Additive Functions with Respect to Numeration Systems on Regular Languages

Asymptotic formulae for the summatory function of additive arithmetic functions related to numeration systems given by regular languages are derived.This author is supported by the START project Y96-MAT of the Austrian Science Fund.This author warmly thanks P. Grabner for his kind invitation to Graz and his hospitality.Received February 18, 2002; in revised form September 20, 2002 Published online May 15, 2003     

On the Algebra Generated by the Bergman Projection and a Shift Operator I

Let be a domain with smooth boundary and let α be a C ²- diffeomorphism on satisfying the Carleman condition .We denote by the C*-algebra generated by the Bergman projection of G, all multiplication operators aI and the operator where is the Jacobian of α. A symbol algebra of is determined and Fredholm conditions are given. We prove that the C*-algebra generated by the Bergman projection of the upper half-plane and the operator is isomorphic and isometric to . Submitted: February 11, 2001?Revised: January 27, 2002     

Toeplitz Operators Associated to Unimodular Algebras

We introduce a class of function algebras, that we call unimodular, and study Toeplitz operators on the Hardy spaces associated to representing measures on these algebras.We show that our class of function algebras is very extensive and that a number of important results for Toeplitz operators and their associated C*-algebras extend to the very general setting we consider. Submitted: July 1. 2001.     

Dirichlet and Neumann boundary conditions: What is in between?

Given an admissible measure μ on where is an open set, we define a realization of the Laplacian in with general Robin boundary conditions and we show that generates a holomorphic C ₀ -semigroup on which is sandwiched by the Dirichlet Laplacian and the Neumann Laplacian semigroups. Moreover, under a locality and a regularity assumption, the generator of each sandwiched semigroup is of the form . We also show that if contains smooth functions, then μ is of the form (where σ is the (n-1)-dimensional Hausdorff measure and β a positive measurable bounded function on ); i.e. we have the classical Robin boundary conditions. RID="h1" ID="h1"Dédié à Philippe Bénilan RID="*" ID="h1"This work is part of the DGF-Project: "Regularit?t und Asymptotik für elliptische und parabolische Probleme".     

Maximal <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>‐regularity for elliptic operators with VMO‐coefficients

In this paper we show that the parabolic problem associated with elliptic operators A having coefficients in has the property of maximal -regularity. RID="h1" ID="h1"In Memoriam Philippe Bénilan     

Asymptotics of Differentiated Bernstein Polynomials

It is shown that the m th-order derivative of the n th-order Bernstein polynomial of a function f satisfying a certain Lipschitz condition, can be written for n\rightarrow +∈fty as a singular integral of Gauss—Weierstrass type, m times differentiated (in a certain sense) under the integral sign. The theorem is applied to yield an overdifferentiation formula, involving p times differentiated Bernstein polynomials of functions that are not C ^p . December 1, 1998. Dates revised: July 22, 1999 and January 11, 2000. Date accepted: February 1, 2000.     

Perturbation of orthonormal bases inL 2-spaces

The paper improves and generalizes a classical result from Paley and Wiener in their book on Fourier transforms. Paley and Wiener gave conditions on functionsh _n that imply that the sequence (1+h _n (x))e ^inx is a Riesz basis forL ²[–,]. These conditions involve theL ²-norm of the second derivativesh _n . The new results replace the differential operatoryy by more general differential operators inL ²-spaces, in particular, by the Hermite differential operator inL ²(R), ande ^inx by arbitrary orthonormal bases.     

A functional model for polynomially posinormal operators

In this paper we introduce a class of operators naturally extending the classes of hyponormal and posinormal operators. For this class we construct a generating family of eigendistributions, unitary invariants and a functional model.     

Banach space duality of absolute Schur algebras

Matrix Schur product is the entry-wise product of matrices of the same size. It was shown by P. Chaisuriya and S.-C. Ong [1] that (forr1) infinite matrices [a _jk ] such that [|a _jk | ^r ] B(l ² form a Banach algebra under the norm [a _jk ] _r =[|a _jk | ^r ]^1/r and the Schur product. In this paper we demonstrate the existence of Banach space duality within the class of these algebras which is analogous to the classical duality between the spaces of compact, trace class, and bounded operators onl ². Also we obtain a general functional calculus on these algebras, which is used to determine the spectrum and to justify the notion of -norm introduced in [1].This author's research was supported by a sabbatical leave and a Division grant from Colby College.This author's research was supported by a research professorship at Central Michigan University.This author's research was supported by a visiting professorship at Central Michigan University.     

Mean curvature one surfaces in hyperbolic space, and their relationship to minimal surfaces in Euclidean space

We describe local similarities and global differences between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. We also describe how to solve global period problems for constant mean curvature 1 surfaces in hyperbolic 3-space, and we give an overview of recent results on these surfaces. We include computer graphics of a number of examples.