of fast numerical algorithms for different classes of Fredholm integral equations of second kind. The Krein-Sobolev functional-differential nonlinear equation for the resolvent provides the ground for a unified approach. New results concerning the general analysis of the Krein-Sobolev equation and the convergence and stability of related numerical schemes are also presented.     

Some remarks on holomorphic functions and taylor series in C

Some previous results on convergence of Taylor series in C^n [3] are improved by indicating outside the domain of convergence the points where the series diverges and simplifying some proofs. These results contain the Cauchy-Hadamard theorem in C. Some Cauchy integral formulas of a holomorphic function on a closed ball in C^n are constructed and the Taylor series expansion is deduced.     

Logarithmic residues, generalized idempotents, and sums of idempotents in Banach algebras

In a commutative Banach algebraB the set of logarithmic residues (i.e., the elements that can be written as a contour integral of the logarithmic derivative of an analyticB-valued function), the set of generalized idempotents (i.e., the elements that are annihilated by a polynomial with non-negative integer simple zeros), and the set of sums of idempotents are all the same. Also, these (coinciding) sets consist of isolated points only and are closed under the operations of addition and multiplication. Counterexamples show that the commutativity condition onB is essential. The results extend to logarithmic residues of meromorphicB-valued functions.     

Spline approximations to spherically symmetric distributions

Summary We discuss the problem of approximating a functionf of the radial distancer in ^d on 0r< by a spline function of degreem withn (variable) knots. The spline is to be constructed so as to match the first 2n moments off. We show that if a solution exists, it can be obtained from ann-point Gauss-Christoffel quadrature formula relative to an appropriate moment functional or, iff is suitably restricted, relative to a measure, both depending onf. The moment functional and the measure may or may not be positive definite. Pointwise convergence is discussed asn. Examples are given including distributions from statistical mechanics.The work of the first author was supported in part by the National Science Foundation under grant DCR-8320561.