Abstract
For a given Tchebycheff space, it is analyzed the relationship between the ability of enlarging the domain of definition in order to get an extension of the Tchebycheff space and the fact that the space has a strictly total positive basis. In this sense, we improve some results known in the mathematical literature. Some criteria for obtaining strictly totally positive bases of Tchebycheff spaces are provided. It is shown that complete Tchebycheff systems which can be extended to a larger domain in a given space are closely related to strictly totally positive systems. We also provide a characterization of all extensible complete Tchebycheff systems in terms of the matrix of change of basis with respect to a particular basis of the space. A collection of examples are provided in order to show the limits of application of the mentioned results.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ando, T., Totally positive matrices, Linear Algebra Appl. 90 (1987), 165–219.
Carnicer, J. M., and Peña, J. M., Totally positive bases for shape preserving curve design and optimality of B-splines. Computer-Aided Geom. Design 11 (1994), 633–654.
Carnicer, J. M., and Peña, J. M., On transforming a Tchebycheff system into a strictly totally positive system, J. Approx. Theory 81 (1995), 274–295.
Carnicer, J. M., and Peña, J. M., Characterizations of the optimal Descartes’ rules of signs, submitted for publication.
Gasca, M., and Peña, J. M., Total positivity and Neville elimination, Linear Algebra Appl. 165 (1992), 25–44.
Gasca, M., and Peña, J. M., Total positivity, QR-factorization and Neville elimination, SIAM J. on Matrix Anal. Appl. 14 (1993), 1132–1140.
Karlin, S., Total positivity, Vol. I, Stanford University Press, Standford, 1968.
Karlin, S., and Studden, W., Tchebycheff systems: with applications in analysis and statistics, Wiley, New York, 1966.
Mühlbach, G., A Recurrence Formula for Generalized Divided Differences and Some Applications, J. Approx. Th. 9 (1973), 165–172.
Németh, A. B., About the extension of the domain of definition of the Tchebycheff systems defined on intervals of the real axis, Mathematica ( Clui ) 11 (1969) , 307–310.
Pólya, G., On the mean value theorem corresponding to a given linear homogeneous differential equation, Trans. Amer. Math. Soc. 24 (1922), 312–324.
Pólya, G., Szegö, G., Problems and Theorems in Analysis vol. II, Springer Verlag, New York, 1976.
Schumaker, L. L., On Tchebycheffian Spline Functions, J. Approx. Th. 18 (1976), 278–303.
Schumaker, L. L., Spline Functions: Basic Theory, John Wiley and Sons, New York, 1981.
Sommer, M., and Strauss, H., A Characterization of Descartes Systems in Haar Subspaces, J. Approx. Th. 57 (1989), 104–116.
Zalik, R. A., On transforming a Tchebycheff system into a complete Tchebycheff system, J. Approx. Th. 20 (1977), 220–222.
Zalik, R. A., and Zwick, D., On Extending the Domain of Definition of Čebyšev and Weak Čebyšev Systems, J. Approx. Th. 57 (1989), 202–210.
Zielke, R., Discontinuous Čebyšev systems, Lecture Notes in Mathematics 707, Springer Verlag, New York, 1979.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Carnicer, J.M., Peña, J.M. (1996). Tchebycheff spaces and total positivity. In: Gasca, M., Micchelli, C.A. (eds) Total Positivity and Its Applications. Mathematics and Its Applications, vol 359. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8674-0_14
Download citation
DOI: https://doi.org/10.1007/978-94-015-8674-0_14
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4667-3
Online ISBN: 978-94-015-8674-0
eBook Packages: Springer Book Archive