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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Abstract

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

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

Dates
First available in Project Euclid: 16 February 2010

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1266330120

Mathematical Reviews number (MathSciNet)
MR2666311

Zentralblatt MATH identifier
1195.42050

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

Keywords
uncertainty principle linear operators Hardy's theorem

Citation

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. https://projecteuclid.org/euclid.rmi/1266330120


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