Exact constants in Jackson-type inequalities for the best mean square approximation in L 2(ℝ) and exact values of mean 𝜈-widths of the classes of functions

Abstract

On the classes of functions $$ {L}_2^r\left(\mathbb{R}\right) $$ , where r ∈ℤ + , for the characteristics of smoothness $$ {\Lambda}_k\left(f,t\right)={\left\{\left(1/t\right){\int}_0^t\left\Vert {\varDelta}_h^k(f)\left\Vert {}^2\right. dh\right.\right\}}^{\kern0em 1/2},t\in \left(0,\infty \right),k\in \mathbb{N} $$ , the exact constants in the Jackson-type inequalities have been obtained in the case of the best mean square approximation by entire functions of the exponential type in the space L 2(ℝ). The exact values of mean 𝜈-widths of the classes of functions defined by Λ k (f) and the majorants Ψ satisfying some conditions are calculated.