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.

     

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.     

Some C*-algebras properties preserved by tracial approximation

We show that the Riesz interpolation property of the K ₀-monoid of C*-algebras in the class Ω is inherited by simple unital C*-algebras in the class TAΩ, and the property of being an admissible target algebras of finite type in the class of Ω is inherited by unital C*-algebras in the class TAΩ.     

On best rational approximation of analytic functions

This paper contains some theorems related to the best approximation ρ_n(f;E) to a function f in the uniform metric on a compact set by rational functions of degree at most n. We obtain results characterizing the relationship between ρ_n(f;K) and ρ_n(f;E) in the case when complements of compact sets K and E are connected, K is a subset of the interior Ω of E, and f is analytic in Ω and continuous on E.     

On 3-monotone approximation by piecewise polynomials

We consider 3-monotone approximation by piecewise polynomials with prescribed knots. A general theorem is proved, which reduces the problem of 3-monotone uniform approximation of a 3-monotone function, to convex local L₁ approximation of the derivative of the function. As the corollary we obtain Jackson-type estimates on the degree of 3-monotone approximation by piecewise polynomials with prescribed knots. Such estimates are well known for monotone and convex approximation, and to the contrary, they in general are not valid for higher orders of monotonicity. Also we show that any convex piecewise polynomial can be modified to be, in addition, interpolatory, while still preserving the degree of the uniform approximation. Alternatively, we show that we may smooth the approximating piecewise polynomials to be twice continuously differentiable, while still being 3-monotone and still keeping the same degree of approximation.     

On pointwise bounded approximation

Let G be an open subset in the extended complex plane and let A（G） denote the algebra of all functions analytic on G and continuous on G. We call a domain multi-nicely connected if there is a circular domain W and a conformal map ~ from W onto G such that the boundary value function of φ is univalent almost everywhere with respect to the arclength on aW. Suppose that every component of G is finitely connected and none of the components of G have single point boundary components. We show that for every bounded analytic function on G to be the pointwise limit of a bounded sequence of functions in A（G）, it is necessary and sufficient that each component of G is multi-nicely connected and the harmonic measures of G are mutually singular. This generalizes the corresponding result of Davie for the case when the components of G are simply connected.     

GENERALIZED I-NONEXPANSIVE MAPS AND INVARIANT APPROXIMATION RESULTS IN p-NORMED SPACES

We extend the concept of R-subcommuting maps due to Shahzad[17,18] to the case of non-starshaped domain and obtain a common fixed point result for this class of maps on non-starshaped domain in the setup of p-normed spaces. As applications, we establish noncommutative versions of various best approximation results for generalized I-nonexpansive maps on non-starshaped domain. Our results unify and extend that of Al-Thagafi, Dotson, Habiniak,Jungck and Sessa, Latif, Sahab, Khan and Sessa and Shahzad.     

On an approximation property of Pisot numbers

Let 1<q<2 be a real number, m≥1 be a rational integer and l_m(q)={|P(q)|,P∈Z[X],P(q)≠0,H(P)≤m}, where Z[X] denotes the set of polynomials P with rational integer coefficients and H(P) is the height of P. The value of l_m(q) was determined for many particular Pisot numbers ([3] and [7]). In this paper we determine the infimum and the supremum of the numbers l_m(q) for any fixed m. We also determine the greatest limit point for the case m=1. This revised version was published online in June 2006 with corrections to the Cover Date.     

On tractability of approximation in special function spaces

We consider approximation problems for a special space of d$d$ variate functions. We show that the problems have small number of active variables, as it has been postulated in the past using concentration of measure arguments. We also show that, depending on the norm for measuring the error, the problems are strongly polynomially or quasi-polynomially tractable even in the model of computation where functional evaluations have the cost exponential in the number of active variables.     

On weighted polynomial approximation with monotone weights

We construct an even weight monotone on the right half line such that the logarithmic integral of the largest -convex minorant of converges and the polynomials are dense in .     

On strong approximation applied to post-widder operators

We introduce modified Post-Widder operators in polynomial weighted spaces of differentiable functions and we study strong approximation for them.     

On relations between weak approximation properties and their inheritances to subspaces

It is shown that for the separable dual X^∗ of a Banach space X, if X^∗ has the weak approximation property, then X^∗ has the metric weak approximation property. We introduce the properties W^∗D and MW^∗D for Banach spaces. Suppose that M is a closed subspace of a Banach space X such that M^⊥ is complemented in the dual space X^∗, where for all m∈M}. Then it is shown that if a Banach space X has the weak approximation property and W^∗D (respectively, metric weak approximation property and MW^∗D), then M has the weak approximation property (respectively, bounded weak approximation property).     

On feasible sets defined through Chebyshev approximation

LetZ be a compact set of the real space with at leastn + 2 points;f,h1,h2:Z continuous functions,h1,h2 strictly positive andP(x,z),x(x ₀,...,x _n ) ⁿ⁺¹,z , a polynomial of degree at mostn. Consider a feasible setM {x ⁿ⁺¹z Z, –h ₂(z) P(x, z)–f(z)h ₁(z)}. Here it is proved the null vector 0 of ⁿ⁺¹ belongs to the compact convex hull of the gradients ± (1,z,...,z ⁿ ), wherez Z are the index points in which the constraint functions are active for a givenx* M, if and only ifM is a singleton.This work was partially supported by CONACYT-MEXICO.     

On lower bounds in radial basis approximation

We investigate the approximation by manifolds _n() generated by linear combinations of n radial basis functions on R^d of the form (|–a|), where is the thin-plate spline type function. We obtain exact asymptotic estimates for the approximation of Sobolev classes W^r(B^d) in the space L(B^d) on the unit ball B^d. AMS subject classification 41A25, 41A63, 65D07, 41A15     

Tracial Limit of C^＊-algebras

A new limit of C^*-algebras, the tracial limit, is introduced in this paper. We show that a separable simple C^*-algebra A is a tracial limit of C^*-algebras in Z-(k) if and only if A has tracial topological rank no more than k. We present several known results using the notion of tracial limits.     

On Runge approximation in a Banach space

We show that on some open sets, more general than balls, Runge approximation is possible in certain Banach spaces, and also in certain complex Banach manifolds. We also show that there is an entire holomorphic curve in Hilbert space on which there is a bounded holomorphic function on the trace of a ball that has no bounded holomorphic extension to even a smaller concentric ball. Using the same technique we also prove that a form of Runge approximation better than an error function is not always possible.     

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.     

NEAR MINIMAX POLYNOMIAL APPROXIMATION

In this paper a representation for linear polynomial projection is given. Then it isshown that the near minimax approximation is equivalant to a problem of the best approxima-tion of hivariste functions. And some properties for near minimax are obtained.     

Lipschitz continuity of the best approximation operator in vector-valued Chebyshev approximation

When G is a finite dimensional Haar subspace of C(X,R^k), the vector-valued continuous functions (including complex-valued functions when k is 2) from a finite set X to Euclidean k-dimensional space, it is well-known that at any function f in C(X,R^k) the best approximation operator satisfies the strong unicity condition of order 2 and a Lipschitz (Hőlder) condition of order . This note shows that in fact the best approximation operator satisfies the usual Lipschitz condition of order 1.