Piecewise LL-splines of order 4: Interpolation and L2L2 error bounds for splines in tension

Piecewise  LL-splines   are generalizations of LL-splines, in the sense that they satisfy different differential equations in different mesh intervals. Prenter attempted in [P.M. Prenter, Piecewise LL-Splines, Numer. Math. 18 (2) (1971) 243–253] to obtain results on piecewise LL-splines by generalizing the results of Schultz and Varga on LL-splines in [M.H. Schultz, R.S. Varga, LL-Splines, Numer. Math. 10 (1967) 345–369]. We show that the results of Prenter are erroneous, and provide correct ones for piecewise LL-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 L2L2 error bounds for the special case of splines in tension with variable tension parameters.