Bandwidth Versus Time Concentration: The Heisenberg–Pauli–Weyl Inequality

The main result is that for quite general weight functions v, w\[ \| f \|_2 ≤ K( {\| {vf} \|_p + \| {w f} \|_q } )\] for all tempered distributions f for which, roughly speaking, the right side makes sense, where 1 ≤ p, q ≤ ∞, K is a constant independent of f, and f is the Fourier transform of f. As a corollary, if θ ,φ ≥ 0 satisfy θ > 1 / p^# and φ > 1 / q^#, where t^# = 2t / (t - 2), there exists K = K(p,q,θ ,φ ) such that \[ (1) \| f \|_2 ≤ K( {\| {|x|^θ f} \|_p + \| {|y|^φ f} \|_q } ) \] for all f. In this case the inequality is equivalent to \| f \|₂ ≤ Kα - α (1 - α )α - 1 \| |x|^θ f \|ₚ^α \| |y|^φ f \|q1 - α where α satisfies α (θ - 1 / p^# ) = (1 - α )(φ - 1 / q^# ). Hence it generalizes the classical uncertainty principle inequality (which is the case $p = q = 2$ and θ = φ = 1) and an inequality due to Hirschman (the case $p = q = 2$ and θ ,φ > 0). Also (1) is trivially true when θ = 0 and $p = 2$ or φ = 0 and $q = 2$ and it is shown that it is not possible apart from these three cases. One of the approaches to the main inequality is as follows: Suppose s, t ∈ [1,2] and E, F are subsets of R of finite measure. For all f ∈ L²\[ \| f \|_2 ≤ K\{ {( {∫E' {|f(x)|^s dx} } )^{{1 / s}} + ( {∫F' {| f(y)|^t dy} } )^{{1 / t}} } \}\] where $K = K(s,t,E,F)$ is independent of f and ' denotes complementation.