# Physics Reports vol.97

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However there are examples of data sets for which there is no solution. Note that a method has been proposed in [15] for sampling points from a given curve f which supports the BHS scheme, in the sense that there is always a solution. For example, the BHS scheme has a solution for the data in Figure 13, generated by the method of [15], and as we can see in the figure, it is hard to distinguish the BHS (grey) curve from the original (black) curve, similar to the quintic in Figure 10(b). 52 Michael S.

242, Universitat Dortmund (November 2003). 19. -B. and Zhou, X. , The lower estimate for linear positive operators, I, Constr. Approximation 11 (1995), 53-66. 20. -B. and Zhou, X. , The lower estimate for linear positive operators, II, Results Math. 25 (1994), 315-330. 21. , Jacobi polynomials, II. An analytic proof of the product formula, SIAM J. Math. Anal. 5 (1974), 125-137. 22. , Oberhettinger, F. and Soni, R. , Springer-Verlag, Berlin, 1966. 23. Paltanea, R. , Approximation Theory Using Positive Linear Operators, Birkhauser-Verlag, Boston, 2004.

Subdivision schemes in Computer-Aided Geometric Design, in: Advances in Numerical Analysis - Vol. II, Wavelets, Subdivision Algorithms and Radial Basis Functions (W. ), Clarendon Press, Oxford, 1992, pp. 36-104. 7. Dyn, N. , Analysis of asymptotically equivalent binary subdivision schemes, J. Math. Analysis and Applications 193 (1995), 594-621. 8. , Gregory, J. , A 4-point interpolatory subdivision scheme for curve design, Computer Aided Geometric Design 4 (1987), 257-268. 9. , Osher, S. , Uniformly high order accurate essentially non-oscillatory schemes III, J.