  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 wide 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 $t\in (0,\pi/2)$. 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.
