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Distinguishing smooth functions by a finite number of point values, and a version of the Takens embedding theorem. (English) Zbl 1058.35045

Summary: 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.

MSC:

35B41 Attractors
35Q53 KdV equations (Korteweg-de Vries equations)
35Q30 Navier-Stokes equations
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