Abstract

Ordinarily, having an n-dimensional operator relation implies operator relations for all dimensions less than n as well. However, for Fourier transforms this is not so. Only the effect of one-dimensional Fourier transforms in two-dimensional first order systems has been solved previously.¹ A partial Fourier transform applied to a nonsingular linear transformation produces a Ohio' transformation, i.e., a matrix pivot. This implies a new definition of the symplectic group SP(n, C) for n ≥ 4 as the group generated by n-dimensional nonsingular linear transformations and the n one-dimensional Fourier transforms. This also provides an alternative method for generalizing low-dimensional results to higher dimensions. Several apparently new determinantal identities generalizing Chio's theorem are immediate consequences.

© 1992 Optical Society of America