Gabor Analysis in Weighted Amalgam Spaces

Gabor frames {e2πinβ·xg(x− kα)}n,k∈Zd provide series representations not only of functions in L(R) but of the entire range of spaces M ν known as the modulation spaces. Membership of a function or distribution f in the modulation space is characterized by a sequence-space norm of the Gabor coefficients of f depending only on the magnitudes of those coefficients, and the Gabor series representation of f converges unconditionally in the norm of the modulation space. This paper shows that Gabor expansions also converge in the entire range of amalgam spaces W (L, Lν), which are not modulation spaces in general but, along with the modulation spaces, play important roles in time-frequency analysis and sampling theory. It is shown that membership of a function or distribution in the amalgam space is characterized by an appropriate sequence space norm of the Gabor coefficients. However, this sequence space norm depends on the phase of the Gabor coefficients as well as their magnitudes, and the Gabor expansions converge conditionally in general. Additionally, some converse results providing necessary conditions on g are obtained.