caa a22 4500 469041595 CHVBK 20180323132812.0 cr unu---uuuuu 170328e19920301xx s 000 0 eng 10.1007/BF02875370 doi (NATIONALLICENCE)springer-10.1007/BF02875370 Rajagopal A. Jawaharlal Nehru Centre for Advanced Scientific Research, Indian Institute of Science Campus, 560 012, Bangalore, India aut Bivariate averaging functions, translation and scale autocorrelations, Fourier and Mellin transforms, the Wiener-Khinchine theorem and their inter-relationships [Elektronische Daten] [A Rajagopal] 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. Indian Academy of Sciences, 1993 Bivariate averaging function nationallicence uncertainty principle nationallicence Fourier and Mellin transforms nationallicence autocorrelations nationallicence the Wiener-Khinchine theorem nationallicence 02 nationallicence 03.65 nationallicence 05.50 nationallicence Pramana Springer India 38/3(1992-03-01), 233-247 0304-4289 38:3<233 1992 38 12043 https://doi.org/10.1007/BF02875370 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF02875370 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Rajagopal A. Jawaharlal Nehru Centre for Advanced Scientific Research, Indian Institute of Science Campus, 560 012, Bangalore, India aut NATIONALLICENCE 773 0- Pramana Springer India 38/3(1992-03-01), 233-247 0304-4289 38:3<233 1992 38 12043 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer