Point Placement for Observation of the Heat Equation on the Sphere

  • Joseph A. Wolf
Part of the Progress in Systems and Control Theory book series (PSCT, volume 15)


In [14] Wallace and I studied discrete observability for invariant evolution equations on compact homogeneous spaces, e.g. for the heat equation on the sphere. The observations there were given by simultaneous measurements, corresponding to function evaluations. The initial data was observed as a limit of truncations deduced from a finite number of measurements. That procedure naturally involves two types of errors. Observations in the form of evaluation functionals are restricted to a finite part of the Fourier Peter Weyl expansion; that restriction implicitly involves a convolution. This discretization error occurs in the head of the approximation. The other error is the truncation error; that is the error in the tail of the approximation. In a later paper [15], we showed that the head (discretization) error depends linearly on the tail (truncation) error, and we investigated the consequences of various spectral growth conditions on the rate of vanishing of the tail error. Here I return to the rate of dependence of the head error on the tail error. This is a matter of placing the observation points, and the results to date are just the outcome of numerical experiments.


Heat Equation Observation Point Irreducible Unitary Representation Product Array Riemannian Symmetric Space 
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  1. 1.
    R. S. Cahn and J. A. Wolf, Zeta functions and their asymptotic expansions for compact symmetric spaces of rank one, Comm. Math. Helv. 51 (1976), 1–21.CrossRefGoogle Scholar
  2. 2.
    E. Cartan, Sur la détermination d’un Système orthogonal complet dans un espace de Riemann symétrique clos, Rendiconti Palermo 53 (1929), 217–252.CrossRefGoogle Scholar
  3. 3.
    A. Erdélyi, Editor, Higher Transcendental Functions, Volume II (Bateman Manuscript Project), Robert E. Krieger Publ. Co., Malabar, Fl., 1981.Google Scholar
  4. 4.
    F. A. Grünbaum, Mathematical Aspects of Computerized Tomography (Proceedings, Oberwolfach, 1980), Springer-Verlag, 1981.Google Scholar
  5. 5.
    S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, 1962.Google Scholar
  6. 6.
    S. I. Konayev, Quadratures of Gaussian type for a sphere invariant under the icosahedral group with inversion, Math. Note 25 (1979), 629-634.Google Scholar
  7. 7.
    V. I. Lebedev, Quadratures on a sphere, Zh. Vychisl. Mat. Fiz. 16 (1976), 293–306.Google Scholar
  8. 8.
    A. K. Louis, Optimal sampling in nuclear magnetic resonance zeugmatography, preprint, Fachb. Angew. Math. u Informatik, Saarbrücken (1981).Google Scholar
  9. 9.
    A. D. McLaren, Optimal numerical integration on a sphere, Math. Cornput. 17 (1963), 361–383.CrossRefGoogle Scholar
  10. 10.
    L. A. Shepp, Computerized tomography and nuclear magnetic resonance, J. Computerized Tomography 4 (1980), 94–107.CrossRefGoogle Scholar
  11. 11.
    S. L. Sobolev, On mechanical quadrature formulae on the surface of a sphere, Siberskii Mat. Zh. 3 (1962), 769–796.Google Scholar
  12. 12.
    A. M. Tarn, Optimal Choice of Directions for the Reconstruction of an Object from a Finite Number of its Plane Integrals, thesis, University of California, Berkeley, 1982.Google Scholar
  13. 13.
    N. Ya. Vilenkin, Special Functions and the Theory of Group Representations (Transl. Math. Monographs, Vol. 22), American Math. Society, 1968.Google Scholar
  14. 12.
    D. I. Wallace and J. A. Wolf, Observability of evolution equations for invariant differential operators, J. Math. Systems, Estimation, and Control 1 (1991), 29–44.Google Scholar
  15. 15.
    D. I. Wallace and J. A. Wolf, Acuity of observation for invariant evolution equations, in “Computation and Control II,” (Proceedings, Bozeman, 1990), Birkhäuser, Progress in Systems and Control Theory 11 (1991), 325–350.CrossRefGoogle Scholar
  16. 16.
    J. A. Wolf, Observability and group representation theory, in “Computation and Control,” (Proceedings, Bozeman, 1988), Birkhäuser, Progress in Systems and Control Theory 1 (1989), 385–391.Google Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Joseph A. Wolf
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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