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.

Keywords: spline, wavelet, biorthogonal systems, decomposition matrix, reconstruction matrix, subdivision scheme, knot insertion and removal algorithms, spline curve

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