Algorithms for the construction of third-order local exponential splines with equidistant knots

We construct new local exponential splines with equidistant knots corresponding to a third-order linear differential operator $\mathcal L_3(D)$ of the form
$$ \mathcal L_3(D)=(D-\beta)(D-\gamma)(D-\delta)\quad (\beta,\gamma,\delta\in \mathbb R). $$
We also establish upper order estimates for the error of approximation by these splines in the uniform metric on the Sobolev class of three times differentiable functions $W_{\infty}^{\mathcal L_3}$. In particular, for the differential operator $\mathcal L_3(D)=D(D^2-\beta^2)$, we give a general scheme for the construction of local splines with additional knots, which leads in one case to known shape-preserving splines and in another case to new local interpolation splines exact on the kernel of $\mathcal L_3(D)$.