Pilipovic spaces is a family of function spaces which contains any standard Fourier invariant Gelfand-Shilov space and are densely embedded within the Schwartz space. These spaces can be defined through Hermite series expansions involving powers of the harmonic oscillator, with certain conditions imposed on the Hermite coefficients. Recent investigations show that Pilipovic spaces, which are non-trivial and not equal to Gelfand-Shilov spaces, are better designed for several fundamental continuity investigations and also possess better mapping properties under certain mappings like harmonic oscillator propagators of complex orders, which appear in certain inverse problems in statistical physics.

In this talk, we aim to characterize Pilipovic space in terms of certain estimates of the involved functions and suitable choices of their fractional Fourier transforms. The proof relies on a multi-dimensional version of Phragmen-Lindelöf's theorem. For the analysis, we recall some important links between Bargmann transforms and short-time Fourier transforms with Gaussian windows, and discuss compositions of such transforms with fractional Fourier transforms.

This talk is based on joint work with Joachim Toft [in: J. Funct. Anal. (2023)].

https://univienna.zoom.us/j/62077153839?pwd=T3pxeHNRNEU0RlFoY1J2cnIzbzU5dz09