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## Completeness of Spaces of Harmonic Functions under Restricted Supremum Norms

### Computational Methods and Function Theory (2004-08-01) 4: 43-45 , August 01, 2004

Let *E* be a subset of a domain Ω in Euclidean space. This note verifies a conjecture of Arcozzi and Björn concerning the completeness of the space of harmonic functions *u* on Ω that are bounded on *E*, where the supremum norm is taken with respect to the restriction of *u* to *E*.

## Antiproximinal convex bounded sets in the space c0(Γ) equipped with the day norm

### Mathematical Notes (2006-03-01) 79: 299-313 , March 01, 2006

We construct a convex smooth antiproximinal set in any infinite-dimensional space *c*_{0}(Γ) equipped with the Day norm; moreover, the distance function to the set is Gâteaux differentiable at each point of the complement.

## Uniform approximations of Stieltjes functions by orthogonal projection on the set of rational functions

### Mathematical Notes (1999-03-01) 65: 302-307 , March 01, 1999

Let μ be a positive Borel measure having support supp μ ⊂ [1, ∞) and satisfying the condition*f*(t−1)^{−1}dμ(t)<∞. In this paper we study the order of the uniform approximation of the function
$$\widehat\mu = \smallint \tfrac{{d\mu (t)}}{{t - z}}, z \in \mathbb{C},$$
on the disk |*z*|≤1 and on the closed interval [−1, 1] by means of the orthogonal projection of
$$\widehat\mu $$
on the set of rational functions of degree*n*. Moreover, the poles of the rational functions are chosen depending on the measure μ. For example, it is shown that if supp μ is compact and does not contain 1, then this approximation method is of best order. But if supp μ=[1,*a*],*a*>1, the measure μ is absolutely continuous with respect to the Lebesgue measure, and
$$\mu '\left( t \right) _\frown ^\smile \left( {t - 1} \right)^\alpha $$
for*t*∈[1,*a*] and some α>0, then the order of such an approximation differs from the best only by
$$\sqrt n $$
.

## On a Continuous Mapping and Sharp Triangle Inequalities

### Inequalities and Applications 2010 (2012-01-01) 161: 125-136 , January 01, 2012

This is a survey on some recent results concerning the sharp triangle inequalities. Our results refine and generalize the corresponding ones in (Kato et al. in Math. Inequal. Appl. 10(2), 451–460 (2007)) and (Mitani et al. in J. Math. Anal. Appl. 10(2), 451–460 (2007)).

## Generalized measures of noncompactness of sets and operators in Banach spaces

### Acta Mathematica Hungarica (2010-11-01) 129: 227-244 , November 01, 2010

New measures of noncompactness for bounded sets and linear operators, in the setting of abstract measures and generalized limits, are constructed. A quantitative version of a classical criterion for compactness of bounded sets in Banach spaces by R. S. Phillips is provided. Properties of those measures are established and it is shown that they are equivalent to the classical measures of noncompactness. Applications to summable families of Banach spaces, interpolations of operators and some consequences are also given.

## Gaussian Version of a Theorem of Milman and Schechtman

### Positivity (1997-03-01) 1: 1-5 , March 01, 1997

Using Gordon's inequalities, we give a short proof of the existence of an embedding T:l2k to l∞n such that ||T||||T-1|| ≤ c √k/ln(1+n/k). In the same way, we give a new proof of a theorem of Milman and Schechtman (1995).

## On the singularities of the gradient of the solution to the Dirichlet-Neumann problem outside a plane cut

### Mathematical Notes (2005-03-01) 77: 335-347 , March 01, 2005

The problem considered in this paper deals with the Laplace equation outside a cut of sufficiently arbitrary form. The Dirichlet condition is given on one side of the cut and the Neumann condition on the other. Using the integral representation for the solution, we obtain explicit asymptotic formulas describing the singularity of the gradient of the solution at the edges of the cut. We discuss the effect of the disappearance of the singularity.

## Convergence Domains for Some Iterative Processes in Banach Spaces Using Outer or Generalized Inverses

### Journal of Computational Analysis and Applications (1999-01-01) 1: 87-104 , January 01, 1999

We provide a semilocal Ptak–Kantorovich-type analysis for inexact Newton-like methods using outer and generalized inverses to approximate a locally unique solution of an equation in a Banach space containing a nondifferentiable term. We use Banach-type lemmas and perturbation bounds for outer as well as generalized inverses to achieve our goal. In particular we determine a domain Ω such that starting from any point of Ω our method converges to a solution of the equation. Our results can be used to solve undetermined systems, nonlinear least-squares problems, and ill-posed nonlinear operator equations in Banach spaces. Finally, we provide two examples to show that our results compare favorably with earlier ones.

## Greedy algorithms in Banach spaces

### Advances in Computational Mathematics (2001-04-01) 14: 277-292 , April 01, 2001

We study efficiency of approximation and convergence of two greedy type algorithms in uniformly smooth Banach spaces. The Weak Chebyshev Greedy Algorithm (WCGA) is defined for an arbitrary dictionary *D* and provides nonlinear *m*-term approximation with regard to *D*. This algorithm is defined inductively with the *m*th step consisting of two basic substeps: (1) selection of an *m*th element ϕ
_{m}^{c}
from *D*, and (2) constructing an *m*-term approximant *G*_{m}^{c}
. We include the name of Chebyshev in the name of this algorithm because at the substep (2) the approximant *G*_{m}^{c}
is chosen as the best approximant from Span(ϕ
_{1}^{c}
,...,ϕ
_{m}^{c}
). The term Weak Greedy Algorithm indicates that at each substep (1) we choose ϕ
_{m}^{c}
as an element of *D* that satisfies some condition which is “*t*_{m}-times weaker” than the condition for ϕ
_{m}^{c}
to be optimal (*t*_{m}=1). We got error estimates for Banach spaces with modulus of smoothness ρ(*u*)≤γ*u*^{q}, 1<*q*≤2. We proved that for any *f* from the closure of the convex hull of *D* the error of *m*-term approximation by WCGA is of order (1+*t*_{1}^{p}+⋅⋅⋅+*t*_{m}^{p})^{−1/p}, 1/*p*+1/*q*=1. Similar results are obtained for Weak Relaxed Greedy Algorithm (WRGA) and its modification. In this case an approximant *G*^{r}_{m} is a convex linear combination of 0,ϕ_{1}^{r},...,ϕ^{r}_{m}. We also proved some convergence results for WCGA and WRGA.

## Asymptotic Solution of a Nonlinear Initial–Boundary Value Problem for the Diffusion Equation with a Small Parameter Multiplying the Time Derivative

### Computational Mathematics and Modeling (2014-01-01) 25: 9-26 , January 01, 2014

We prove the existence and construct the form of the asymptotic solution for an initial–boundary value problem that describes diffusive filling of thin shells with a real gas. Such problems arise in the manufacturing of laser targets, where a thin spherical shell is filled with hydrogen isotopes to a high pressure [1–3]. In this article we apply the methods of [4], but the nonlinearity of the boundary conditions requires new approaches to the problem.