Weighted best local \(\|\cdot\|\)-approximation in Orlicz spaces (Q2895246)

The authors obtain results in the problem of the existence of best multipoint local approximation with respect to the Luxemburg norm of a smooth function from an N-dimensional subspace of an Orlicz space for a suitable integer \(N\). For this purpose, the authors introduce the concept of balanced \(n\)-tuple and balanced integer with respect to the Luxemburg norm, provide an example of balanced integers and prove the uniformly boundness of the nets of the best approximations in Orlicz spaces relative to the Luxemburg norm. Finally, it is proved the existence and uniqueness of the best multipoint local approximation from subspaces with balanced dimension in Orlicz spaces with respect to the Luxemburg norm, approximation which is obtained by a Hermite interpolation. The generalization of this existence and uniqueness result for the case of several variables is stated without proof for both Luxemburg and Orlicz norm.