On uniform Lebesgue constants of local exponential splines with equidistant knots

For a linear differential operator $\mathcal {L}_r$ of arbitrary order $r$ with constant coefficients and real pairwise different roots of the characteristic polynomial, we study Lebesgue constants (the norms of linear operators from $C$ to $C$) of local exponential splines corresponding to this operator with a uniform arrangement of knots; such splines were constructed by the authors in earlier papers. In particular, for the third-order operator $\mathcal {L}_3=D(D^2-\beta^2)$ ($\beta>0$), we find the exact values of Lebesgue constants for two types of local splines and compare these values with Lebesgue constants of exponential interpolation splines.