## Big question

It is commonly believed that a signal cannot vary on time scales that are smaller than its bandwidth would indicate. As was shown by Sir Michael Berry, Yakir Aharonov and others, this is not so: there are signals that locally oscillate faster than expected. Why is that so, what is the significance and how can we make super-oscillatory signals or wave functions?

## Basic idea

I showed that superoscillations can exist (and not violate Shannon's channel capacity theorem) because they trade signal-to-noise ratio for bandwidth. With Paulo Ferreira, I also calculated the most efficient method for designing such signals. We can now easily make acoustic superoscillations. I am currently mostly interested in the behaviours of superoscillations in quantum wave functions.

## Selected publications

- P.J.S.G. Ferreira, A. Kempf, M.J.C.S. Reis, Construction of Aharonov–Berry's superoscillations, J. Phys. A40, 5141 (2007)
- P.J.S.G. Ferreira and A. Kempf, Superoscillations: Faster than the Nyquist Rate, IEEE Trans. Signal Processing, 54, 3732 (2006)

## Projects

- Quantum gravity I
- Quantum gravity II
- Cosmology
- Relativistic quantum information
- Quantum computing
- Communication engineering
- Shannon sampling theory/data compression
- Mathematical biology
- Radar signal design for maximum information return
- The Casimir effect in layered superconductors
- Combinatorics in quantum field theory
- Further interests: Consciousness