About B-splines. Twenty answers to one question: What is the cubic B-spline for the knots -2,-1,0,1,2?

Authors

  • Ewald Quak SINTEF Applied Mathematics, Norway

DOI:

https://doi.org/10.33993/jnaat451-1099

Keywords:

cubic B-splines, curves, surfaces, Approximation Theory, Numerical Analysis
Abstract views: 879

Abstract

In this composition an attempt is made to answer one simple question only: What is the cubic B-spline for the knots -2,-1,0,1,2? The note will take you on a most interesting trip through various fields of Mathematics and finally convince you on how little we know.

Downloads

Download data is not yet available.

References

C. de Boor, Best approximation properties of spline functions of odd degree, J. Math. Mech. 12 (1963), pp. 747-750, http://dx.doi.org/10.1512/iumj.1963.12.12051 DOI: https://doi.org/10.1512/iumj.1963.12.12051

C. de Boor, On calculating with B-splines, J. Approx. Theory 6 (1972), pp. 50-62, http://dx.doi.org/10.1016/0021-9045(72)90080-9 DOI: https://doi.org/10.1016/0021-9045(72)90080-9

C. de Boor, A practical guide to splines, Springer, New York, 1978. DOI: https://doi.org/10.1007/978-1-4612-6333-3

C. de Boor, K. Hollig, and S. Riemenschneider, Box splines, Springer, New York, 1993. DOI: https://doi.org/10.1007/978-1-4757-2244-4

C. de Boor and R.E. Lynch, On splines and their minimum properties, J. Math. Mech., 15 (1966), pp. 953–970, http://dx.doi.org/10.1512/iumj.1966.15.15063 DOI: https://doi.org/10.1512/iumj.1966.15.15063

V. Brun, Gauss’ fordelingslov, Norsk Matematisk Tidsskrift, 14 (1932), pp. 81-92.

P.L. Butzer, M. Schmidt and E.L. Stark, Observations on the history of central B-splines, Archive for History of Exact Sciences, 39 (1988), pp. 137-156, http://dx.doi.org/10.1007/BF00348440 DOI: https://doi.org/10.1007/BF00348440

A.S. Cavaretta, W. Dahmen and C.A. Micchelli, Stationary subdivision, Memoirs of AMS 93, 1991, http://dx.doi.org/10.1090/memo/0453 DOI: https://doi.org/10.1090/memo/0453

G.M. Chaikin, An algorithm for high-speed curve generation, Comp. Graphics and Image Proc. 3 (1974), pp. 346-349, http://dx.doi.org/10.1016/0146-664X(74)90028-8 DOI: https://doi.org/10.1016/0146-664X(74)90028-8

C.K. Chui, Multivariate Splines, CBMS 54, SIAM, Philadelphia, 1988, http://dx.doi.org/10.1137/1.9781611970173 DOI: https://doi.org/10.1137/1.9781611970173

C.K. Chui, An Introduction to Wavelets , Academic Press, Boston, 1992. DOI: https://doi.org/10.1063/1.4823126

M.G. Cox, The numerical evaluation of B-splines. J. Inst. Math. Applics. 10 (1972), pp. 134-139, http://dx.doi.org/10.1093/imamat/10.2.134 DOI: https://doi.org/10.1093/imamat/10.2.134

H.B. Curry, and I.J. Schoenberg, On Polya frequency functions IV. The fundamental spline and their limits, J. d’Analyse Math. bf 17 (1966), pp. 71-107, http://dx.doi.org/10.1007/BF02788653 DOI: https://doi.org/10.1007/BF02788653

I. Daubechies, Ten Lectures on Wavelets, CBMS 61, SIAM, Philadelphia, 1992, http://dx.doi.org/10.1137/1.9781611970104.fm DOI: https://doi.org/10.1137/1.9781611970104

P. J. Davis, Interpolation and Approximation 2nd edition, Dover, New York, 1975 (originally published 1963).

T. DeRose, M. Kass, and T. Truong, Subdivision surfaces in character animation, Computer Graphics 32 (1998), pp. 85–94, http://dx.doi.org/10.1145/280814.280826 DOI: https://doi.org/10.1145/280814.280826

J. Duchon, J., Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces, RAIRO Analyse numerique,10 (1976), pp. 5-12. DOI: https://doi.org/10.1051/m2an/197610R300051

N. Dyn, Interpolation of scattered data by radial functions, in Topics in Multivariate Approximation, C.K. Chui, L.L. Schumaker, and F.I. Utreras (eds), Academic Press, New York, 1987, pp. 47-61. DOI: https://doi.org/10.1016/B978-0-12-174585-1.50009-9

N. Dyn, Interpolation and approximation by radial and related functions, in Approximation Theory VI, Vol. 1, C.K. Chui, L.L. Schumaker, and J.D. Ward (eds.), Academic Press, New York, 1989, pp. 211-234.

N. Dyn, Analysis of convergence and smoothness by the formalism of Laurent polynomials, in Tutorials on Multiresolution in Geometric Modelling, A. Iske, E. Quak, and M.S. Floater (eds.), Springer, Heidelberg, 2002, pp. 51–68, http://dx.doi.org/10.1007/978-3-662-04388-2_3 DOI: https://doi.org/10.1007/978-3-662-04388-2_3

F. Eggenberger, and G. Polya Uber die Statistik verketteter Vorgange, Z. Angew. Math. Mech. 1 (1923), pp. 279-289. 82 Ewald Quak 46, http://dx.doi.org/10.1002/zamm.19230030407 DOI: https://doi.org/10.1002/zamm.19230030407

G. Farin, Curves and Surfaces for Computer-Aided Geometric Design, 4th edition, Academic Press, San Diego, 1998.

J.E. Fjelstaad, Bemerking til Viggo Brun: Gauss’ fordelingslov, Norsk Matematisk Tidsskrift 19 (1937), pp. 69-71.

