where ‖ · ‖K denotes the sup norm on K ⊆ R. It is well known that the best approximant r∗ mn exists and is unique within Rmn (cf. [Me, §§9.1, 9.2] or [Ri, §5.1]). The unique existence also holds in the special case (n = 0) of best polynomial approximants. Since fα(x) := |x|α is an even function on [−1, 1], the same is true for its unique approximant r∗ mn = r ∗ mn(fα, [−1, 1]; ·), and consequen...

In this paper, the problem of best uniform polynomial approximation to a continuous function on a compact set X is approached through the partitioning of X and the definition of norms corresponding to the partition and each of the standard Lp norms 1 g p < oo. For computational convenience, a pseudo norm is defined corresponding to each partition. When the partition is chosen appropriately, the...

In this paper, a parameter uniform numerical method based on Shishkin mesh is suggested to solve a system of second order singularly perturbed differential equations with a turning point exhibiting boundary layers. It is assumed that both equations have a turning point at the same point. An appropriate piecewise uniform mesh is considered and a classical finite difference scheme is applied on t...

In this paper we describe an experimental technique for computing realistic values of the parameter–uniform order of convergence and error constant in the maximum norm associated with a parameter–uniform numerical method for solving singularly perturbed problems. We employ the technique to compute Reynolds– uniform error bounds in the maximum norm for the numerical solutions generated by a fitt...

We introduce and study the noncommutative modular function spaces of measurable operators affiliated with a semifinite von Neumann algebra and show that they are complete with respect to their modular. We prove that these spaces satisfy the uniform Opial condition with respect to ρ̃-a.e.-convergence for both the Luxemburg norm and the Amemiya norm. Moreover, these spaces have the uniform Kadec–K...

exists for each x, y ∈ S(E). In this case E is called smooth. The norm of E is said to be Fréchet differentiable if for each x ∈ S(E) the limit is attained uniformly for y ∈ S(E). The normofE is called uniformlyFréchet differentiable if the limit is attained uniformly for x, y ∈ S(E). It is well known that (uniform) Fréchet differentiability of the norm of E implies (uniform) Gâteaux differenti...

Recently, using DiGiorgi-type techniques, Caffarelli and Vasseur [1] showed that a certain class of weak solutions to the drift diffusion equation with initial data in L gain Hölder continuity provided that the BMO norm of the drift velocity is bounded uniformly in time. We show a related result: a uniform bound on BMO norm of a smooth velocity implies uniform bound on the C norm of the solutio...

Here we are concerned with the best approximation by polynomials to rational functions in the uniform norm. We give some new theorems about the best approximation of 1/(1 + x) and 1/(x − a) where a > 1. Finally we extend this problem to that of computing the best approximation of the Chebyshev expansion in uniform norm and give some results and conjectures about this.  2005 IMACS. Published by...

We introduce a new concept of perturbation of closed linear subspaces and operators in Banach spaces called uniform λ−adjustment which is weaker than perturbations by small gap, operator norm, q−norm, and K2−approximation. In arbitrary Banach spaces some of the classical Fredholm stability theorems remain true under uniform λ−adjustment, while other fail. However, uniformly λ−adjusted subspaces...