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.
Highlights
There has been an increased interest in the generalization of integral transforms to the quaternionic and Clifford settings
If f = g in (3.1) we have that the quaternionic Segal–Bargmann transform realizes an isometry from L2(R, H) onto the slice hyperholomorphic Bargmann-Fock space FS2,lνice(H), as proved in a different way in [17, Thm. 4.6]
Through the 1D quaternion short-time Fourier transform (QSTFT) we can prove in another way that the eigenfunctions of the 1D quaternion Fourier transform are given by the Hermite functions
Summary
There has been an increased interest in the generalization of integral transforms to the quaternionic and Clifford settings Such kind of transforms are widely studied, since they help in the analysis of vector-valued signals and images. We introduce an extension of the short-time Fourier transform in a quaternionic setting in dimension one. To this end, we fix a property that relates the complex short-time Fourier transform and the complex Segal–Bargmann transform: Vφf (x, ω). To achieve our aim we use the quaternionc analogue of the Segal–Bargmann transform studied in [17] This integral transform is used in [18] to study some quaternionic Hilbert spaces of Cauchy–Fueter regular functions. We show that the 1D QSTFT follows a Lieb’s uncertainty principle, some classical uncertainty principles for quaternionic linear operators in quaternionic Hilbert spaces were considered in [27]
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