Solotaree's approximation problem is that of obtaining the best uniform approximation on the interval-1,+1] of a real polynomial of degree n by one of degree n ? 2 or less. Without loss of generalitywe take the approximated polynomial to be x n +rx n?1 , r 0. We treat r as a parameter and seek to compute the coeecients of the best approximations, for small xed values of n, as piecewise algebrai...

Best rational approximations are notoriously difficult to compute. However, the difference between the best rational approximation to a function and its Carathéodory-Fejér (CF) approximation is often so small as to be negligible in practice, while CF approximations are far easier to compute. We present a robust and fast implementation of this method in the Chebfun software system and illustrate...

In this paper we present two novel generic schemes for approximation algorithms for optimization NP-hard graph problems constrained to partial k-trees. Our first scheme yields deterministic polynomialtime algorithms achieving typically an approximation factor of k/ log n, where k = polylog(n). The second scheme yields randomized polynomialtime algorithms achieving an approximation factor of k/ ...

A general formula is developed which gives the "best" approximation for any linear functional on the Hardy space H1. Some "best" approximations are given for interpolation, differentiation, and integration and are compared to polynomial approximations.

A new method is presented for near-best approximation of a real function F on [-r, r] by a polynomial of degree m. The method is derived by transplanting the given problem to the unit disk, then applying the Carath6odory-Fej6r theorem. The resulting near-best approximation is constructed from the principal eigenvalue and eigenvector of a Hankel matrix of Chebyshev coefficients of F. It is well ...

We obtain a sucint and nesessery theoreoms simple for compactness andweakly compactness of the best approximate sets by closed unit balls. Also weconsider relations Kadec-Klee property and shur property with this objects.These theorems are extend of papers mohebi and Narayana.