Abstract
In a recent paper [4] we 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. First, observations qua evaLuation functionals are restricted to a finite part of the Fourier Peter Weyl expansion; that restriction implicitly involves a convoLution. See (1.4) and (1.6) below. We think of the resulting error as the error in the head of the approximation. Second, the actual initial data minus the truncations are the usual type of error terms; we think of them as the error in the tail of the approximation. In this paper we show that the error in the head depends linearly on the error in the tail. We then investigate the extent to which smoothness of the initial data function controls the tail error through a set of Sobolev inequalities. We also investigate consequences of polynomial spectral growth conditions on the rate of vanishing of the tail error. Finally, we specialize these results to riemannian symmetric spaces.
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References
R. S. Cahn and J. A. Wolf, Zeta functions and their asymptotic expansions for compact symmetric spaces of rank one, Comm. Math. Heiv. 51 (1976), 1–21.
É. Cartan, Sur la détermination d’un systéme orthogonal complet dans un espace de Riemann symétrique clos, Rendiconti Palermo 53 (1929), 217–252.
S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, 1962.
D. I. Wallace and J. A. Wolf, Observability of evoLution equations for invariant differential operators, J. Math. Systems, Estimation, and Control, to appear in 1990.
N. R. Wallach, Harmonic Analysis on Homogeneous Spaces, Marcel Dekker, 1973.
G. Warner, Harmonic Analysis on Semisimple Lie Groups, I, Springer Verlag, 1972.
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.
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© 1991 Springer Science+Business Media New York
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Wallace, D.I., Wolf, J.A. (1991). Acuity of Observation for Invariant Evolution Equations. In: Bowers, K., Lund, J. (eds) Computation and Control II. Progress in Systems and Control Theory, vol 11. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0427-5_22
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DOI: https://doi.org/10.1007/978-1-4612-0427-5_22
Publisher Name: Birkhäuser, Boston, MA
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Online ISBN: 978-1-4612-0427-5
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