On an extension problem, generalized fourier analysis, and an entropy formula

Abstract

Let h(t) be an n × n matrix valued function on the interval |t| ⩽ τ with summable entries. Let ĥ denote the Fourier transform of h and let e denote the n × n identity matrix. Necessary and sufficient conditions for the existence of an extension u of h to the full line such that e-u admits either a left or a right canonical factorization and the inverse transform of (e-u)−1-e vanishes for |t| ⩾ τ are presented and discussed. The connections between these extensions and a generalized Fourier transform are then explored in detail with the help of the theory of triangular factorization. It is then shown that if an allied finite Wiener-Hopf operator based on h is positive, then h admits exactly one extension of the type alluded to above. This extension is then characterized in terms of an entropy integral.