International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 39   | 1 | 2 |

Section 1.3.2.4.4.3. Heisenberg's inequality, Hardy's theorem

G. Bricognea

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1.3.2.4.4.3. Heisenberg's inequality, Hardy's theorem

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The result just obtained, which also holds for [\bar{\scr F}], shows that the `peakier' [G_{\bf A}], the `broader' [{\scr F}[G_{\bf A}]]. This is a general property of the Fourier transformation, expressed in dimension 1 by the Heisenberg inequality (Weyl, 1931[link]):[\eqalign{&\left({\int} x^{2}|\,f(x)|^{2} \,\hbox{d}x\right) \left({\int} \xi^{2}|{\scr F}[\,f]( \xi)|^{2} \,\hbox{d}\xi \right)\cr &\quad \geq {1 \over 16\pi^{2}} \left({\int} |\,f(x)|^{2} \,\hbox{d}x\right)^{2},}]where, by a beautiful theorem of Hardy (1933)[link], equality can only be attained for f Gaussian. Hardy's theorem is even stronger: if both f and [{\scr F}[\,f]] behave at infinity as constant multiples of G, then each of them is everywhere a constant multiple of G; if both f and [{\scr F}[\,f]] behave at infinity as constant multiples of [G \times \hbox{monomial}], then each of them is a finite linear combination of Hermite functions. Hardy's theorem is invoked in Section 1.3.4.4.5[link] to derive the optimal procedure for spreading atoms on a sampling grid in order to obtain the most accurate structure factors.

The search for optimal compromises between the confinement of f to a compact domain in x-space and of [{\scr F}[\,f]] to a compact domain in ξ-space leads to consideration of prolate spheroidal wavefunctions (Pollack & Slepian, 1961[link]; Landau & Pollack, 1961[link], 1962[link]).

References

Hardy, G. H. (1933). A theorem concerning Fourier transforms. J. London Math. Soc. 8, 227–231.
Landau, H. J. & Pollack, H. O. (1961). Prolate spheroidal wave functions, Fourier analysis and uncertainty (2). Bell Syst. Tech. J. 40, 65–84.
Landau, H. J. & Pollack, H. O. (1962). Prolate spheroidal wave functions, Fourier analysis and uncertainty (3): the dimension of the space of essentially time- and band-limited signals. Bell Syst. Tech. J. 41, 1295–1336.
Pollack, H. O. & Slepian, D. (1961). Prolate spheroidal wave functions, Fourier analysis and uncertainty (1). Bell Syst. Tech. J. 40, 43–64.
Weyl, H. (1931). The Theory of Groups and Quantum Mechanics. New York: Dutton. [Reprinted by Dover Publications, New York, 1950.]








































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