23. November 2021
Joachim Toft (Linnaeus/Växjö University)
Small test function and large distribution spaces, and their images under the Bargmann transform
We consider a family of function spaces, defined by estimates of H^N f, which include all Fourier Invariant standard Gelfand-Shilov (FIGS) spaces. Here H is the harmonic oscillator. Note that any FIGS is contained in the Schwartz space, and the duals, or corresponding distribution spaces to FIGS contains the set of tempered distributions. The first time function and distribution spaces were characterised in terms of the harmonic oscillator seems to be done by Pilipovic in the 80th. For this reason, the function spaces that appeared by such characterisations are (here) called Pilipovic spaces. The smallest spaces are significantly smaller than the non-trivial FIGS-spaces, and the family of corresponding distribution spaces contain spaces larger any FIGS distribution space.
We deduce:
- Characterizations in terms of Hermite expansions, as well as power series expansions via the Bargmann transform.
- Modulation spaces with almost no restrictions on involved weight functions.
- A ”Paley-Wiener related” property”, linking Gröchenig’s S_C space to the considered family of functions and distributions.
Parts of the talk is based on collaborations with M. Cappiello, C. Fernandez, A. Galbis and L. Rodino.