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

Abstract

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.

On the polyanalytic short-time Fourier transform in the quaternionic setting

“…We note that for n = 0 we have ψ 0 (t) = 2 1/4 e −πt 2 , which is exactly the window function that we considered in [23].…”

“…It is used in several applications such as the predictions of sound source position emanated by fault machine [38] and the interpretation of ultrasonic waveforms [35]. The short-time Fourier transform has been studied in quaternionic and Clifford settings in [11,22,23]. In particular in [23] we gave a definition of a quaternionic short-time Fourier transform (QSTFT) in dimension one for a Gaussian window.…”

“…The short-time Fourier transform has been studied in quaternionic and Clifford settings in [11,22,23]. In particular in [23] we gave a definition of a quaternionic short-time Fourier transform (QSTFT) in dimension one for a Gaussian window.…”

“…Remark 16. For n = 0 in (6.3) and (6.4) we obtain the definition of the 1Dquaternion short-time Fourier transform with respect to the Gaussian window g(t) = 2 1/4 e −πt 2 , see [23,Def. 5.1].…”

“…Moreover, we have already observed that B 1 ϕ = Bϕ, which is the quaternionic analogue of the Bargmann transform. Therefore, with formulas (6.3) and (6.4) we are working in a more general setting with respect to the paper [23].…”