Forward and converse theorems of polynomial approximation for exponential weights on \([-1,1]\). I (Q1369239)

The author proves Jackson theorems in weighted \(L_p\)-spaces on \([-1,1]\), \(0<p\leq \infty\), where the weights have not a behaviour like Jacobi weights, but are exponential functions of the form \(\exp(-Q(x))\), \(Q\) an even function that grows faster than \((1-x^2)^{-\alpha}\) near \(\pm 1\) for some \(\alpha >0\). These Jackson theorems are given in terms of moduli of continuity defined in the paper and require many previous technical results. The author announces a second part for the relationship of moduli of the continuity with the theory of \(K\)-functionals as well as for the study of the converse theorems.