The theoretical analysis for an iterative envelope algorithm

Cubic spline interpolating the local maximal/minimal points is often employed to calculate the envelopes of a signal approximately. However, the undershoots occur frequently in the cubic spline envelopes. To improve them, in our previous paper we proposed a new envelope algorithm, which is an iterative process by using the Monotone Piecewise Cubic Interpolation. Experiments show very satisfying results. But the theoretical analysis on why and how it works well was not given there. This paper establishes the theoretical foundation for the algorithm. We will study the structure of undershoots, prove rigorously that the algorithm converges to an envelope without undershoots with exponential rate of convergence, which can be used to determine the number of iterations needed in the algorithm for a good envelope in applications.