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Distinguishing smooth functions by a finite number of point values, and a version of the Takens embedding theorem



We prove a general result showing that a finite-dimensional collection of smooth functions whose differences cannot vanish to infinite order can be distinguished by their values at a finite collection of points; this theorem is then applied to the global attractors of various dissipative parabolic partial differential equations. In particular for the one-dimensional complex Ginzburg-Landau equation and for the Kuramoto-Sivashinsky equation, we show that a finite number of measurements at a very small number of points (two and four, respectively) serve to distinguish between different elements of the attractor: this gives an infinite-dimensional version of the Takens time-delay embedding theorem. (C) 2004 Elsevier B.V. All rights reserved

Topics: QA, QC
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