Diagonal recovery from noisy Fourier--Hermite coefficients: exact orders of the error

We consider the recovery of functions of the weighted Wiener classes $W_s^{\mu }$ from perturbed Fourier–Hermite coefficients in the metric of $L_{\mathrm{\infty }}(\mathbb{R})$. For the summation methods of the class ${\mathcal{S}}^{\theta }$ we find error estimates on $W_s^{\mu }$. The estimates take into account the behaviour of the Hermite functions both in the oscillatory region and near the turning point: the maxima of functions with different indices are attained at different points, and the supremum of a sum over a block of indices is substantially smaller than the sum of the suprema. Owing to this, in particular, we establish a common lower bound for all finite diagonal methods; for $s\mathrm{\ne }4\mathrm{/}3$ and $p\mathrm{\ne }4\mathrm{/}3$ it matches the upper bound and gives the exact order of the error in this class, while at the critical points the gap between the bounds does not exceed $(\log (e\mathrm{/}\delta ){)}^{1\mathrm{/}4}$ up to a constant. If, in addition, $s>4\mathrm{/}3$ or $p<4\mathrm{/}3$, then this order is also optimal among all recovery operators that use the whole sequence of data.