The Taikov type inequalities in rigged Hilbert spaces

For the chain $H_+\subset H_0\subset H_-$ of rigged Hilbert spaces we solve the problem of finding sharp Taikov type inequalities that estimate the value of generalized element $\alpha\in H_-$ on smooth elements $u\in H_+$ in terms of the norm of the element $u$ in $H_+$. Also, we solve the problem of approximating generalized elements $\alpha\in H_-$ by regular elements $x\in H_0$ on the class $W_+ = \{u\in H_+\,:\,\|u\|_{H_+}\le 1\}$. We consider some applications of these results, in particular, to finding sharp Taikov-type inequalities for partial derivatives of multi-variate periodic functions.