Interpolatory tension splines with automatic selection of tension factors

TLDR: In this paper, a shape-preserving interpolation method was developed in terms of piecewise exponential functions with a tension factor associated with each interval, where knots coincide with data points, and the interpolant is formulated in term of its values and first derivatives at these points, which enables the efficient computation of the minimum tension factor for which the interpolants satisfies locally defined properties such as monotonicity and convexity.

Abstract: A powerful and versatile method of shape-preserving interpolation is developed in terms of piecewise exponential functions with a tension factor associated with each interval. Knots coincide with data points, and the interpolant is formulated in terms of its values and first derivatives at these points. For a given set of derivatives, this enables the efficient computation of the minimum tension factor for which the interpolant satisfies locally defined properties such as monotonicity and convexity, as well as more general bounds on function values and derivatives, in each interval. A local derivative-estimation procedure results in a $C^1 $ interpolant satisfying the constraints with minimum tension, and an iterative procedure can be used to obtain a $C^2 $ spline fit which satisfies the constraints. Test results are presented which show both methods to produce visually pleasing interpolants to various data sets.