Superoscillations of Prescribed Amplitude and Derivative

Superoscillations occur when a bandlimited signal oscillates at a rate higher than its maximum frequency. We show that it is possible to construct superoscillations by constraining not only the value of the signal but also that of its derivative. This allows a better control of the shape of the superoscillations. We find that for any given bandwidth, no matter how small, there exists a unique signal of minimum energy that satisfies any combination of amplitude and derivative constraints, on a sampling grid as fine as desired. We determine the energy of the signal, for any grid, regular or irregular. When the set of derivative constraints is empty the results reduce to minimum energy interpolation. In the absence of amplitude constraints, we obtain pure derivative-constrained extremals. The flexibility gained by having two different types of constraints makes it possible to design superoscillations based only on amplitudes, based only on derivatives, or based on both. In the last case, the amplitude and derivative sampling grids can be interleaved or aligned. We explore this flexibility to build superoscillations that cost less energy. Illustrating examples are given.