For the class W ∞ ℒ2 = {f:f′ ∈ AC,∥f″ + α 2f∥∞≤1} 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 ∞ ℒ2 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.