Approximation by local $L$-splines corresponding to a linear differential operator of the second order

For the class of functions $W_\infty^{\mathcal L_2}=\{f:f'\in AC,\|\mathcal L_2(D)f\|_\infty\le1\}$, where $\mathcal L_2(D)$ is a linear differential operator of the second order whose characteristic polynomial has only real roots, we construct a noninterpolating linear positive method of exponential spline approximation possessing extremal and smoothing properties and locally inheriting the monotonicity of the initial data (the values of a function $f\in W_\infty^{\mathcal L_2}$ at the points of a uniform grid). The approximation error is calculated exactly for this class of functions in the uniform metric.