Functions of variable bandwidth via time-frequency analysis tools

This paper is motivated by the apparent lack of a precise mathematical description of the very “natural” idea of variable bandwidth for a function, defined on the real line. Different existing concepts suffer from serious shortcomings. We will present a new and mathematically well justified function space model, which is based on the theory of coorbit spaces, in a time-frequency context. Making use of the flexibility of this approach we define a family of Hilbert spaces which can be viewed as generalised modulation spaces. More precisely, we define a family of Banach spaces of functions with variable bandwidth (VB-functions) by imposing a weighted mixed-norm condition on the short-time Fourier transform of their elements, “punishing” the contributions outside a strip of variable width described by the band-width function b. Similar bandwidth functions define the same space with equivalent norms. Any “good” Gabor frame (e.g. with Schwartz windows) can be used to characterize the membership of a function in such a space. Moreover, various classes of functions, e.g. those obtained by a time-variant filter, are shown to belong to such a space. In addition, error estimates are given when approximating certain subclasses.