The Fractional Fourier Transform (FrFT) has widespread applications in areas like signal analysis, Fourier optics, diffraction theory, etc. The Holomorphic Fractional Fourier Transform (HFrFT) proposed in the present paper may be used in the same wid
e range of applications with improved properties. The HFrFT of signals spans a one-parameter family of (essentially) holomorphic functions, where the parameter takes values in the bounded interval $tin (0,pi/2)$. At the boundary values of the parameter, one obtains the original signal at $t=0$ and its Fourier transform at the other end of the interval $t=pi/2$. If the initial signal is $L^2 $, then, for an appropriate choice of inner product that will be detailed below, the transform is unitary for all values of the parameter in the interval. This transform provides a heat kernel smoothening of the signals while preserving unitarity for $L^2$-signals and continuously interpolating between the original signal and its Fourier transform.
In this paper, we give a new construction of the adapted complex structure on a neighborhood of the zero section in the tangent bundle of a compact, real-analytic Riemannian manifold. Motivated by the complexifier approach of T. Thiemann as well as c
ertain formulas of V. Guillemin and M. Stenzel, we obtain the polarization associated to the adapted complex structure by applying the imaginary-time geodesic flow to the vertical polarization. Meanwhile, at the level of functions, we show that every holomorphic function is obtained from a function that is constant along the fibers by composition with the imaginary-time geodesic flow. We give several equivalent interpretations of this composition, including a convergent power series in the vector field generating the geodesic flow.
