A local duality principle for the Baire classes of functions

A local dual of a Banach space X is a closed subspace of X^∗ that satisfies the properties that the principle of local reflexivity assigns to X as a subspace of X∗∗. We show that, for every ordinal 1?α?ω₁, the spaces Bα[0,1] of bounded Baire functions of class α are local dual spaces of the space M[0,1] of all Borel measures. As a consequence, we derive that each annihilator Bα^⊥[0,1] is the kernel of a norm-one projection.     

An approximate Herbrand’s theorem and definable functions in metric structures

We develop a version of Herbrand's theorem for continuous logic and use it to prove that definable functions in infinite‐dimensional Hilbert spaces are piecewise approximable by affine functions. We obtain similar results for definable functions in Hilbert spaces expanded by a group of generic unitary operators and Hilbert spaces expanded by a generic subspace. We also show how Herbrand's theorem can be used to characterize definable functions in absolutely ubiquitous structures from classical logic.     

Best one-sided L 1-approximation by harmonic functions

Let , where B is the open unit ball in (), and let denote the collection of functions h in which are harmonic on B and satisfy on . A function h ^* in is called a best harmonic one-sided L ¹-approximant to f if for all h in . This paper characterizes such approximants and discusses questions of existence and uniqueness. Corresponding results for approximation on the cylinder are also established, but the proofs in this case are more difficult and rely on recent work concerning tangential harmonic approximation. The characterizations are quite different in nature from those recently obtained for harmonic L ¹-approximation without a one-sidedness condition. Received: 25 September 1997     

Gleason's problem for harmonic Bergman and Bloch functions on half-spaces

On the setting of the half-spaceR ^n–1×R ₊, we investigate Gleason's problem for harmonic Bergman and Bloch functions. We prove that Gleason's problem for the harmonicL ^p -Bergman space is solvable if and only ifp>n. We also prove that Gleason's problem for the harmonic (little) Bloch space is solvable.     

Rough isometry and Dirichlet finite harmonic functions on Riemannian manifolds

We prove that the dimension of harmonic functions with finite Dirichlet integral is invariant under rough isometries between Riemannian manifolds satisfying the local conditions, expounded below. This result directly generalizes those of Kanai, of Grigor'yan, and of Holopainen. We also prove that the dimension of harmonic functions with finite Dirichlet integral is preserved under rough isometries between a Riemannian manifold satisfying the same local conditions and a graph of bounded degree; and between graphs of bounded degree. These results generalize those of Holopainen and Soardi, and of Soardi, respectively. Received: 23 July 1998 / Revised version: 10 February 1999     

The dual basis functions for the Bernstein polynomials

An explicit formula for the dual basis functions of the Bernstein basis is derived. The dual basis functions are expressed as linear combinations of Bernstein polynomials. This revised version was published online in June 2006 with corrections to the Cover Date.     

Inclusion relations and convolution properties of a certain class of analytic functions associated with the Ruscheweyh derivatives

Let A denote the class of functions f(z) with

f(0)=f^′(0)−1=0,     

On strong and total Lagrange duality for convex optimization problems

We give some necessary and sufficient conditions which completely characterize the strong and total Lagrange duality, respectively, for convex optimization problems in separated locally convex spaces. We also prove similar statements for the problems obtained by perturbing the objective functions of the primal problems by arbitrary linear functionals. In the particular case when we deal with convex optimization problems having infinitely many convex inequalities as constraints the conditions we work with turn into the so-called Farkas-Minkowski and locally Farkas-Minkowski conditions for systems of convex inequalities, recently used in the literature. Moreover, we show that our new results extend some existing ones in the literature.     

Inclusion and neighborhood properties for certain classes of multivalently analytic functions of complex order associated with the convolution structure

Making use of the familiar convolution structure of analytic functions, in this paper we introduce and investigate two new subclasses of multivalently analytic functions of complex order. Among the various results obtained here for each of these function classes, we derive the coefficient inequalities and other interesting properties and characteristics for functions belonging to the class introduced here.     

