The Unimodality of Initial B-Spline Approximations in Spline Fitting

Finding optimal knots is a challenging problem in spline fitting due to a lack of prior knowledge regarding optimal knots. The unimodality of initial B-spline approximations associated with given data is a promising characteristic of locating optimal knots and has been applied successfully. The initial B-spline approximations herein are required to approximate given data well enough and characterized by the unimodality if jumps from the highest-order derivatives of the approximations at some interior knots are local maxima. In this paper, we prove the unimodality of the initial B-spline approximations that are constructed under two assumptions: Data points are sampled uniformly and sufficiently from B-spline functions, and initial knots are chosen as the parameters of sampling points. Our work establishes the theoretical basis of the unimodality of initial B-spline approximations and pioneers the theoretical study of locating optimal knots.