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Uncertainty principles and asymptotic behavior. (English)
Appl. Comput. Harmon. Anal. 16, No. 1, 19-43 (2004).
Summary: Various uncertainty principles for univariate functions are studied, including classes of such principles not considered before. For many uncertainty principles for periodic functions, the lower bound on the uncertainty is not attained. By considering Riemann sums, we show that for functions whose Fourier coefficients are sampled from the Gaussian with spacing $h$, the uncertainty approaches the lower bound as $h \to 0$ with order $O(h^2)$, whereas earlier work had shown at best $O(h)$. We deduce that there is a sequence of trigonometric polynomials of degree $k$ whose uncertainty approaches the lower bound with order $O(1/k^2)$ as $k\to \infty$. We also establish a general uncertainty principle for $n$ pairs of operators on a Hilbert space, $n=2,3,\dots$, which allows us to extend the above univariate uncertainty principles to such principles for functions of $n$ variables. Furthermore, we deduce an uncertainty principle for functions on the sphere $\Bbb S^n$ in $\Bbb R^{n+1}$, $n=2,3,\dots$, generalizing known results for radial functions and for real-valued functions on $\Bbb S^2$. By considering the above work on univariate uncertainty principles, we can similarly derive, for all our multivariate uncertainty principles, sequences of functions for which the lower bound on the uncertainty is approached.