Sampling Theory, Signal Processing, and Data Analysis
Modulation spaces form a class of function spaces that measure smoothness in terms of phase-space (time-frequency) concentration instead of differences and derivatives. This class of function spaces was introduced by Hans G. Feichtinger in the early 1980s and has become an important area of research in recent years. One main reason for the modern flourishing of modulation spaces is that, simply put, they are the "right" function spaces for time-frequency analysis and phase-space analysis. They
... ccur naturally throughout time-frequency analysis: in time-frequency expansions, in the formulation of fine uncertainty principles, in the construction of Weyl-Heisenberg frames (Gabor frames), and as symbol classes for pseudodifferential operators. The fact that the modulation spaces provide a good way to quantify time-frequency concentration and to describe time-varying systems is currently being exploited in such engineering applications as wireless communications. A quick search in the Mathematical Reviews yields 49 articles (including a few book chapters) that deal with modulation spaces. The vitality of this research direction is evidenced by the fact that 36 of these articles appeared after 2000. Ironically, the two most influential groups of articles are not included in this list. Feichtinger's original preprint from 1983, "Modulation spaces on locally compact abelian groups," did not appear in print until recently. Yet the impressive development of time-frequency analysis would not be possible without Feichtinger's fundamental work, which develops the basic theory of modulation spaces not only on JRd, but simultaneously on all locally compact abelian groups. The second set of articles missing from this list are "Wiener type algebras of pseudodifferential operators" and "An algebra of pseudodifferential operators" from 1994 and 1995. In these articles, Johannes Sjostrand introduced a particular modulation space as a non-standard symbol class for pseudodifferential operators. However, as he was unaware of the existing theory of modulation spaces, he named his function space differently, and so his work escapes a search for modulation spaces. Yet his papers have opened a new direction in time-frequency analysis. Both groups of papers, by Feichtinger and by Sjostrand, continue to have a profound influence on research and are true landmarks in the field.