Realizable algorithm for approximating Hilbert–Schmidt operators via Gabor multipliers

In this work, we consider new computational aspects to improve the approximation of Hilbert–Schmidt operators via generalized Gabor multipliers. One aspect is to consider the approximation of the symbol of an Hilbert–Schmidt operator as L2 projection in the spline-type space associated to a Gabor multiplier. This gives the possibility to employ a selection procedure of the analysis and synthesis function, interpreted as time-frequency lag; hence, with the related algorithm it is possible to handle both underspread and overspread operators. In the numerical section, we exploit the case of approximating overspread operators having compact and smooth spreading function and discontinuous time-varying systems. For the latter, the approximation of discontinuities in the symbol is not directly achievable in the generalized Gabor multipliers setting. For this reason, another aspect is to further process the symbol through a Hough transform, to detect discontinuities and to smooth them using a new class of approximants. This procedure creates a bridge between features detection techniques and harmonic analysis methods, and in specific cases it almost doubles the accuracy of approximation.