The pointwise dimensions of Moran measures

In this paper,we get the formulas of upper(lower) pointwise dimensions of some Moran measures on Moran sets in Rd under the strong separation condition.We also obtain formulas for the dimension of the Moran measures.Our results extend the known results of some self-similar measures and Moran measures studied by Cawley and Mauldin.     

On pointwise estimates for the Littlewood-Paley operators

In a recent paper we proved pointwise estimates relating some classical maximal and singular integral operators. Here we show that inequalities essentially of the same type hold for the Littlewood-Paley operators.

     

The strong approximation property and the weak bounded approximation property

We show that the strong approximation property (strong AP) (respectively, strong CAP) and the weak bounded approximation property (respectively, weak BCAP) are equivalent for every Banach space. This gives a negative answer to Oja's conjecture. As a consequence, we show that each of the spaces _c0$c_0$ and _?1${?}_1$ has a subspace which has the AP but fails to have the strong AP.     

On asymptotic pointwise nonexpansive mappings in modular function spaces

We prove the existence of fixed points of asymptotic pointwise nonexpansive mappings in modular function spaces.     

Pointwise strong and very strong approximation of Fourier series

We present an estimation of the and H _u ^λφ f means as approximation versions of the Totik type generalization (see [6, 7]) of the result of G. H. Hardy, J. E. Littlewood, considered by N. L. Pachulia in [5]. Some results on the norm approximation will also be given.      

On Dirichlet and Neumann problems for harmonic functions

The aim of the paper is to examine some aspects of the boundary value problems for harmonic functions in half-spaces related to approximation theory. M. V. Keldyshmentioned curious fact on richness in some sense of the solutions of Dirichlet problem in upper half-plane for a fixed continuous boundary data on the real axis. This can be considered as a model version for the Dirichlet problem with continuous boundary data, defined except a single boundary point, with no restrictions imposed on solutions near that point.Some extensions and multi-dimensional versions of Keldysh’s richness are obtained and related questions on existence, representation and richness of solutions for the Dirichlet and Neumann problems discussed.     

Clifford algebra approach to pointwise convergence of Fourier series on spheres

We offer an approach by means of Clifford algebra to convergence of Fourier series on unit spheres of even-dimensional Euclidean spaces. It is based on generalizations of Fueter’s Theorem inducing quaternionic regular functions from holomorphic functions in the complex plane. We, especially, do not rely on the heavy use of special functions. Analogous Riemann-Lebesgue theorem, localization principle and a Dini’s type pointwise convergence theorem are proved. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

On the invariant translation approximation property for discrete groups

Recently J.Roe considered the question of whether for a discrete group the reduced group -algebra is the fixed point algebra of Ad acting on the uniform Roe algebra . is said to have the invariant translation approximation property in this case. We consider a slight generalization of this property which, for exact , is equivalent to the operator space approximation property of . We also give a new characterization of exactness and a short proof of the equivalence of exactness of and exactness of for discrete groups.

     

A pointwise estimate on the 1-order approximation of G{varepsilon}x0

On the basis of Avellaneda & Hua-Lin (1991, Commun. PureAppl. Math., 44 897–910), a pointwise error estimate onthe 1-order approximation of Green function G_x0 defined in R²is shown at first. Then based on this estimate and using asymptoticexpansion method, an improved approximation of G_x0 and its pointwiseerror estimate are obtained.     

ALTERNATION THEORY IN APPROXIMATION BY POLYNOMIALS HAVING BOUNDED COEFFICIENTS

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A generalization of Dirichlet approximation theorem for the affine actions on real line

In this paper, we obtain strong density results for the orbits of real numbers under the action of the semigroup generated by the affine transformations T₀(x)=x/a and T₁(x)=bx+1, where a,b>1. These density results are formulated as generalizations of the Dirichlet approximation theorem and improve the results of Bergelson, Misiurewicz, and Senti. We show that for any x,u>0 there are infinitely many elements γ in the semigroup generated by T₀ and T₁ such that |γ(x)−u|<C(t1/|γ|−1), where C and t are constants independent of γ, and |γ| is the length of γ as a word in the semigroup. Finally, we discuss the problem of approximating an arbitrary real number by the ratios of prime numbers and the ratios of logarithms of prime numbers.     

Numerical analysis for the approximation of optimal control problems with pointwise observations

In this paper, we study the numerical methods for optimal control problems governed by elliptic PDEs with pointwise observations of the state. The first order optimality conditions as well as regularities of the solutions are derived. The optimal control and adjoint state have low regularities due to the pointwise observations. For the finite dimensional approximation, we use the standard conforming piecewise linear finite elements to approximate the state and adjoint state variables, whereas variational discretization is applied to the discretization of the control. A priori and a posteriori error estimates for the optimal control, the state and adjoint state are obtained. Numerical experiments are also provided to confirm our theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

Pointwise simultaneous approximation by left gamma quasi-interpolants

Recently some classical operator quasi-interpolants were introduced to obtain much faster convergence.We consider left Gamma quasi-interpolants and give a pointwise simultaneous approximation equivalence theorem with ω2r(4)λ(f,f)∞ by means of unified the classical modulus and Ditzian-Totick modulus.     

Three-space problems for the bounded compact approximation property

In this paper, the notion of the bounded compact approximation property (BCAP) of a pair [Banach space and its subspace] is used to prove that if X is a closed subspace of L∞ with the BCAP, then L∞/X has the BCAP. We also show that X* has the λ-BCAP with conjugate operators if and only if the pair (X, Y) has the λ-BCAP for each finite codimensional subspace Y∈X. Let M be a closed subspace of X such that M⊥ is complemented in X*. If X has the (bounded) approximation property of order p, then M has the (bounded) approximation property of order p.     

On pointwise <Emphasis Type="Italic">R</Emphasis>-subweakly commuting maps and best approximations

The main purpose of this paper is to prove some common fixed point theorems for pointwise R-subweakly commuting maps on non-starshaped domains in P-normed spaces and locally convex topological vector spaces.As applications,invariant approximation results ale established.This work provides extension as well as substantial improvement of several results in the existing literature.     

Rate of pointwise approximation for locally bounded functions by Szász operators

In this paper the asymptotic behavior of Szász operators for locally bounded functions f is studied at points x where f(x+) and f(x−) exist. An asymptotic estimate of this type approximation is obtained by using some techniques and results of probability theory. The estimate essentially is the best possible for continuous points of f.     

On the optimal approximation of bounded linear functionals in Hilbert spaces of analytic functions

Optimal numerical approximation of bounded linear functionals by weighted sums in Hilbert spaces of functions analytic in a circleK _r , in a circular annulusK _r1,r2 and in an ellipseE _r is investigated by Davis' method on the common algebraic background for diagonalising the normal equation matrix. The weights and error functional norms for optimal rules with nodes located angle-equidistant on the concentric circleK _s or on the confocal ellipseE _s and in the interval [–1,1] for an arbitrary bounded linear functional are given explicitly. They are expressed in terms of a complete orthonormal system in the Hilbert space.     

On meromorphic approximation

Let G be a bounded N-connected domain, the boundary of which consists of closed analytic Jordan curves. We assume that 0G. For any nonnegative integers n and m, denote by _n,m the class of all meromorphic functions on G that can be represented in the form h=p/qz ^m , where p belongs to the Smirnov class E (G), q is a polynomial of degree at most n, q0. Theorems giving necessary and sufficient conditions for a function belonging to the class _n,m to be an element of best approximation to a continuous function f on in the space L () by functions in the class _n,m are proved. Some questions concerning orthogonal polynomials and the theory of Hankel operators are also considered.