Local interpolation curve with a prescribed degree of smoothness which preserves the constant sign of curvature

A local interpolation method is described, whereby the curve is kept monotonic and its curvature sign fixed, provided that the initial points enable such a curve to be constructed. The algorithm allows the straight parts on the curve to be separated and provides continuity of the derivatives of a given degree. It is shown that, if the function $f^{(q)}(x)$ is continuous in the interval $[a,b]$, $q=0,1,2$, then the interpolation function of the appropriate degree of smoothness converges to the function $f(x)$ on a sequence of meshesat least at the rat $\|\Delta\|^q$, where $\|\Delta\|=\max_i|\Delta x_i|$.