For the class \(W_\infty ^{\mathcal{L}_2 } = \left\{ {f:f' \in AC,\left\| {f'' + \alpha ^2 f} \right\|\infty \leqslant 1} \right\}\) 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 \(W_\infty ^{\mathcal{L}_2 } \) 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.