Range restricted interpolation using Gregory's rational cubic splines

The construction of range restricted univariate and bivariate interpolants to gridded data is considered. We apply Gregory's rational cubic C¹ splines as well as related rational quintic C² splines. Assume that the lower and upper obstacles are compatible with the data set. Then the tension parameters occurring in the mentioned spline classes can be always determined in such a way that range restricted interpolation is successful.