On the Quaternionic Short-Time Fourier and Segal–Bargmann Transforms

In this paper, we study a special one-dimensional quaternion short-time Fourier transform (QSTFT). Its construction is based on the slice hyperholomorphic Segal–Bargmann transform. We discuss some basic properties and prove different results on the QSTFT such as Moyal formula, reconstruction formula and Lieb’s uncertainty principle. We provide also the reproducing kernel associated with the Gabor space considered in this setting.