On Characterization of the Gabor Wavelet Transform of Analytic Functionals

The wavelet transformation is usually considered for the square integrable functions. Since the Gabor function is of exponential type, we can consider the Gabor wavelet transformation to the analytic functionals. As the Gabor function is Gaussian type, the Gabor wavelet transformation is similar to the windowed Fourier transformation. That is, the Gabor wavelet transformation is closely related to the Fourier transformation. Therefore our previous results on Fourier transformation are useful when we consider the Gabor transformation. In this paper, we will review our previous results and will consider the relationship between the Fourier transform and the Gabor wavelet transform of analytic functional on the sphere based on our previous results.