B. Friedman, A simple urn model, Comm. Pure Appl. Math. 2 (1949), pp. 59-70, http://dx.doi.org/10.1002/cpa.3160020103 DOI: https://doi.org/10.1002/cpa.3160020103

R.N. Goldman, Urn models, approximations, and splines. J. Approx. Theory 54 (1988), pp. 1-66, http://dx.doi.org/10.1016/0021-9045(88)90116-5 DOI: https://doi.org/10.1016/0021-9045(88)90116-5

R.N. Goldman, Recursive Triangles , in Computation of Curves and Surfaces, W. Dahmen, M. Gasca and C. M. Micchelli (eds), Kluwer, Dordrecht, 1989, pp. 27-72, http://dx.doi.org/10.1007/978-94-009-2017-0_2 DOI: https://doi.org/10.1007/978-94-009-2017-0_2

J.C. Holladay, A smoothest curve approximation, Math. Tables Aid. Comput. 11 (1957), pp. 233-243, http://dx.doi.org/10.2307/2001941 DOI: https://doi.org/10.1090/S0025-5718-1957-0093894-6

J. Hoschek, and D. Lasser, Fundamentals of Computer Aided Geometric Design, AKPeters, Wellesley, 1993.

S. Karlin, and Z. Ziegler, Chebyshevian spline functions, SIAM J. Numer. Anal.3 (1966), pp. 514-543, http://dx.doi.org/10.1137/0703044 DOI: https://doi.org/10.1137/0703044

M.G. Krein , and G. Finkelstein, Sur les fonctions de Green completement non-negatives des operateurs differentiels ordinaires, Doklady Akad. Nauk. SSSR 24 (1939), pp. 202-223.

M.J. Marsden, An identity for spline functions with applications to variation-diminishing spline approximation, J. Approx Theory 3 (1970), pp. 7-49, http://dx.doi.org/10.1016/0021-9045(70)90058-4 DOI: https://doi.org/10.1016/0021-9045(70)90058-4

C.A. Micchelli, Mathematical Aspects of Geometric Model ling, CBMS 65, SIAM, Philadelphia, 1995. DOI: https://doi.org/10.1137/1.9781611970067

E. Quak, Nonuniform B-splines and B-wavelets, in Tutorials on Multiresolution in Geometric Modelling, A. Iske, E. Quak, and M.S. Floater (eds.), Springer, Heidelberg, 2002, pp. 101-146, http://dx.doi.org/10.1007/978-3-662-04388-2_6 DOI: https://doi.org/10.1007/978-3-662-04388-2_6

G. de Rham, Un peu de mathematique a propos d’une courbe plane, Elemente der Mathematik 2 (1947), pp. 73-76, pp. 89-97.

C. Runge, Uber empirische Funktionen und die Interpolation zwischen aquidistanten Ordinaten , Z. f. Math. u. Phys. 46 (1901), pp. 224-243.

M.A. Sabin, The use of piecewise forms for the numerical representation of shape, Dissertation, Hungarian Academy of Sciences, 1977.

M.A. Sabin, Subdivision of box-splines; Eigenanalysis and artifacts of subdivision curves and surfaces, in Tutorials on Multiresolution in Geometric Modelling, A. Iske, E. Quak, and M. S. Floater (eds.), Springer, Heidelberg, 2002, pp. 3-23, pp. 69-92, http://dx.doi.org/10.1007/978-3-662-04388-2_4 DOI: https://doi.org/10.1007/978-3-662-04388-2_4

I.J. Schoenberg, Contributions to the problem of approximation of equidistant data by analytic functions, Part A: On the problem of smoothing or graduation. A first class of analytic approximation formulae. Part B: On the second problem of osculatory interpolation. A second class of analytic approximation formulae. Quart. Appl. Math. 4 (1946), pp. 45-99 and pp. 112-141. DOI: https://doi.org/10.1090/qam/16705

I.J. Schoenberg, On spline functions , in Inequalities I, O. Shisha (ed.), Academic Press, New York, 1967.

I.J. Schoenberg, Cardinal Spline Interpolation, CBMS 12, SIAM, Philadelphia, 1973, http://dx.doi.org/10.1137/1.9781611970555 DOI: https://doi.org/10.1137/1.9781611970555

I.J. Schoenberg, and A. Whitney, On Polya frequency functions III. The positivity of translation determinants with an application to the interpolation problem by spline curves, Trans. Amer. Math. Soc. 74 (1953), pp. 246-259. DOI: https://doi.org/10.1090/S0002-9947-1953-0053177-X

L.L. Schumaker, Spline Functions: Basic Theory, Wiley, New York, 1981.

A. Sommerfeld, Eine besonders anschauliche Ableitung des Gaussischen Fehlergesetzes, in Festschrift Ludwig Boltzmann gewidmet zum 60.Geburtstage, 20. Februar 1904, Barth, Leipzig, 1904, pp. 848-859.

J.L. Walsh, J.H. Ahlberg and E.N. Nilson, Best approximation properties of the spline fit, J. Math. Mech. 11 (1962), pp. 225-234, http://dx.doi.org/10.1512/iumj.1962.11.11015 DOI: https://doi.org/10.1512/iumj.1962.11.11015

Downloads

Published

2016-09-19

How to Cite

Quak, E. (2016). About B-splines. Twenty answers to one question: What is the cubic B-spline for the knots -2,-1,0,1,2?. J. Numer. Anal. Approx. Theory, 45(1), 37–83. https://doi.org/10.33993/jnaat451-1099

Issue

Section

Articles