On Lagrange and Hermite Interpolation in Rk.
The present paper shows that for any sequences of real numbers, each with infinitely many distinct elements, , j=1,...,s, the rational combinations of are always dense in .
We give a generalization of box splines. We prove some of their properties and we give applications to interpolation and approximation of functions.
The approximation in the uniform norm of a continuous function f(x) = f(x₁,...,xₙ) by continuous sums g₁(h₁(x)) + g₂(h₂(x)), where the functions h₁ and h₂ are fixed, is considered. A Chebyshev type criterion for best approximation is established in terms of paths with respect to the functions h₁ and h₂.
Recently it was proved for 1 < p < ∞ that , a modulus of smoothness on the unit sphere, and , a K-functional involving the Laplace-Beltrami operator, are equivalent. It will be shown that the range 1 < p < ∞ is optimal; that is, the equivalence does not hold either for p = ∞ or for p = 1.