Best approximation of the differentiation operator in the space \(L_2\) on the semiaxis
In the space \(L_2(0,\infty)\) one considers the class \(\mathcal Q_2^2(0,\infty)\) of functions \(f\in L_2(0,\infty)\) continuously differentiable on \([0,\infty)\) such that \(f'\) is locally absolutely continuous, \(f''\in L_2(0,\infty)\) and \(\|f''\|\leq 1.\) Denote by \(\mathcal B=\mathcal B_2(0,\infty)\) the set of bounded linear operators on \(L_2(0,\infty)\) and let \(\mathcal B(N)=\{S\in\mathcal B : \|S\|\leq N\}\) for some \(N>0.\) Let also \[ E(N)=\inf_{S\in\mathcal B(N)}\sup_{f\in\mathcal Q^2_2}\|f'-Sf\|_{L_2(0,\infty)}\,. \tag{\(*\)} \] The authors consider the problem of finding the value \(E(N)\) and an extremal operator for which the lower bound in (\(*\)) is attained. The problem is completely solved by considering for \(f\in L_2(0,\infty)\) the differential problem \[ h^3Y'''(x)-\frac{2h^2}{3}\, Y''(x)+\frac{2h}{3}\, Y'(x) +Y(x)=f(x)\,, \] \[ Y\in W^3_2,\;\; Y''(0)=0,\;\; h>0\,. \] The operator \(T^*_h\) defined by the formula \[ \left(T^*_hf\right)(x)=Y'(x)-\frac{h}{3}\, Y''(x)-\frac{h^2}{3}\, Y'''(x)\,,\;\; f\in L_2(0,\infty)\,, \] is a bounded linear operator having the norm \(\|T^*_hf\|=1/(h\sqrt2).\) Also \(T^*_h\) is extremal for (\(*\)) with \(N=1/(h\sqrt2)\) and \(E\left(\frac{1}{h\sqrt2}\right)=\frac{h}{\sqrt2}.\)
scientific article; zbMATH DE number 6351412
scientific article