Stability estimates for adaptive focused time-frequency transforms

This contribution is a follow-up of a recent paper by the authors on adaptive, non-linear time-frequency transforms, focusing on the STFT based transforms. The adaptivity is provided by a focus function, that depends on the analyzed function or signal, and that adapts dynamically the time-frequency resolution of the analysis. Sticking to the continuous case setting, this work provides new stability results for the transform (stability with respect with the focus function). It also investigates in some details focus functions based upon regularized R{\'e}nyi entropies and show corresponding continuity results.

A class of non linear adaptive time-frequency transforms

This paper introduces a couple of new time-frequency transforms, designed to adapt their scale to specific features of the analyzed function. Such an adaptation is implemented via so-called focus functions, which control the window scale as a function of the time variable, or the frequency variable. In this respect, these transforms are non-linear, which makes the analysis more complex than usual.Under appropriate assumptions, some norm control can be obtained for both transforms in L^2(R) spaces, which extend the classical continuous frame norm control and guarantees well-definedness on L^2. Given the non-linearity of the transforms, the existence of inverse transforms is not guaranteed anymore, and is an open question. However, the results of this paper represent a first step towards a more general theory.Besides mathematical results, some elementary examples of time and frequency focus functions are provided, which can serve as staring point for concrete applications.