Gabor expansion on orthogonal bases

In its classical form, the Gabor expansion constitutes a superposition of sliding window functions multiplied by a Fourier kernel. Herein, we propose a generalised formulation whereby the multiplying kernel is presumed to be complete and orthogonal over a prescribed segment of the independent variable, but is otherwise arbitrary. Basis kernels such as Legendre, Chebyshev and Walsh are reasonable candidates. The expected application is to areas such as wave analysis and picture processing.