Mean-square approximation of functions on the whole axis by algebraic polynomials with the Chebyshev-Hermite weight

We derive exact inequalities of Jackson–Stechkin type between the value $E_{n-1}(f^{(s)})_{2}$ of the best mean-square approximation on $\mathbb{R}$ with the weight $\rho(x)=e^{-x^2}$ of successive derivatives $f^{(s)}$, $s=0,1,...,r$, of functions $f\in L_{2,\rho}^{(r)}(\mathbb{R})$ and average values of $m$th-order generalized moduli of continuity of the $r$th derivatives. The exact values of some extremal approximation characteristics in the space $L_{2,\rho}(\mathbb{R})$ are found for classes of functions defined in terms of these moduli of continuity.