A convolution-integral representation for a class of linear operators

Abstract

Let be the complex vector space consisting of all complex-valued functions of a non-negative real variable t. For each positive number u, the shift operator Iu is the mapping of into itself defined by the formulaA linear operator T which maps a subspace of into itself is said to be a V-operator (13) if:(a) for each x in , the complex-conjugate function x* is in ;(b) both and \ are invariant under the shift operators;(c) every shift operator commutes with T.(Property (a) ensures that every function x in can be uniquely expressed as x1 + ix2, where x1 and x2 are real functions in .)