Abstract
A class of generalized polynomials is considered consisting of the null spaces of certain differential operators with constant coefficients. This class strictly contains ordinary polynomials and appropriately scaled trigonometric polynomials. An analog of the classical Bernstein operator is introduced and it is shown that generalized Bernstein polynomials of a continuous function converge to this function. A convergence result is also proved for degree elevation of the generalized polynomials. Moreover, the geometric nature of these functions is discussed and a connection with certain rational parametric curves is established.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Alfeld, M. Neamtu and L.L. Schumaker, Circular Bernstein–Bézier polynomials, in: Mathematical Methods for Curves and Surfaces, eds. M. Dæhlen, T. Lyche and L.L. Schumaker (Vanderbilt University Press, Nashville, TN, 1995) pp. 11–20.
G. Casciola and S. Morigi, Spline curves in polar and Cartesian coordinates, in: Curves and Surfaces with Applications in CAGD, eds. A. LeMéhauté, C. Rabut and L. L. Schumaker (Vanderbilt University Press, Nashville, TN, 1997) pp. 61–68.
G. Casciola, S. Morigi and J. Sánchez-Reyes, Degree elevation for p-Bézier curves, Comput. Aided Geom. Design 15(1998) 313–322.
E. Cohen and L.L. Schumaker, Rates of convergence of control polygons, Comput. Aided Geom. Design 2(1985) 229–235.
P.J. Davis, Interpolation and Approximation(Dover, New York, 1975).
P. de Casteljau, Splines focales, in: Curves and Surfaces in Geometric Design, eds. P.-J. Laurent, A. Le Méhauté and L.L. Schumaker (A.K. Peters, Wellesley, MA, 1994) pp. 91–103.
G. Farin, Triangular Bernstein–Bézier patches, Comput. Aided Geom. Design 3(1986) 83–128.
G. Farin, Curves and Surfaces for Computer-Aided Geometric Design. A Practical Guide(Academic Press, San Diego, CA, 1997).
D.E. Gonsor and M. Neamtu, Non-polynomial polar forms, in: Curves and Surfaces II, eds. P.-J. Laurent, A. Le Méhauté and L.L. Schumaker (A.K. Peters, Wellesley, MA, 1994) pp. 193–200.
D.E. Gonsor and M. Neamtu, Null spaces of differential operators, polar forms, and splines, J. Approx. Theory 86(1996) 81–107.
T.N.T. Goodman and S.L. Lee, B-splines on the circle and trigonometric B-splines, in: Approximation Theory and Spline Functions, eds. S.P. Singh, J.H.W. Burry and B. Watson (Reidel, Dordrecht, 1984) pp. 297–325.
T.N.T. Goodman and A. Sharma, Trigonometric interpolation, Proc. Edinburgh Math. Soc. (2) 35(1992) 457–472.
L. Hörmander, An Introduction to Complex Analysis in Several Variables(North-Holland, Amsterdam, 1990).
S. Karlin and W.J. Studden, Tchebycheff Systems: with Applications in Analysis and Statistics(Wiley, New York, 1966).
P.E. Koch, Jackson-type estimates for trigonometric splines, in: Industrial Mathematics Week, Trondheim, August 1992, Department of Mathematical Sciences, Norwegian Institute of Technology (NTH), Trondheim (1992) pp. 117–124.
P.E. Koch, T. Lyche, M. Neamtu and L.L. Schumaker, Control curves and knot insertion for trigonometric splines, Adv. Comput. Math. 3 (1995) 405–424.
P.P. Korovkin, Linear Operators and Approximation Theory(Hindustan Publ. Corp., Delhi, 1960).
G.G. Lorentz, Bernstein Polynomials(University of Toronto Press, Toronto, 1953).
S. Morigi, Modelling with p-spline curves and surfaces, Dissertation, University of Bologna, Italy (1997).
M. Neamtu, H. Pottmann and L.L. Schumaker, Dual focal splines and rational curves with rational offsets, Math. Engrg. Indust. 7 (1998) 139–154.
M. Neamtu, H. Pottmann and L.L. Schumaker, Designing NURBS cam profiles using trigonometric splines, J. Mech. Design 120(1998) 175–180.
H. Prautzsch and L. Kobbelt, Convergence of subdivision and degree elevation, Adv. Comput. Math. 2(1994) 143–154.
J. Sánchez-Reyes, Single-valued curves in polar coordinates, Comput. Aided Design 22(1990) 19–26.
K. Yosida, Functional Analysis(Springer, Berlin, 1968).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Morigi, S., Neamtu, M. Some results for a class of generalized polynomials. Advances in Computational Mathematics 12, 133–149 (2000). https://doi.org/10.1023/A:1018908917139
Issue Date:
DOI: https://doi.org/10.1023/A:1018908917139