Expansion by polynomials with respect to Freud-type weights

The weights Wα(x)=exp(−12|x|α),α>1, form an important subclass of Freud weights. For the expansion by orthogonal polynomials with respect to Wα(x)Wα(x), we prove analogues of the Hausdorff–Young and the Hardy–Littlewood inequalities. The special cases of expansion by Hermite functions (α=2)(α=2), which are new as well, are given explicitly.