A version of the Uncertainty Principle for functions with lacunary Fourier transforms ✩

We prove estimates of the L2 norms on relatively dense subsets of the real line of functions with Fourier transforms supported on lacunary sets of intervals. This result is a real line analogue of Zygmund’s theorem on lacunary trigonometric series. The results also hold in higher dimensions.  2003 Elsevier Inc. All rights reserved