I have a set of points $(x_0,y_0)$ ... $(x_N,y_N)$ with the $x_i$ increasing and the $y_i$ such that $\frac{y_{i+1} - y_{i}}{x_{i+1} - x_{i}} > \frac{y_{i} - y_{i-1}}{x_{i} - x_{i-1}}$
Is there a well-known interpolation scheme that would give me a smooth function $f$ that is continuous, differentiable, and convex?
I don't want to constrain the derivative at the $x_i$, I just want the function to be as smooth as possible without introducing inflexion points. Ideally the derivative should be as smooth as possible as well.
Edit: For instance, in the following charts, I am trying to fit a set of points that are clearly convex. However the kernel interpolation I am performing is not:
It looks like Gregory's Shape Preserving Spline Interpolation https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19850020252.pdf provides a spline that fits the requirements. Strangely it doesn't seem to be implemented in any interpolation packages I found... Maybe it's got a different name...