Convergence of Quartic Interpolating Splines

The problem of interpolation by quartic splines according to Marsden's scheme is considered. It is shown that the calculation of an interpolating spline in terms of the coefficients of expansion of its second derivative in $L_1$-normalized quadratic B-splines yields a system of linear equations for the chosen parameters. The matrix of the system is pentadiagonal and has a column diagonal dominance, which makes it possible to efficiently calculate the required parameters and establish the convergence of the spline interpolation process according to Marsden's scheme for any function from the class $C^1$ on an arbitrary sequence of grids without any constraints. In Marsden's scheme, it is assumed that a knot grid is given and the interpolation nodes are chosen strictly in the middle. The established results are transferred to the case of interpolation by quartic splines according to Subbotin's scheme (the data grid and knot grid are swapped). Here the system of equations for the coefficients of expansion of the third derivative in $L_\infty$-normalized B-splines has a diagonal dominance, and the interpolation process converges for any interpolated function from the class $C^3$.