Constrained Interpolation via Cubic Hermite Splines

In industrial designing and manufacturing, it is often required to generate a smooth function approximating a given set of data which preserves certain shape properties of the data such as positivity, monotonicity, or convexity, that is, a smooth shape preserving approximation.
It is assumed here that the data is sufficiently accurate to warrant interpolation, rather than least squares or other approximation methods. The shape preserving interpolation problem seeks a smooth curve/surface passing through a given set of data, in which we priorly know that there is a shape feature in it and one wishes the interpolant to inherit these features. One of the hidden features in a data set may be its boundedness. Therefore, we have a data set, which is bounded, and we already know that. This happens, for example, when the data comes from a sampling of a bounded function or they reflect the probability or efficiency of a process.
Scientists have proposed various shape-preserving interpolation methods and every approach has its own advantages and drawbacks. However, anyone confesses that splines play a crucial role in any shape-preserving technique and every approach to shape-preserving interpolation, more or less, uses splines as a cornerstone.
This study concerns an interpolation problem, which must preserve boundedness and needs a smooth representation of the data so the cubic Hermite splines are employed.