One-sided approximation by entire functions

Let f:R→Rf:R→R have an n  th derivative of finite variation Vf(n)Vf(n) and a locally absolutely continuous (n-1)(n-1)st derivative. Denote by E±(δ,f)pE±(δ,f)p the error of onesided approximation of f   (from above and below, respectively) by entire functions of exponential type δ>0δ>0 in Lp(R)Lp(R)-norm. For 1≤p≤∞1≤p≤∞ we show the estimateE±(δ,f)p⩽Cn1-1/pπ1/pVf(n)δ-n-1p,with constants Cn>0Cn>0.