Bounds on Space-Time-Frequency Dimensionality

We bound the number of electromagnetic signals which may be observed over a frequency range 2W for a time T within a region of space enclosed by a radius R. Our result implies that broadband fields in space cannot be arbitrarily complex: there is a finite amount of information which may be extracted from a region of space via electromagnetic radiation. Three-dimensional space allows a trade-off between large carrier frequency and bandwidth. We demonstrate applications in super-resolution and broadband communication.