L1L1-approximation to Laplace transforms of signed measures

For functions that are piecewise equal to one-sided Laplace transforms of signed measures, we construct interpolations that are entire functions with interpolation points that are zeros of Laguerre–Pólya entire functions.If the interpolated function ff is sufficiently regular, these interpolations are best approximations in L1L1-norm to ff by functions of fixed exponential type. This is demonstrated for the example fa,b(x)=eax(1+ebx)−1 with 0