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General Approach to Discrete Splines
by
Boris I. Kvasov
Suranaree University of Technology, Thailand
The tools of generalized splines and generalized B-splines (GB-splines for short) are widely used in solving problems of shape preserving approximation. By introducing various parameters into the spline structure, one can preserve characteristics of the initial data such as convexity, monotonicity, presence of linear and planar sections, etc. Here, the main challenge is to develop algorithms that choose parameters automatically. Recently, in [1, 2] a difference method for constructing shape preserving hyperbolic tension splines as solutions of multipoint boundary value problems was developed. Such an approach permits us to avoid the computation of hyperbolic functions and has substantial other advantages. However, the extension of a mesh solution will be a discrete hyperbolic tension spline.
The contents of this paper is as follows. In section 1 we give a general definition of a discrete generalized spline and prove sufficient conditions for its existence and uniqueness. Next, we construct a minimum length local support basis of the new splines, denoted as discrete GB-splines; see section 2. The local approximation properties are discussed in section 3, while in section 4 we consider recurrence formulae for calculations with discrete GB-splines. The properties of GB-spline series are summarized in section 5. Section 6 provides some examples of defining functions that conform to the sufficient conditions derived earlier in the paper. We conclude in section 7 with graphical examples to illustrate the tension features of discrete generalized splines and to discuss their possibility in applications.
Date received: January 8, 1999
Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabp-20.