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Transactions of the American Mathematical Society

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$n$-unisolvent sets and flat incidence structures

Author: Burkard Polster
Journal: Trans. Amer. Math. Soc. 350 (1998), 1619-1641
MSC (1991): Primary 41A05, 51H15; Secondary 05B15, 51B15
MathSciNet review: 1407710
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Abstract: For the past forty years or so topological incidence geometers and mathematicians interested in interpolation have been studying very similar objects. Nevertheless no communication between these two groups of mathematicians seems to have taken place during that time. The main goal of this paper is to draw attention to this fact and to demonstrate that by combining results from both areas it is possible to gain many new insights about the fundamentals of both areas. In particular, we establish the existence of nested orthogonal arrays of strength $n$, for short nested $n$-OAs, that are natural generalizations of flat affine planes and flat Laguerre planes. These incidence structures have point sets that are ``flat'' topological spaces like the Möbius strip, the cylinder, and strips of the form $I \times \mathbb{R}$, where $I$ is an interval of $\mathbb{R}$. Their circles (or lines) are subsets of the point sets homeomorphic to the circle in the first two cases and homeomorphic to $I$ in the last case. Our orthogonal arrays of strength $n$ arise from $n$-unisolvent sets of half-periodic functions, $n$-unisolvent sets of periodic functions, and $n$-unisolvent sets of functions $I\to \mathbb{R}$, respectively.

Associated with every point $p$ of a nested $n$-OA, $n>1$, is a nested $(n-1)$-OA-the derived $(n-1)$-OA at the point $p$. We discover that, in our examples that arise from $n$-unisolvent sets of $n-1$ times differentiable functions that solve the Hermite interpolation problem, deriving in our geometrical sense coincides with deriving in the analytical sense.

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  • 1. D. Betten, Topologische Geometrien auf dem Möbiusband, Math. Z. 107 (1968), 363-379. MR 38:6607
  • 2. Ph. C. Curtis, Jr., $n$-parameter families and best approximation, Pacific J. Math. 9 (1959), 1013-1027. MR 21:7385
  • 3. W. Forst, Variationsmindernde Eigenschaften eines speziellen Kreinschen Kernes, Math. Z. 148 (1976), 67-70. MR 53:5886
  • 4. D. G. Glynn, A geometrical representation theory for orthogonal arrays, Bull. Austr. Math. Soc. 49 (2) (1994), 311-324. MR 94m:05039
  • 5. H. Groh, Topologische Laguerreebenen I, Abh. Math. Sem. Univ. Hamburg 32 (1968), 216-231. MR 38:2660
  • 6. H. Groh, Topologische Laguerreebenen II, Abh. Math. Sem. Univ. Hamburg 34 (1970), 11-21. MR 41:2506
  • 7. P. Hartman, Unrestricted $n$-parameter families, Rend. Circ. Mat. Palermo (2) 7 (1958), 123-142. MR 21:4211
  • 8. W. Heise, Optimal codes, n-arcs and Laguerre geometry, Acta Informatica 6 (1976), 403-406. MR 54:12356
  • 9. D. J. Johnson, The trigonometric Hermite-Birkhoff interpolation problem, Trans. Amer. Math. Soc. 212 (1976), 365-374. MR 54:5712
  • 10. S. Karlin - W. J. Studden, Tchebycheff Systems with Applications in Analysis and Statistics, Interscience, New York, 1966. MR 34:4757
  • 11. G. G. Lorentz - K. Jetter - S. D. Riemenschneider, Birkhoff interpolation, Addison-Wesley, Reading, MA, 1983, and Cambridge University Press, New York, 1984. MR 84g:41002
  • 12. R. A. Lorentz, Simultaneous Approximation and Birkhoff Interpolation II: The periodic case, J. Approx. Theory 44 (1985), 21-29. MR 86j:41018
  • 13. R. A. Lorentz, Some new periodic Chebysheff Subspaces, J. Approx. Theory 61 (1990), 13-22. MR 81a:41006
  • [14] R. M. Mathsen, $\lambda (n)$-parameter families, Canad. Math. Bull. 12 (1969), 185-191. MR 39:7191
  • [15] T. S. Motzkin, Approximation by curves of a unisolvent family, Bull. Amer. Math. Soc. 55 (1949), 789-793. MR 11:101f
  • [16] F. R. Moulton, A simple non-desarguesian plane geometry, Trans. Amer. Math. Soc. 3 (1902), 192-195.
  • [17] B. Polster, Integrating and differentiating two-dimensional incidence structures, Arch. Math 64 (1995), 75-85. MR 95i:51015
  • [18] B. Polster, Integrating completely unisolvent functions, J. Approx. Theory 82 (1995), 434-439. MR 96g:26016
  • [19] J. R. Rice, The Approximation of Functions I, Addison-Wesley, Reading, MA, 1964. MR 29:3795
  • [20] J. R. Rice, The Approximation of Functions II, Addison-Wesley, Reading, MA, 1969. MR 39:5989
  • [21] H. Salzmann, Topological planes, Adv. Math. 2 (1967), 1-60. MR 36:3201
  • [22] H. Salzmann - D. Betten - T. Grundhöfer - H. Hähl - R. Löwen - M. Stroppel, Compact projective planes, de Gruyter, Berlin, 1995. MR 97b:51009
  • [23] L. L. Schumaker, Spline functions: Basic theory, Wiley, New York, 1981. MR 82j:41001
  • [24] G. F. Steinke, Topological affine planes composed of two Desarguesian halfplanes and projective planes with trivial collineation group, Arch. Math. 44 (1985), 472-480. MR 86j:51023
  • [25] G. F. Steinke, Semiclassical topological flat Laguerre planes obtained by pasting along two parallel classes, J. Geom. 32 (1988), 133-156. MR 89k:51021
  • [26] G. F. Steinke, Topological circle geometries, in: Handbook of Incidence Geometry (F. Buekenhout, ed.), Elsevier, pp. 1325-1353, 1995. MR 96i:51015
  • [27] L. Tornheim, On $n$-parameter families of functions and associated convex functions, Trans. Amer. Math. Soc. 69 (1950), 457-467. MR 12:395d
  • [28] M. L. H. Willems, Optimal codes, Laguerre and special Laguerre $i$-structures, Eur. J. Comb. 4 (1983), 87-92. MR 85a:51007
  • [29] M. L. H. Willems - J. A. Thas, A note on the existence of special Laguerre i-structures and optimal codes, Eur. J. Comb. 4 (1983), 93-96. MR 85a:51008

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Additional Information

Burkard Polster
Affiliation: Department of Pure Mathematics, The University of Adelaide, Adelaide, SA 5005, Australia

Keywords: Non-linear interpolation, unisolvent set, Chebyshev system, Laguerre plane, Laguerre $m$-structure, orthogonal array, topological incidence geometry
Received by editor(s): October 3, 1994
Received by editor(s) in revised form: July 20, 1996
Additional Notes: This research was supported by a Feodor Lynen fellowship.
Article copyright: © Copyright 1998 American Mathematical Society

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