C^4 interpolation and smoothing exponential splines based on a sixth order differential operator with two parameters

In this work, a class of interpolation and smoothing exponential splines with respect to a sixth order differential operator with two parameters is constructed. All the square matrices involved in the construction are proved to be tridiagonal symmetric and diagonally dominant, which results in algorithms for computing this class of exponential splines. The obtained splines have C^4 continuity and are the minimum solution of the combination of interpolation and a generalized smoothing energy integral. The performances of the resulting splines in financial data from the S P500 index and the effect of fitting multi-exponential decay data are given. Numerical experiments show that the resulting splines have more freedom to adjust the shape and control the energy of the curves and perform better than previous methods in fitting multi-exponential decay data. And cross validation and generalized cross validation for determining an appropriate smoothing parameter are also developed.