Applied and computational harmonic analysis
The simultaneous study of signals in time and frequency domain, according to the uncertainty principle, and on different scales represents one of the main applications of Harmonic Analysis. The basic paradigm consists in the decomposition (analysis) of a signal in elementary packets, obtained by translations, dilations or modulations of a fixed window. The subsequent phase entails the processing and reconstruction (synthesis) of signals. The current research is mostly addressed to study overcomplete expansions (mainly wavelet and Gabor frames), where some redundancy makes the design of the system more flexible. More generally, but in the same spirit, one of the most recent and charming research themes concerns the classification of reproducing groups, beyond classical wavelet theory. The
applications are extremely widespread, from mobile communication to sensing and image processing. More recently, such techniques have been provided to be very useful in the study of partial differential equations, specially certain integral-differential equations arising in telecommunications problems, and also in mathematical physics (sparse representations of Schroedinger-type propegators, mathematical formalism of quantum mechanics, path integrals).