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We consider projections from the space of continuous functions on polyhedron with n + 2 vertices in n onto subspace of affine functions. We give necessary and sufficient condition for a projection to be minimal. Then, for each n we find the polyhedron, for which the norm of a minimal projection is maximal. This is a generalization of the result from Shekhtman's and Skrzypek's paper [9], where only the case n = 2 was considered. It is worth noticing that our proof in the even case is significantly different from our proof in the odd case. [ABSTRACT FROM AUTHOR]

In this paper, the study of best simultaneous Lp-approximation by algebraic polynomials on a union of intervals, I, is considered. We obtain convergence of the best simultaneous approximations when the measure of I tends to zero. We also get an interpolation result. [ABSTRACT FROM AUTHOR]

In this paper, we prove an a posteriori and an a priori convergence theorem for Newton-Kantorovich approximations starting from an initial point x0. We apply these results to operators that are analytic at interior points of a closed ball centered at x0 and of radius R. We obtain some theorems on approximate zeros and on approximate zeros of second kind for these operators, which improve previous results. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of whether the best approximation to any x in a Hilbert space X from the set where C is a closed convex subset of X , S is a closed convex cone that does not necessarily have nonempty interior, Y is a Banach space, and g : X ? Y is a continuous S -convex function, can be characterized by the best approximation to a perturbation x - l of x from the set C for some l ? X . We provide a global approach to this problem by presenting a dual global constraint qualification, which is less restrictive than the Slater type (or interior-point) conditions, guaranteeing the strong conical hull intersection property (CHIP). We then show that the strong CHIP characterizes the perturbation property under a mild closure condition. The closure condition, for instance, holds whenever the explicit constraint set is described by finitely many linear inequality constraints. We also establish easily verifiable dual conditions that are equivalent to the best approximation from the set K . We finally demonstrate that our results recapture the corresponding known results in the particular case where Y is a finite dimensional space. [ABSTRACT FROM AUTHOR]

This paper is devoted to the numerical analysis of abstract parabolic differential equations in -spaces. The presentation uses general approximation scheme and is based on semigroup theory and a functional analysis approach. For the solutions of the first and second order of accuracy difference schemes, the almost coercive inequality in spaces with the factor are obtained. In the case of UMD space En we establish a coercive inequality in under the condition of R-boundedness. [ABSTRACT FROM AUTHOR]