Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-06-08T19:20:57.461Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximation by Λ-Splines on the Circle

Published online by Cambridge University Press:  20 November 2018

T. N. T. Goodman ,
S. L. Lee  and
A. Sharma
T. N. T. Goodman
Affiliation:
University of Dundee, Dundee, Scotland
S. L. Lee
Affiliation:
University of Dundee, Dundee, Scotland
A. Sharma
Affiliation:
University of Dundee, Dundee, Scotland
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let Λ = {λ0,…, λn} denote a set of distinct integers and let Π(Λ) denote the set of all generalized polynomials of the form

For any given ζ on the unit circle U with

we consider the set Zk of points 1, ζ, ζ2, …, ζk−1 where

We shall denote by or the class of Λ-splines S(z) which satisfy the following conditions:

  • (i) S(z) ∊ Cn−1(U)

  • (ii) S(z)|Av ∊ Π(Λ) where

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 6 , 01 December 1985 , pp. 1085 - 1111
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Ahlberg, J. H., Nilson, E. N. and Walsh, J. L., Properties of analytic splines, I: Complex polynomial splines, J. of Analysis and Appl. 33 (1971), 234–257.Google Scholar
2. de Boor, C and Fix, G. J., Spline approximation by quasi-interpolants, J. Approx. Theory 8 (1973), 19–45.Google Scholar
3. Han-Lin, Chen, Interpolation and approximation on the unit circle, I, Math. Comp. No. 5/80 ISBN 82-7151-035-5 Quasi interpolant splines on the unit circle, J. Approx. Theory 38 (1983), 312–318.Google Scholar
4. Goodman, T. N. T. and Lee, S. L., B-splines on the circle and trigonometric B-splines, Proc. Conference on Approx. Theory, St. John's (Newfoundland). To appear.CrossRefGoogle Scholar
5. Hirschman, I. I. and Widder, D. V., Generalized Bernstein polynomials, Duke Math J. 76 (1949), 433–438.Google Scholar
6. Lorentz, G. G., Bernstein polynomials (University of Toronto Press, Toronto, 1953).Google Scholar
7. Marsden, M. J., An identity for spline functions with applications to variation diminishing spline approximation, J. Approx, theory 3 (1970), 7–49.Google Scholar
8. Micchelli, C. A. and Sharma, A., Spline functions on the circle: Cardinal L-sp lines revisited, Can. J. Math. 32 (1980), 1459–1473.Google Scholar
9. Schumaker, L., Spline functions I (John Wiley & Sons, New York, 1981).Google Scholar
10. Schoenberg, I.J., On trigonometric spline interpolation, J. Math. Mech. 13 (1964), 795–826.Google Scholar
11. Schoenberg, I.J., On polynomial spline functions on the circle (I and II), Proceedings of the Conference on Constructive Theory of Functions (Budapest, 1972), 403–433.Google Scholar
12. Schoenberg, I.J., On variation diminishing approximation methods. In On numerical approximation, 249–274 MRC Symposium (U. of Wisconsin Press, Madison, 1959).Google Scholar