Quadratic/linear rational spline histopolation

The histopolation with quadratic/linear rational splines of class is studied. Such kind of splines keep the sign of its second derivative on the whole interval and, consequently, the given histogram should be strictly convex or strictly concave. The grid points of the histogram and suitable number of the spline knots between them are supposed to place arbitrarily. The uniqueness of such an histopolant is established. It is shown that the histopolant may not exist but some sufficient conditions for the existence are given. Presented numerical results confirm their adequacy.