Location of best simultaneous approximations

Amir and Ziegler have proved that a real normed space E of dimension ≥ 3 is an inner product space if and only if, for any x,y ε E and any two dimensional subspace M of E, the expression max(‖x — w‖,‖y — w‖) attains its minimum in some point w of each segment [u,v], with u and v, respectively, best approximations to x and y from M. We extend this result to expressions π(‖x — w‖,‖y — w‖), where π denotes a monotonic norm in R²     

Stabilized approximations of strongly continuous semigroups

This paper introduces stabilization techniques for intrinsically unstable, high accuracy rational approximation methods for strongly continuous semigroup. The methods not only stabilize the approximations, but improve their speed of convergence by a magnitude of up to 1/2.     

On the computational complexity of best Chebyshev approximations

The best Chebyshev approximation of degree n to a continuous function f on [0, 1] is the unique polynomial ϕ of degree less than or equal to n such that the maximum difference of f and ϕ on [0, 1] is minimized. On the basis of a formal model of computation, it is shown that the question of whether the best Chebyshev approximations of polynomial-time computable functions on [0, 1] are always polynomial-time computable depends on the relationship among well-known discrete complexity classes. In particular, P = NP implies that these best approximations are polynomial-time computable, and EXP ≠ NEXP implies that these best approximations are not polynomial-time computable. It is also pointed out that the fact that the popular Remes algorithm converges fast does not conflict with the above result, since the Remes algorithm requires, in each iteration, the finding of maximal points of continuous functions on an interval [a, b], which is, in general, provably intractable.     

Functions with prescribed best linear approximations

A common problem in applied mathematics is that of finding a function in a Hilbert space with prescribed best approximations from a finite number of closed vector subspaces. In the present paper we study the question of the existence of solutions to such problems. A finite family of subspaces is said to satisfy the Inverse Best Approximation Property (IBAP) if there exists a point that admits any selection of points from these subspaces as best approximations. We provide various characterizations of the IBAP in terms of the geometry of the subspaces. Connections between the IBAP and the linear convergence rate of the periodic projection algorithm for solving the underlying affine feasibility problem are also established. The results are applied to investigate problems in harmonic analysis, integral equations, signal theory, and wavelet frames.     

Zeros of Bernoulli-type functions and best approximations

The zero sets of (D+a)ⁿg(t) with in the (t,a)-plane are investigated for and .The results are used to determine entire interpolations to functions , which give representations for the best approximation and best one-sided approximation from the class of functions of exponential type η>0 to .     

Strongly unique spline approximations with free knots

A necessary and a sufficient alternation condition for strongly unique best spline approximations with free knots is given. In the case of simple knots these conditions coincide, and strongly unique best approximations and strongly unique local best approximations are the same. The numerical consequences are discussed.     

Distribution of alternation points in best rational approximations

We study the convergence of counting measures of alternation point sets in best rational approximations to the equilibrium measure. It is shown that, for any prescribed nondecreasing sequence of denominator degrees, there exists a function analytic on [0, 1] and a sequence of numerator degrees such that the corresponding sequence of measures does not converge to the equilibrium measure of the interval.     

CHARACTERIZATION OF BEST APPROXIMATIONS IN METRIC LINEAR SPACES

Let (X,d) be a real metric linear space, with translation-invariant metric d and G a linear subspace of X. In this paper we use functionals in the Lipschitz dual of X to characterize those elements of G which are best approximations to elements of X. We also give simultaneous characterization of elements of best approximation and also consider elements of ε-approximation.     

Characterization of best Chebyshev approximations with prescribed norm

In this paper a new characterization of smooth normed linear spaces is discussed using the notion of proximal points of a pair of convex sets. It is proved that a normed linear space is smooth if and only if for each pair of convex sets, points which are mutually nearest to each other from the respective sets are proximal.