Uncertainty principles invariant under the fractional Fourier transform. (English) Zbl 0765.43004

Mustard, David

J. Aust. Math. Soc., Ser. B 33, No. 2, 180-191 (1991).

The fractional Fourier transform of an \(L^ 2({\mathbf R})\) function \(f\) may be defined by writing \(f\) as a sum \(\sum_{n \geq 0} a_ n h_ n,\) where \((a_ n)\) is a square summable sequence and the \(h_ n\) are the Hermite functions. Then \({\mathcal F}_ \theta f = \sum_{n \geq 0} a_ n e^{i \theta n} h_ n.\) For a function of norm \(1\) and mean \(0,\) whose Fourier transform also has mean \(0,\) the Heisenberg uncertainty principle states that \[ \sigma_ H(f) = \| xf(x) \|_ 2 \| t\widehat{f}(t) \| _ 2 \geq C, \] where the constant \(C\) depends on where one puts the factor \(2\pi\) in the Fourier transform. The measure of “spread” \(\sigma_ H\) is not invariant under the fractional Fourier transform. Measures of spread \(\sigma_ k\) are developed which are invariant under this transform. The first of these, \(\sigma_ 1,\) improves the Heisenberg measure of spread in that \(\sigma_ H(f) \geq \sigma_ 1(f) \geq C.\)

Reviewer: M.Cowling (Kensington)

MSC:

43A30

Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.

Keywords:

fractional Fourier transform; Hermite functions; Heisenberg uncertainty principle; Heisenberg measure