Time-frequency analysis and coorbit spaces of operators

Abstract

To analyse the time-frequency content of an ordered data set, it is desirable to consider cross-correlation of time-frequency contents of data points, while maintaining the order of such correlations. To this end, we introduce an operator-valued short-time Fourier transform for certain classes of operators with operator windows, and show that the transform acts in a way analogous to the way the short-time Fourier transform for functions acts, in particular giving rise to a family of vector-valued reproducing kernel Banach spaces, the so-called coorbit spaces, as spaces of operators. As a result of this structure, the operators generating equivalent norms on the function modulation spaces are fully classified. We show that these classes of operators have the same atomic decomposition properties as the function spaces, and use this to give a characterisation of the spaces using localisation operators.