Generalised mean averaging interpolation by discrete cubic splines (Q1331257)

Discrete cubic splines are continuous piecewise cubic polynomial functions which satisfy discrete smoothness conditions at the joints by matching the first and second central differences (in terms of a value \(h\), say). As \(h\to 0\), the discrete cubic spline corresponds to the usual cubic spline. Mean averaging interpolation by usual and discrete cubic splines have been studied in [the reviewer, J. Approximation Theory 22, 105-110 (1978; Zbl 0386.41001) and the reviewer and \textit{P. Powar}, Approximation theory and applications, Proc. Int. Conf. 75th Birthday G. G. Lorentz, St. John's/Newfoundland 1984, Res. Notes Math. 133, 35-45 (1985; Zbl 0573.41003)]. The author studies here existence, uniqueness, and convergence properties of a discrete cubic spline satisfying certain discrete interpolatory conditions.