Mean Approximation of Functions on the Real Axis by Algebraic Polynomials with Chebyshev–Hermite Weight and Widths of Function Classes

We obtain sharp Jackson–Stechkin type inequalities on the sets $L^r_{2,\rho}(\mathbb{R})$ in which the values of best polynomial approximations are estimated from above via both the moduli of continuity of $m$th order and $K$-functionals of $r$th derivatives. For function classes defined by these characteristics, the exact values of various widths are calculated in the space $L_{2,\rho}(\mathbb{R})$. Also, for the classes $W^r_{2,\rho}(\mathbb{K}_m,\Psi)$, where $r=2,3,\dots$, the exact values of the best polynomial approximations of the intermediate derivatives $f^{(\nu)}$, $\nu=1,\dots,r-1$, are obtained in $L_{2,\rho}(\mathbb{R})$.