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

AbstractPiecewise L-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