This ARI Guest Talk by Joachim Toft is based on joint work with Christine Pfeuffer and Nenad Teofanov.

In the opening section, Joachim illustrates how sound can be mathematically modeled using a blend of local and global Fourier analysis. This approach raises various questions concerning energy estimates for such representations. The so-called modulation spaces provide a natural setting for these estimates.

The central part of the talk explores a broad class of modulation spaces in greater depth. Toft presents recent results concerning extensive families of equivalent (quasi-)norms within these spaces, shedding light on their structural properties.

In the final segment, he discusses how these new norm equivalences can refine classical results — particularly in the context of pseudo-differential operators (interpreted as non-stationary filters in signal analysis) and localization operators, also known as Toeplitz operators.

About the Speaker
Joachim Toft is a professor of mathematics at Linnaeus University. His research focuses on pseudo-differential calculus and its connections to various analytical fields, including Fourier analysis, time-frequency analysis, harmonic analysis, operator theory, generalized functions, and microlocal analysis. Website: https://lnu.se/en/staff/joachim.toft/