Bivariate averaging functions, translation and scale autocorrelations, Fourier and Mellin transforms, the Wiener-Khinchine theorem and their inter-relationships

Abstract

We first draw attention to the fact that the position operator,$\hat X$, its translation generator,$\hat P$, and its scale generator,$\hat D$, form an important group of triplet of operators that appear in the Heisenberg uncertainty relation stated in its most general form. The pair$(\hat X,\hat P)$ forms the phase-space and they have led to Fourier transform pair, the autocorrelation function, the Wiener-Khinchine theorem, and the Wigner function with many different applications to wave phenomena. The importance of the pairs$(\hat X,\hat D)$ and$(\hat P,\hat D)$ has been pointed out by Moses and Quesada (1972, 1973, 1974) who showed that we must then consider a Mellin transform pair, a scale autocorrelation function, and a corresponding Wiener-Khinchine theorem. In the present paper, we define and explore properties of a bivariate averaging function defined in a new “phase-space” involving the Mellin transform variable and its partner which can either be the position or momentum, analogous to the Wigner function. The not-necessarily positive feature of the bivariate averaging functions is traced to the general Heisenberg uncertainty mentioned above. The properties and their inter-relationships among the averaging functions are given. We hope this will be of use in discussing physical phenomena involving fractals, turbulence, and near phase transitions where the scaling properties are of importance.