Uniform approximation of functions on an interval (Q1092245)

The following results are obtained: Theorem 1. If \(\phi\in S(r,k)\) then a function \(f\in W^ rH_ K[\phi]\) exists such that for every \(n\in N\), a polynomial \(P_ n\) exists which satisfies \(| f(x)-P_ n(x)| \leq C \rho^ r_ n(x)\phi (\rho_ n(x)),\) \(x\in [0,1]\), \(C=C(r,k)>0\). Theorem 2. For any function \(\phi \in \Phi^ K\) there is a function \(f\in W^ rH_ K^{\phi}\) such that a) for all \(n\in N\), \(E_ n(f)\leq n^{-2r}\phi (n^{-2}),\) b) \(w_ K(f^{(r)},t)\geq C\phi (t),\) \(t\in [0,1/K]\), \(c=c(r,K)>0\). Theorem 3. Let \(a_ n\) be an increasing sequence of natural numbers, such that \(\sum^{\infty}_{n=1}1/na=\infty\). Then a function f on [0,1] exists for which f \(\in\) \(c^ r[0,1]\) and \(E_ n(f)=O(n^{-2r}/a_ n)\) (r\(\geq 2)\).