The Fourier Transformation and Elements of Harmonic Analysis

This paper reconsiders the contribution of Henry Ludwell Moore to dynamic economics through the use of harmonic analysis. We show that Moore’s analysis is innovative in its use of the Fourier transformation for the identification of cycles with different periodicities. This enables Moore to identify cycles of longer length with more precision than would be the case for the standard methodology. We are able to replicate the main features of his results and confirm the existence of a rainfall cycle with a periodicity similar to that of the business cycle (eight years). However, we find that the evidence for a longer (thirty-three-year) rainfall cycle is weaker than Moore indicates. We also argue that a central theme of Moore’s analysis—the relationship among rainfall, agricultural productivity, and the business cycle—marks an early precursor of the “real business cycle” approach. George Stigler’s (1962) dismissal of Moore’s work on cycles as “a complete failure” is therefore, in our opinion, unfair. Instead, we argue that, although his work is certainly flawed, it nevertheless deserves a place in both the history of business cycle theory and empirical economics.

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The Fourier Transformation and Elements of Harmonic Analysis

Theorems and Problems in Functional Analysis - Problem Books in Mathematics ◽

10.1007/978-1-4613-8153-2_9 ◽

1982 ◽

pp. 204-218

Author(s):

A. A. Kirillov ◽

A. A. Gvishiani

Keyword(s):

Harmonic Analysis ◽

Fourier Transformation

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New Perspectives on Henry Ludwell Moore’s Use of Harmonic Analysis

10.31219/osf.io/27aer ◽

2021 ◽

Author(s):

Paul Turner ◽

Justine Wood

Keyword(s):

Business Cycle ◽

Harmonic Analysis ◽

Fourier Transformation ◽

Real Business Cycle ◽

Business Cycle Theory ◽

Empirical Economics ◽

History Of ◽

Complete Failure ◽

The Business Cycle ◽

The Relationship

This paper reconsiders the contribution of Henry Ludwell Moore to dynamic economics through the use of harmonic analysis. We show that Moore’s analysis is innovative in its use of the Fourier transformation for the identification of cycles with different periodicities. This enables Moore to identify cycles of longer length with more precision than would be the case for the standard methodology. We are able to replicate the main features of his results and confirm the existence of a rainfall cycle with a periodicity similar to that of the business cycle (eight years). However, we find that the evidence for a longer (thirty-three year) rainfall cycle is weaker than Moore indicates. We also argue that a central theme of Moore’s analysis, the relationship between rainfall, agricultural productivity and the business cycle, marks an early precursor of the ‘Real Business Cycle’ approach. Stigler’s (1962) dismissal of Moore’s work on cycles as ‘a complete failure’ is therefore, in our opinion rather unfair. Instead, we argue that, although his work is certainly flawed, it nevertheless deserves a place in both the history of business cycle theory and empirical economics.

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The Fourier Transformation and Elements of Harmonic Analysis

Theorems and Problems in Functional Analysis - Problem Books in Mathematics ◽

10.1007/978-1-4613-8153-2_14 ◽

1982 ◽

pp. 309-324

Author(s):

A. A. Kirillov ◽

A. A. Gvishiani

Keyword(s):

Harmonic Analysis ◽

Fourier Transformation

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Spherical harmonic analysis and Fourier transformation

Studia Geophysica et Geodaetica ◽

10.1007/bf01630849 ◽

1986 ◽

Vol 30 (1) ◽

pp. 13-27

Author(s):

Jaromír Bordovský ◽

Z. Nádeník

Keyword(s):

Harmonic Analysis ◽

Spherical Harmonic ◽

Fourier Transformation ◽

Spherical Harmonic Analysis

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Simulations of high-resolution images of amorphous silicon films

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s042482010014419x ◽

1986 ◽

Vol 44 ◽

pp. 544-545

Author(s):

G. Y. Fan ◽

J. M. Cowley

Keyword(s):

Electron Microscope ◽

High Frequency ◽

Fourier Transformation ◽

Amorphous Materials ◽

Low Frequency ◽

Chromatic Aberration ◽

Structure Information ◽

Frequency Components ◽

High Resolution Images ◽

Spatial Frequency Spectrum

It is well known that the structure information on the specimen is not always faithfully transferred through the electron microscope. Firstly, the spatial frequency spectrum is modulated by the transfer function (TF) at the focal plane. Secondly, the spectrum suffers high frequency cut-off by the aperture (or effectively damping terms such as chromatic aberration). While these do not have essential effect on imaging crystal periodicity as long as the low order Bragg spots are inside the aperture, although the contrast may be reversed, they may change the appearance of images of amorphous materials completely. Because the spectrum of amorphous materials is continuous, modulation of it emphasizes some components while weakening others. Especially the cut-off of high frequency components, which contribute to amorphous image just as strongly as low frequency components can have a fundamental effect. This can be illustrated through computer simulation. Imaging of a whitenoise object with an electron microscope without TF limitation gives Fig. 1a, which is obtained by Fourier transformation of a constant amplitude combined with random phases generated by computer.

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