In this presentation we are going to take a look at the concept of variable bandwidth in the time-frequency setting. As band-limited functions theory is well-developed and rounded, one may ask: Why variable bandwidth? Recall what the basic drawback of Shannon sampling theorem is: a satisfying approximation requires a lot of out-of-interest-area samples, which is highly impractical. The localization problem is better addressed via time-frequency analysis methods. Besides, a time-limited part of a band-limited function is no longer band-limited, due to the uncertainty principle; so, the concept of variable bandwidth is a very natural choice (as suggested by Slepian in his Shannon lecture).
Various natural approaches found in the literature, trying to describe the idea of functions of variable bandwidth in one or the other way have serious shortcomings. This might also be the reason why one finds little theoretical work on this topic. For instance, proposed solutions do not apply on a larger class of signals/functions, no function space structure is achieved, not to talk about closed spaces or complete spaces. All of this, and a lot more, is achieved when we address variable bandwidth with time-frequency analysis tools.