Trigonometric Splines; A Survey with New Results.
by
Tom Lyche
University of Oslo

Over the years many generalizations of the classical univariate polynomial splines have been proposed, see the book by Schumaker and references there-inn. However, further development of these generalized univariate splines has for the most part been slow, since there seem to be few problems that cannot be solved as well with polynomial splines as with any of the generalized splines. Lately it has been demonstrated that certain problems can be better solved by a class of splines known as trigonometric splines, than by the classical polynomial splines. A specific example is data-fitting on the sphere. As shown by Schumaker and Traas[1991], by using trigonometric splines we can use fast and simple tensor product methods which can handle the problem with smoothness at the poles. The resulting surface has a continuous tangent plane everywhere and can be decomposed in a wavelet fashion, see Lyche and Schumaker [1999]. Another application is to the class of so called Pythagorean-Hodograph curves which are rational curves with rational offsets. In addition trigonometric splines have been suggested for CAM design see Neamtu, Pottmann, and Schumaker[1998], and trajectory generation , see Srinivasan, L., Rastegar[1997]. Trigonometric splines can be used to define single valued curves in polar coordinates, (Sanchez-Reys[1990, 1992]), circular Bernstein Bézier polynomials (Alfeld, Neamtu, Schumaker[1998]), and piecewise rational curves with rational offsets (Neamtu, Pottmann, Schumaker[1997]). There is also an interesting connection with circle splines, see Goodman Lee[1984].

We present a fairly complete survey of trigonometric splines. This includes a new elementary introduction to trigonometric B-splines, knot insertion- and removal algorithms, blossoming, quasi-interpolants, total positivity, variation diminishing properties, and L^p-stability. There are new results on knot insertion and removal, and total positivity of trigonometric B-splines normalized to form a partition of unity.

Date received: February 8, 1999