# Re-expansions on compact Lie groups

## TL;DR

This paper investigates re-expansion problems on compact Lie groups, establishing conditions for Fourier coefficient summability of central extensions, extending classical results from tori to more general groups, with specific results for SU(2).

## Contribution

It extends classical re-expansion results to compact Lie groups using root systems and Fourier analysis, providing necessary and sufficient conditions for summability.

## Key findings

- Derived conditions for Fourier coefficient summability on compact Lie groups.
- Extended classical re-expansion results from tori to general groups.
- Provided specific sufficient conditions for SU(2).

## Abstract

In this paper we consider the re-expansion problems on compact Lie groups. First, we establish weighted versions of classical re-expansion results in the setting of multi-dimensional tori. A natural extension of the classical re-expansion problem to general compact Lie groups can be formulated as follows: given a function on the maximal torus of a compact Lie group, what conditions on its (toroidal) Fourier coefficients are sufficient in order to have that the group Fourier coefficients of its central extension are summable. We derive the necessary and sufficient conditions for the above property to hold in terms of the root system of the group. Consequently, we show how this problem leads to the re-expansions of even/odd functions on compact Lie groups, giving a necessary and sufficient condition in terms of the discrete Hilbert transform and the root system. In the model case of the group SU(2) a simple sufficient condition is given.