Continuous Stockwell transform: Coherent states and localization operators
Mária Hutníková, Anna Mišková · Journal of Mathematical Physics · 2015
There are several methods of interest in signal analysis and image processing. The most widely used are the wavelet transform, the short-time Fourier transform, the shearlet transform, and the Stockwell transform. Putting a filter in the corresponding reproducing formula, one gets the well-known Calderón-Toeplitz and Gabor-Toeplitz localization operators widely studied in the context of time scale, time-frequency as well as shearlet analysis. Using Vasilevski’s technique of Hilbert-space decomposition (applied for the space of continuous Stockwell transforms of L2(ℝ)-functions), we get the structural results of the transform space and we study Toeplitz operators in this context with many desirable properties of localization operators. We find their unitary equivalent images for the case of separable generating symbols and show that the Wick symbols for these operators are associated with a well-defined calculus. Also, certain algebras generated by these operators are described in detail. Thus, the results complete the full picture about properties and behavior of Toeplitz localization operators related to four most commonly used signal transforms.