Approximation by local trigonometric splines

For the class $W_\infty^{\mathscr L_2}=\{f:f'\in AC,\ \|f''+\alpha^2f\|_\infty\leqslant1\}$ of 1-periodic functions, we use the linear noninterpolating method of trigonometric spline approximation possessing extremal and smoothing properties and locally inheriting the monotonicity of the initial data, i.e., the values of a function from $W_\infty^{\mathscr L_2}$ at the points of a uniform grid. The approximation error is calculated exactly for this class of functions in the uniform metric. It coincides with the Kolmogorov and Konovalov widths.