Abstract
We introduce control curves for trigonometric splines and show that they have properties similar to those for classical polynomial splines. In particular, we discuss knot insertion algorithms, and show that as more and more knots are inserted into a trigonometric spline, the associated control curves converge to the spline. In addition, we establish a convex-hull property and a variation-diminishing result.
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P. Alfeld, M. Neamtu and L.L. Schumaker, Circular Bernstein-Brzier polynomials, in:Mathematical Methods for Curves and Surfaces, eds. M. Dæhlen, T. Lyche and L.L. Schumaker (Vanderbilt University Press, Nashville, 1995) pp. 11–20.
W. Boehm, Inserting new knots into B-spline curves, Comp. Aided Design 12 (1980) 199–201.
C. de Boor,A Practical Guide to Splines (Springer, New York, 1978).
E. Cohen, T. Lyche and R. Riesenfeld, DiscreteB-splines and subdivision techniques in computer-aided geometric design and computer graphics, Comp. Graphics Image Proc. 14 (1980) 87-111.
E. Cohen and L.L. Schumaker, Rates of convergence of control polygons, Comp. Aided Geom. Design 2 (1985) 229–235.
W. Dahmen, Subdivision algorithms converge quadratically, J. Comp. Appl. Math. 16 (1986) 145–158.
G. Farin,Curves and Surfaces for Computer Aided Geometric Design (Academic Press, 1988).
T.N.T. Goodman and S.L. Lee, Interpolatory and variation-diminishing properties of generalized B-splines, Proc. Royal Soc. Edinburgh 96A (1984) 249-259.
D. Gonsor and M. Neamtu, Non-polynomial polar forms, in:Curves and Surfaces in Geometric Design, eds. P.-J Laurent, A. Le Méhauté and L.L. Schumaker, (AKPeters, Wellesley, MA, 1994) pp. 193–200.
J. Hoschek and D. Lasser,Computer Aided Geometric Designs (AKPeters, Wellesley, MA, 1993).
P.E. Koch, Jackson-type estimates for trigonometric splines, in:Industrial Mathematics Week, Trondheim, Department of Mathematical Sciences, Norwegian Institute of Technology (NTH), Trondheim (1992) pp. 117-124.
P.E. Koch and T. Lyche, Bounds for the error in trigonometric Hermite interpolation, in: Quantitative Approximation, eds. R. DeVore and K. Scherer (Academic Press, New York, 1980) pp. 185–196.
P.E. Koch, T. Lyche and L.L. Schumaker, Quasi-interpolation with trigonometric splines, preprint (1994).
J.M. Lane and R.F. Riesenfeld, A geometric proof for the variation diminishing property of B-spline approximation, J. Approx. Theory 37 (1983) 1–4.
T. Lyche, A recurrence relation for Chebyshevian B-splines, Constr. Approx. 1 (1985) 155–173.
T. Lyche, Discrete B-splines and conversion problems, in: Computation of Curves and Surfaces, eds. W. Dahmen, M. Gasca and C. Micchelli (Kluwer, Dordrecht, 1990) pp. 117–134.
T. Lyche and L.L. Schumaker, L-spline wavelets, in:Wavelets: Theory, Algorithms, and Applications, eds. C. Chui, L. Montefusco and L. Puccio (Academic Press, New York, 1994) pp. 197–212.
T. Lyche and R. Winther, A stable recurrence relation for trigonometric B-splines, J. Approx. Theory 25 (1979) 266–279.
I.J. Schoenberg, On trigonometric spline interpolation, J. Math. Mech. 13 (1964) 795–825.
L.L. Schumaker,Spline Functions: Basic Theory (Interscience, New York, 1991; reprinted by Krieger, Malabar, Florida, 1993).
L.L. Schumaker, On recursions for generalized splines, J. Approx. Theory 36 (1982) 16–31.
L.L. Schumaker, On hyperbolic splines, J. Approx. Theory 38 (1983) 144–166.
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Koch, P.E., Lyche, T., Neamtu, M. et al. Control curves and knot insertion for trigonometric splines. Adv Comput Math 3, 405–424 (1995). https://doi.org/10.1007/BF03028369
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DOI: https://doi.org/10.1007/BF03028369