The Fourier and Hilbert transforms under the Bargmann transform
  There is a canonical unitary transformation from $L^2(\R)$ onto the Fock space $F^2$, called the Bargmann transform. We study the action of the Bargmann transform on several classical integral operators on $L^2(\R)$, including the fractional Fourier transform, the fractional Hilbert transform, and the wavelet transform. 
Xing-Tang Dong and Kehe Zhu
math.CV math.FA