Revista Matemática Iberoamericana

Vector-valued distributions and Hardy's uncertainty principle for operators

M. G. Cowling , B. Demange , and M. Sundari

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Suppose that $f$ is a function on $\mathbb{R}^n$ such that $\exp(a |\cdot|^2) f$ and $\exp(b |\cdot|^2) \hat f$ are bounded, where $a,b > 0$. Hardy's Uncertainty Principle asserts that if $ab > \pi^2$, then $f = 0$, while if $ab = \pi^2$, then $f = c\exp(-a|\cdot|^2)$. In this paper, we generalise this uncertainty principle to vector-valued functions, and hence to operators. The principle for operators can be formulated loosely by saying that the kernel of an operator cannot be localised near the diagonal if the spectrum is also localised.

Article information

Rev. Mat. Iberoamericana, Volume 26, Number 1 (2010), 133-146.

First available in Project Euclid: 16 February 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B05: Fourier series and coefficients 47G10: Integral operators [See also 45P05]

uncertainty principle linear operators Hardy's theorem


Cowling, M. G.; Demange, B.; Sundari, M. Vector-valued distributions and Hardy's uncertainty principle for operators. Rev. Mat. Iberoamericana 26 (2010), no. 1, 133--146.

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