Abstract
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.
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