Abstract
We show that the size of the 1-norm condition number of the univariate Bernstein basis for polynomials of degree n is O (2n / √n). This is consistent with known estimates [3], [5] for p = 2 and p = ∞ and leads to asymptotically correct results for the p-norm condition number of the Bernstein basis for any p with 1 ≤ p ≤ ∞.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. de Boor (1976): On local linear fuctionals which vanish at all B-splines but one. In: Theory of Approximation with Applications (A. G. Law, B. N. Sahney, eds.). New York: Academic Press, pp. 120–145.
C. de Boor (1990): The exact condition of the B-spline basis may be hard to determine. J. Approx. Theory, 60:344–359.
Z. Ciesielskii, J. Domsta (1985): The degenerate B-spline basis as basis in the space of algebraic polynomials. Ann. Polon. Math., XXVI:71–79.
R. DeVore, G. G. Lorentz (1993): Constructive Approximation. New York: Springer-Verlag.
T. Lyche (1978): A note on the condition numbers of the B-spline bases. J. Approx. Theory, 22:202–205.
T. Lyche, K. Scherer (1997): On the sup-norm condition number of the multivariate triangular Bernstein basis. In: Multivariate Approximation and Splines (G. Nürnberger, J. W. Schmidt, G. Walz, eds.). ISNM, Vol. 125. Basel: Birkhäuser Verlag, pp. 141–151.
T. Lyche, K. Scherer (2000): On thep-norm condition number of the multivariate triangular Bernstein basis. J. Comput. Appl. Math., 119:259–273.
K. Mørken (1984): On two topics in spline theory: Discrete splines and the equioscillating spline. Master’s thesis, University of Oslo.
J. R. Rice (1964): The Approximation of Functions, Vol. 1. Reading, MA: Addison-Wesley.
K. Scherer (2001): Lower bounds for Bernstein-Bézier condition number. In: Mathematical Methods for Curves and Surfaces: Oslo 2000 (T. Lyche, L. L. Schumaker, eds.). Nashville: Vanderbilt University Press, pp. 433–443.
K. Scherer, A. Yu. Shadrin (1999): New upper bound for the B-spline basis condition number. J. Approx. Theory, 99:217–229.
L. L. Schumaker (1981): Spline Functions: Basic Theory. New York: Wiley.
G. Szegő (1959): Orthogonal Polynomials. New York: American Mathematical Society.
A. F. Timan (1963): Theory of Approximation of Functions of a Real Variable. Oxford, UK: Pergamon Press.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Edward B. Saff.
Rights and permissions
About this article
Cite this article
Lyche, T., Scherer, K. On the L 1-condition number of the univariate Bernstein basis. Constr. Approx. 18, 503–528 (2002). https://doi.org/10.1007/s00365-002-0507-0
Received:
Revised:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s00365-002-0507-0