Piecewise L-splines of order 4: Interpolation and L² error bounds for splines in tension

PiecewiseL -splines are generalizations of L-splines, in the sense that they satisfy different differential equations in different mesh intervals. Prenter attempted in [P.M. Prenter, Piecewise L-Splines, Numer. Math. 18 (2) (1971) 243-253] to obtain results on piecewise L-splines by generalizing the results of Schultz and Varga on L-splines in [M.H. Schultz, R.S. Varga, L-Splines, Numer. Math. 10 (1967) 345-369]. We show that the results of Prenter are erroneous, and provide correct ones for piecewise L-splines of order 4. We prove the existence and uniqueness of such interpolants and establish the first and second integral relations. In addition we obtain new L² error bounds for the special case of splines in tension with variable tension parameters.