7. Conclusions

We for the first time to our best knowledge applied CWT for the theoretical analysis of superoscillations in the time and frequency domain. We discussed the basic properties of the superoscillating signals containing the components with the frequencies larger than the maximum frequency in the signal spectrum. We also considered some possible applications of superoscillations in optics and signal processing. The superoscillating components are extremely weak and short in the time domain. They cannot be identified by the Fourier transform since they require the time-frequency analysis. We discussed the fundamental properties of CWT and DWT and their typical applications. The CWT is a unique tool for the superoscillation studies because it provides the localization of the signal both in time and in the frequency domain. We used the Mexican hat and the Morlet mother wavelets for the CWT of the sinusoidal superoscillating signal because these mother wavelets are similar to the signal oscillations. The theoretical results clearly show that the superoscillation frequency, time duration, and energy contours can be identified by using the CWT of the corresponding signal. Generally, CWT with different mother wavelets can be used for the analysis of superoscillating signals with different structures.
