I have seen some papers where the lineal canonical transform is discussed but I have been strugling to find a proof for the additivy (sometimes is refered as cascability) of its Kernel. See for example: https://en.wikipedia.org/wiki/Linear_canonical_transformation
The Kernel is defined as $$K_A(x,y) = (2\pi b)^{1/2} \exp(i \cdot(\frac{a}{2b}x^2 - \frac{1}{b}xy + \frac{d}{2b}y^2 - \frac{\pi}{4})) $$ where, $A = \begin{pmatrix}a & b \\ c & d \end{pmatrix}$ and $\det(A) = 1$.
The additivity property says that $$\int_{- \infty}^{\infty}K_{A_1}(x, \omega)K_{A_2}(\omega,y) d\omega = K_{A_1A_2}(x,y) $$
Does anybody know a proof for this property?
The proof can be done using standard (strictly speaking, tempered-distributional) integral formulas for integrals of linear chirp functions on the whole real line (which are easily proved by a change of variables- the integrals turn into Gaussian integrals)- however, there should actually be a unimodular constant factor in one side of the equation, as shown in the book 「Linear Canonical Transforms: Theory and Applications」 by Healy, J. J., Kutay, M. A., Ozaktas, H. M., and Sheridan, J. T.