Knot insertion and knot removal matrices for nonpolynomial splines

Continuously differentiable splines of second order on a nonuniform grid are constructed. Formulas of polynomial and nonpolynomial (trigonometric and hyperbolic) are given. Calibration relations expressing the coordinate splines on the original grid as a linear combination of splines of the same type on a refined grid and calibration relations representing the coordinate splines on an enlarged grid as a linear combination of splines of the same type on the original grid are obtained. Knot insertion and knot removal matrices on an interval and on a segment for splines associated with infinite and finite nonuniform grids respectively are constructed.