A Gabor Approach to Functions of Variable Bandwidth

From variants of the so-called Shannon sampling theorem it is well known that a band-limited L2-function can be recovered from samples taken a sufficiently dense lattice (Nyquist criterion). Previous work of NuHAG members (Feichtinger/Groechenig/Strohmer) has shown that it is also possible to recover a band-limited functions from samples taken over irregular point sets. Indeed, there are iterative algorithms, which allow to perform the reconstruction in a very efficient way. When it comes to practical applications of such algorithms to "long/large" signals, the assumption of a fixed global bandwidth may often be rather restrictive. Therefore, a sound mathematical concept of "functions of variable bandwidth" is called for. Despite the fact that such a term has a very natural practical meaning, it is rather tricky to come up with a satisfactory and mathematically sound definition of this concept. Definitions suggested so far seriously lack some of the desired properties (e.g., compatibility with natural concepts of time-variant filtering, or correct behaviour under time-variant stretching operations, to give two simple examples). In the proposed project a new approach to this concept is taken, based on methods from Gabor analysis (the other speciality of NuHAG research, besides irregular sampling), and more generally the theory of coorbit spaces as developed by Feichtinger/Groechenig in the late 80ies. Starting point is the idea that signals having a time- frequency concentration essentially within a certain strip of variable width in the time-frequency plane should have in such a "Hilbert space of functions of variable bandwidth" the same norm as in L2, yet deviations from this strip should be punished and result in a rather big norm: We propose to define such spaces as coorbit spaces (with respect to the Schroedinger representation, or expressend more practically, characterized by the behaviour of the short-time Fourier transforms of its elements), derived from appropriate weighted L2-spaces whose weights reflect the variable bandwidth. It is expected (and first results confirm this) that this new family of reproducing kernel Hilbert spaces shows correct behaviour with respect to the above mentioned properties. In particular, it will be possible to show - maybe under some mild extra conditions - that functions in such a space, sampled according to the local Nyquist rate, can be approximately recovered, again by iterative methods, and that the remaining error term will be controlled completely by the oversampling rate. Numerical examples are supposed to confirm the practical relevance of the theory to be developed.

From variants of the so-called Shannon sampling theorem it is well known that a band-limited L2-function can be recovered from samples taken a sufficiently dense lattice (Nyquist criterion). Previous work of NuHAG members (Feichtinger/Groechenig/Strohmer) has shown that it is also possible to recover a band-limited functions from samples taken over irregular point sets. Indeed, there are iterative algorithms, which allow to perform the reconstruction in a very efficient way. When it comes to practical applications of such algorithms to "long/large" signals, the assumption of a fixed global bandwidth may often be rather restrictive. Therefore, a sound mathematical concept of "functions of variable bandwidth" is called for. Despite the fact that such a term has a very natural practical meaning, it is rather tricky to come up with a satisfactory and mathematically sound definition of this concept. Definitions suggested so far seriously lack some of the desired properties (e.g., compatibility with natural concepts of time-variant filtering, or correct behaviour under time-variant stretching operations, to give two simple examples). In the proposed project a new approach to this concept is taken, based on methods from Gabor analysis (the other speciality of NuHAG research, besides irregular sampling), and more generally the theory of coorbit spaces as developed by Feichtinger/Groechenig in the late 80ies. Starting point is the idea that signals having a time-frequency concentration essentially within a certain strip of variable width in the time-frequency plane should have in such a "Hilbert space of functions of variable bandwidth" the same norm as in L2, yet deviations from this strip should be punished and result in a rather big norm: We propose to define such spaces as coorbit spaces (with respect to the Schroedinger representation, or expressend more practically, characterized by the behaviour of the short-time Fourier transforms of its elements), derived from appropriate weighted L2-spaces whose weights reflect the variable bandwidth. It is expected (and first results confirm this) that this new family of reproducing kernel Hilbert spaces shows correct behaviour with respect to the above mentioned properties. In particular, it will be possible to show - maybe under some mild extra conditions - that functions in such a space, sampled according to the local Nyquist rate, can be approximately recovered, again by iterative methods, and that the remaining error term will be controlled completely by the oversampling rate. Numerical examples are supposed to confirm the practical relevance of the theory to be developed.