SPACE OF TEST FUNCTIONS FOR HIGHER-ORDER FIELD-THEORIES

The fundamental space zeta is defined as the set of entire analytic fu nctions [test functions phi(z)], which are rapidly decreasing on the r eal axis. The variable z corresponds to the complex energy plane. The conjugate or dual space zeta' is the set of continuous linear function als (distributions) on zeta. Among those distributions are the propaga tors, determined by the poles implied by the equations of motion and t he contour of integration implied by the boundary conditions. All prop agators can be represented as linear combinations of elementary (one p ole) functionals. The algebra of convolution products is also determin ed. The Fourier transformed space ($) over tilde zeta contains test fu nctions ($) over tilde phi(x). These functions are extra-rapidly decre asing, so that the exponentially increasing solutions of higher-order equations are distributions on ($) over tilde zeta.