Incompatibility of Frequency Splitting and Spatial Localization: A Quantitative Analysis of Hegerfeldt's Theorem.

We prove quantitative versions of the following statement: If a solution of the 1 + 1 -dimensional wave equation has spatially compact support and consists mainly of positive frequencies, then it must have a significant high-frequency component. Similar results are proven for the 3 + 1 -dimensional wave equation.