Behaviour of exponential splines as tensions increase without bound

Schweikert (J. Math. Phys. 45 (1966), 312-317) showed that for sufficiently high tensions an exponential spline would have no more changes in sign of its second derivative than there were changes in the sign of successive second differences of its knot sequence. Spath (Computing 4 (1969), 225-233) proved the analogous result for first derivatives, assuming uniform tension throughout the spline. Later, Pluess (J. Approx. Theory 17 (1976), 86-96) extended Spath's result to the case where the inter-knot tensions p(i) may not all be the same but tend to infinity at the same asymptotic growth rate, in the sense that p(i) is an element of Theta(p(i)) for all i. This paper extends Pruess's result by showing his hypothesis of uniform boundedness of the tensions to be unnecessary. A corollary is the Fact that for high enough minimum interknot tension, the exponential spline through monotone knots will be a l(2) monotone curve. In addition, qualitative bounds on the difference in slopes between the interpolating polygon and the exponential spline are developed, which show that Gibbs-like behaviour of the spline's derivative cannot occur in the neighbourhood of the knots. (C) 1997 Academic Press.