Algorithmic Approaches to Convex Interpolation in Sublet Spaces: Error Analysis and Shape Preservation with Piecewise Polynomial Splines

This study explores convex interpolation within a Sobolev space by minimizing a semi-norm of a specified order and discretizing the problem in the space of piecewise polynomial spline functions. The shape-preserving condition examined is the positivity of the derivative function of a given order. An algorithm is proposed for computing the resulting spline function, with convergence demonstrated through established theorems. The interpolation error is shown to be of order O (1/n​), where n denotes the number of Lagrangian data points. Additionally, the error analysis extends to rational interpolating functions, deriving the error when the interpolated function is C3 within the interpolation interval. The findings underscore the effectiveness of the proposed method in preserving shape and achieving accurate interpolation results.