Optimal Knots Selection in Fitting Degenerate Reduced Data

The problem of fitting a given ordered sample of data points in arbitrary Euclidean space is addressed. The corresponding interpolation knots remain unknown and as such must be first somehow found. The latter leads to a highly non-linear multivariate optimization task, equally non-trivial for theoretical analysis and for derivation of a computationally efficient numerical scheme. The non-degenerate case of at least four data points can be handled by Leap-Frog algorithm merging generic and non-generic univariate overlapping optimizations. Sufficient conditions guaranteeing the unimodality for both cases of Leap-Frog optimization tasks are already established in the previous research. This work complements the latter by analyzing the degenerate situation i.e. the case of fitting three data points, for which Leap-Frog cannot be used. It is proved here that the related univariate cost function is always unimodal yielding a global minimum assigned to the missing internal-point knot (with no loss both terminal knots can be assumed to be fixed). Illustrative examples supplement the analysis in question.