H. E. Moses’s research while affiliated with United States Air Force and other places

The Mellin transform is used to diagonalize the dilation operator in a manner analogous to the use of the Fourier transform to diagonalize the translation operator. A power spectrum is also introduced for the Mellin transform which is analogous to that used for the Fourier transform. Unlike the case for the power spectrum of the Fourier transform where sharp peaks correspond to periodicities in translation, the peaks in the power spectrum of the Mellin transform correspond to periodicities in magnification. A theorem of Wiener‐Khinchine type is introduced for the Mellin transform power spectrum. It is expected that the new power spectrum will play an important role extracting meaningful information from noisy data and will thus be a useful complement to the use of the ordinary Fourier power spectrum.

In a previous work it was shown that the scale autocorrelation coefficient, which compares a function of time with itself when the scale of time is stretched, could be used to determine ratios of frequencies that occur in phenomena with periodicities. We review the properties of the scale autocorrelation coefficient and apply it to the finding of some of the ratios of the frequencies in atmospheric pressures measured daily for 8766 days. We also apply it to measurements of the international magnetic character figure Ci, which was measured daily for 27,258 days. We regard these applications as the first test of the utility of the scale autocorrelation coefficient. The results indicate that the scale autocorrelation coefficient may indeed prove to be a valuable supplement to the use of the usual autocorrelation coefficient and the Fourier transformation of time-dependent functions.

The irreducible representations of the scale-Euclidean group in three dimensions are introduced, and the general tensor is expanded in terms of these representations. The cases of zero-rank tensor (scalar), rank-1 tensor (vector), and rank-2 tensor are studied in detail. The expansion is shown to be a generalization of the Helmholtz expansion of a vector into rotational and irrotational parts. As in Part I of this work (Concepts: One-Dimensional Problems), the correlations that are introduced are invariant under changes of frames of reference. Correlations are set up between tensors of different ranks and dimensions. A correlation that measures a degree of isotropy is defined.

The group of transformations is introduced which consists of translation, rotation, and dilation of a Euclidean coordinate system. It is shown how physical quantities can be expanded in the irreducible representations of the group. It is then possible to introduce correlations which are invariant under the transformations of the group. The most novel feature is the introduction of a scale autocorrelation which compares a quantity with itself when “stretched”. This autocorrelation is expected to find use in problems involving noise and turbulence. It is shown in the present paper how this new autocorrelation enables one to find relationships between periodicities. This paper is concerned with concepts and the one-dimensional problem. Part II will deal with three-dimensional problems.