Characterization of Local Strict Convexity Preserving Interpolation Methods by C1 Functions Original Research Article

This paper is concerned with the problem of strict convexity preserving interpolation in one variable. It is shown that a strictly convex Hermite interpolant to strictly convex data can always be chosen smooth and even to be a polynomial. Furthermore, two-point Hermite strict convexity preserving interpolation schemes using neither tension parameters nor additional knots are classified according to a number of certain desirable properties like symmetry, quadratic exactness, affine invariance, etc. One of the main results is the characterization of the set of such methods as a one-parameter family of solutions to certain boundary value problems.