Neighborhood properties of certain classes of multivalently analytic functions associated with the convolution structure

In this paper, by making use of the familiar concept of neighborhoods of p-valently analytic functions, we prove coefficient bounds, distortion inequalities and associated inclusion relations for the (n, δ)-neighborhoods of a family of p-valently analytic functions and their derivatives, which is defined by means of a certain general family of non-homogenous Cauchy-Euler differential equations.     

An augmented galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems

Here we present a new solution procedure for Helm-holtz and Laplacian Dirichlet screen and crack problems in IR² via boundary integral equations of the first kind having as an unknown the jump of the normal derivative across the screen or a crack curve T. Under the assumption of local finite energy we show the equivalence of the integral equations and the original boundary value problem. Via the method of local Mellin transform in [5]-[lo] and the calculus of pseudodifferential operators we derive existence, uniqueness and regularity results for the solution of our boundary integral equations together with its explicit behaviour near the screen or crack tips.With our integral equations we set up a Galerkin scheme on T and obtain high quasi-optimal convergence rates by using special singular elements besides regular splines as test and trial functions.     

Bohr radius for locally univalent harmonic mappings

We consider the class of all sense‐preserving harmonic mappings of the unit disk , where h and g are analytic with , and determine the Bohr radius if any one of the following conditions holds:

• 1. h is bounded in .
• 2. h satisfies the condition in with .
• 3. both h and g are bounded in .
• 4. h is bounded and .

We also consider the problem of determining the Bohr radius when the supremum of the modulus of the dilatation of f in is strictly less than 1. In addition, we determine the Bohr radius for the space of analytic Bloch functions and the space of harmonic Bloch functions. The paper concludes with two conjectures.     

Integral means and Dirichlet integral for analytic functions

For normalized analytic functions f in the unit disk, the estimate of the integral means is important in certain problems in fluid dynamics, especially when the functions are non‐vanishing in the punctured unit disk . We consider the problem of finding the extremal function f which maximizes the integral means for f belong to certain classes of analytic functions related to sufficient conditions of univalence. In addition, for certain subclasses of the class of normalized univalent and analytic functions, we solve the extremal problem for the Yamashita functional where denotes the area of the image of under . The first problem was originally discussed by Gromova and Vasil'ev in 2002 while the second by Yamashita in 1990.     

Generalized symmetric functions and invariants of matrices

We generalize the classical isomorphism between symmetric functions and invariants of a matrix. In particular, we show that the invariants over several matrices are given by the abelianization of the symmetric tensors over the free associative algebra. The main result is proved by finding a characteristic free presentation of the algebra of symmetric tensors over a free algebra. The author is supported by research grant Politecnico di Torino n.119, 2004.     

Asymptotic expansion for harmonic functions in the half‐space with a pressurized cavity

In this paper, we address a simplified version of a problem arising from volcanology. Specifically, as a reduced form of the boundary value problem for the Lamé system, we consider a Neumann problem for harmonic functions in the half‐space with a cavity C. Zero normal derivative is assumed at the boundary of the half‐space; differently, at ?C, the normal derivative of the function is required to be given by an external datum g, corresponding to a pressure term exerted on the medium at ?C. Under the assumption that the (pressurized) cavity is small with respect to the distance from the boundary of the half‐space, we establish an asymptotic formula for the solution of the problem. Main ingredients are integral equation formulations of the harmonic solution of the Neumann problem and a spectral analysis of the integral operators involved in the problem. In the special case of a datum g, which describes a constant pressure at ?C, we recover a simplified representation based on a polarization tensor. Copyright © 2016 John Wiley & Sons, Ltd.     

On the structure of function spaces in optimal recovery of point functionals for ENO-schemes by radial basis functions

Summary. Radial basis functions are used in the recovery step of finite volume methods for the numerical solution of conservation laws. Being conditionally positive definite such functions generate optimal recovery splines in the sense of Micchelli and Rivlin in associated native spaces. We analyse the solvability to the recovery problem of point functionals from cell average values with radial basis functions. Furthermore, we characterise the corresponding native function spaces and provide error estimates of the recovery scheme. Finally, we explicitly list the native spaces to a selection of radial basis functions, thin plate splines included, before we provide some numerical examples of our method. Received March 14, 1995