A NEW IDEA TO DEFINE THE δ-FUNCTION OF AN OBSERVABLE IN THE CONTEXT OF WHITE NOISE ANALYSIS

Abstract

In this paper, we present a new idea to define the δ-function of an observable within the canonical framework of white noise analysis. We introduce a class of observables called δ-composable observables. We then define the δ-function of such an observable as the generalized operator determined uniquely by the spectral density of the observable. Surprisingly, we find that the δ-function of a δ-composable observable is always positive as a generalized operator. Finally, we show an example of a δ-composable observable and examine properties of its δ-function.