An inequality associated with the uncertainty principle

Abstract

We prove$\left\| F \right\|_{2,\Omega } \leqslant c({\rm T} \Omega )\left\| f \right\|_{A{}_T}$, whereF is the Fourier transform off,||F||_2,Ω is theL ²-norm ofF on$[ - \Omega ,\Omega ],\left\| f \right\|_{A{}_T}$ is the absolutely convergent Fourier series norm for 2T-periodic functions, and

$$c(T\Omega ) = (\frac{1}{\pi }\int\limits_{ - T\Omega }^{T\Omega } {\frac{{\sin ^2 \gamma }}{{\gamma ^2 }}d\gamma } )^{1/2} $$

Analogous inequalities, depending on prolate spheroidal wave functions, are more difficult to prove and their constants are less explicit.