Abstract
Given a totally ordered setT containing at leastn+1 elements (say a subset ofR 1), the graph of the functiona:T→R n is called a Chebyshev curve (inR n) if the determinant of the matrix (a(t 1),a(t 2), ...,a(t n)) is either positive whenevert 1<t 2<...<t n or negative whenevert 1<t 2<...<t n. For finiteT a characterization of these curves (sequences) has been given by the author.
In this paper the result is extended to non-finiteT. The characterization proved here is an improved (reformulated) version of that given by the author for infiniteT.
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References
Uhrin, B.: A characterization of finite Chebyshev sequences inR n,Linear Algebra Appl. 18 (1977), 59–74.
Uhrin, B.: Theorem of Helly and the existence of generalized polynomials of prescribed values and signs,Seminar Notes, Mathematics No. 3, Hungarian Committee for Systems Analysis, Budapest, 1975, pp. 1–33.
Gantmacher, F. R. and Krein, M. G.:Oszillationsmatrizen, Oszillationskerne und kleine Schwingungen mechanischer Systeme, Academic V., Berlin, 1959.
Karlin, S. and Studden, W. J.:Tchebycheff Systems: with Applications to Analysis and Statistics, Interscience, New York, 1966.
Karlin, S.:Total Positivity, Stanford Univ. Press, Stanford, CA, 1968.
Sturmfels, B.: Cyclic polytopes andd-order curves,Geom. Dedicata 24 (1987), 103–107.
Zielke, R.:Discontinuous Čebyšev Systems, Springer, Berlin, Heidelberg, New York, 1979.
Tihanyi, A. and Uhrin, B.: About the generalized polynomials with prescribed signs on discrete sets,Ann. Univ. Sci. Budapest, Ser. Math. 19 (1976), 165–169.
Wulbert, D.: Oscillation spaces,J. Approx. Theory 69 (1992), 188–204.