Space of test functions for higher‐order field theories

The fundamental space ζ is defined as the set of entire analytic functions [test functions φ(z)], which are rapidly decreasing on the real axis. The variable z corresponds to the complex energy plane. The conjugate or dual space ζ′ is the set of continuous linear functionals (distributions) on ζ. Among those distributions are the propagators, determined by the poles implied by the equations of motion and the contour of integration implied by the boundary conditions. All propagators can be represented as linear combinations of elementary (one pole) functionals. The algebra of convolution products is also determined. The Fourier transformed space ζ̃ contains test functions φ̃(x). These functions are extra‐rapidly decreasing, so that the exponentially increasing solutions of higher‐order equations are distributions on ζ̃.