Speed limit quasi splines and their application to interpolation with bounded first order derivative

The problem of computing spline-like functions for ideal data, subject to two-sided inequality constraints on the first-order derivative, are considered, focusing primarily on constant constraints and then generalizing. Using a variational approach, in which the inequality constraints are not explicitly imposed, a special type of exponential splines we call speed limit quasi splines is introduced. A simple, non-parametric, efficient, and robust iterative solver is suggested, which is suitable for a wide range of inequality constraints. Analysis of the convergence factor of this algorithm is provided and supported by extensive numerical tests.