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

Abstract

Suppose that f is a function on R n such that exp(a | · | 2 )f and exp(b | · | 2 )f are bounded, where a, b > 0. Hardy's Uncertainty Principle asserts that if ab > π 2 , then f = 0, while if ab = π 2 , then f = c exp(−a | · | 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.Date: May 6, 2008.

An Note on Uncertainty Inequalities for Deformed Harmonic Oscillators

“…The present paper is a continuation of our previous papers [1][2][3] to prove some uncertainty principles (UP) for a general class of integral operators, including the Fourier transform, the Fourier-Bessel transform, the Dunkl transform [4], the generalized Fourier transform [5], the deformed Fourier transform [6] and the Clifford transform. Other versions of UP for integral operators have been proved in [7][8][9].…”

Fourier-Like Multipliers and Applications for Integral Operators

“…This class includes the usual Fourier transform, the Fourier-Bessel (or Hankel) transform, the Dunkl transform and the deformed Fourier transform as particular cases. A version of Hardy's and Donoho-Stark's uncertainty principles for integral operators has been proved in [7,8]. In this paper, we consider results of a different nature on the subspaces of functions that are essentially timelimited on S and bandlimited on Σ, or functions that are essentially concentrated on S and bandlimited on Σ, where S and Σ are general subsets of finite measure.…”

“…In particular, T extends to an unitary transform from L 2 (Ω, µ) onto L 2 (Ω, µ), such that 7) and for all f ∈ L 2 (Ω, µ), T −1 f (ξ) = Tf (ξ), ξ ∈ Ω.…”