Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-17T23:24:04.886Z Has data issue: false hasContentIssue false
  • English
  • Français

Spline Functions on the Circle: Cardinal L-Splines Revisited

Published online by Cambridge University Press:  20 November 2018

Charles A. Micchelli  and
A. Sharma
Charles A. Micchelli
Affiliation:
IBM Research Center, Yorktown Heights, New York
A. Sharma
Affiliation:
University of Alberta, Edmonton, Alberta
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.

Although the literature on splines has grown vastly during the last decade [11], the study of polynomial splines on the circle seems to have suffered neglect. The first to study the subject in depth seem to be Ahlberg, Nilson and Walsh [1]. Almost at the same time I. J. Schoenberg [8] studied the problem of interpolation at the roots of unity by splines and its relation to quadrature on the circle. For discrete polynomial splines on the circle we refer to [5]. M. Golomb [3] also considers interpolation by a class of “spline” functions in the complex plane but his point of view is based on minimum norm properties of spline functions. Perhaps the reason for this neglect may be attributed to the fact that one can pass from the circle to the line by means of the transformation z → exp 2-πix. This changes the problem on the circle into periodic interpolation on the line with the difference that instead of interpolation by piecewise polynomial, we now consider piecewise exponential polynomials with complex exponents.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 6 , 01 December 1980 , pp. 1459 - 1473
Copyright
Copyright © Canadian Mathematical Society 1980

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. Friedland, S. and Micchelli, C. A., Bounds on the solution of difference equations and spline interpolation at knots, J. of Linear Algebra and Appl. 21 (1978), 219–251.Google Scholar
3. Golomb, M., Interpolation operators as optimal recovery schemes, in Optimal estimation in approximation theory (Plenum Press, 1977).Google Scholar
4. Karlin, S., Total positivity (Stanford University Press, 1967).Google Scholar
5. Mathur, K. K. and Sharma, A., Discrete polynomial splines on the circle, Acta Math. Scien. Hungarica 33 (1978), 143–153.Google Scholar
6. Micchelli, C. A., Cardinal L-splines, in Studies in spline functions and approximation theory (Academic Press, 1976).Google Scholar
7. Newman, D. J., A review of Müntz-Jackson theorem, in Approximation theory (Academic Press, 1973).Google Scholar
8. 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
9. Schoenberg, I. J., On Micchelli's theory of cardinal L-splines, in Studies in spline functions and approximation theory (Academic Press, 1976).Google Scholar
10. Schoenberg, I. J., Cardinal spline interpolation, Regional Conf. Series in Appl. Math. 12 (SIAM, Philadelphia, 1973).CrossRefGoogle Scholar
11. van Rooij, P. L. J. and Schurer, F., A bibliography on spline functions, in Tagung über Spline-Funktionen (Bibiographisches Institut, Mannheim, 1974), 315–414.Google Scholar
12. Sharma, A. and Tzimbalario, J., A class of cardinal trigonometric splines, SIAM J. of Math. Anal. 7 (1976), 809–819.Google Scholar
13. Tzimbalario, J., Interpolation by complex splines, Trans. Amer. Math. Soc. 243 (1978), 213–222.Google Scholar