Abstract
As a concept and as a tool, the Fourier transform is pervasive in applied mathematics, computing, mathematics, probability and statistics as well as in substantive sciences such as chemistry, geophysics and physics. This chapter presents a review of such applications and then four personal analyses of scientific data based on Fourier transforms. Specific points made include: Fourier analysis is conceptually simple, its concepts often have direct physical interpretations, useful statistical properties are available, and there are various interesting connections between the mathematical and physical concepts.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Reference
Aki, K. and P. G. Richards, Quantitative Seismology I & II. Freeman, San Francisco, 1980.
Amato, I., Nobel prizes 1991 Science 254: 518-51 9, 1991.
Barndorff-Nielsen, O. E. and D. R. Cox, Asymptotic Techniques for Use in Statistics, Chapman and Hall, London, 1989.
Bath, M., Spectral Analysis in Geophysics. Elsevier, Amsterdam, 1994.
Bazin, M. J., P. H. Lucie, and S. Moss de Olivierra, Experimental demonstrations of the mathematical properties of Fourier transforms using diffraction phenomena, Eur. J. Phys. 7: 183-188 (1986) .
Benedetto, J. J. and M. W. Frazier, (eds), Wavelets, CRC Press, Boca Raton, 1994.
Berger, J. and R. Wolpert, Estimating the mean function of a Gaussian process and the Stein effect, J. Mult. Anal. 13: 401-424, (1983) .
Blackman, R. B. and J . W. Tukey, The Measurement of Power Spectra. Dover, New York, 1959.
Bloembergen, N., Nonlinear optics and spectroscopy, Science 215: 10571064 (1982).
Bloomfield, P., Fourier Analysis of Time Series: An Introduction, Wiley, New York, 1976.
Blow, D. M. and F. H. C. Crick, The treatment of errors in the isomorphous replacement method, Acta Cryst . 12: 794-802 (1959) .
Blumich, 8. , Stochastic nmr spectroscopy, Bull. Magnet. Resonance 7: 5-26 (1985) .
Bochner, S., Lectures 011 Fourier Integrals, Princeton Press , Princeton, 1959.
Bochner, S., Harmonic Analysis and the Theory of Probability , Univ. Calif. Press , Berkeley, 1960.
Bolt, B. A. and J. Butcher, Rayleigh wave dispersion for a single layer on an elastic half space, Australian J. Physics 13: 498-504 (1960) .
Bolt, B. A., Y. B. Tsai, K. Yeh and M. K. Hsu, Earthquake strong motions recorded at a large near-source array of digital seismographs, Earthquake Eng . Structural Dynam. 10: 561-573 (1982).
Born, M. and Wolf, E. (1964). Principles of Optics. Second Edition. Macmillan, New York.
Bracewell , R. N. , The Fourier transform, Scientific Ameri., June: 86-95 (1989).
Brillinger, D. R., A search for a relationship between monthly sunspot numbers and certain climatic series, Bull . Inter. Statist. Inst , 43: 293-306 (1969).
Brillinger, D. R., Time Series : Data Analysis and Theory, Holt, New York, 1975.
Brillinger, D. R., Asymptotic normality of finite Fourier transforms of stationary generalized processes, J. Mult, Analysis 12:64-71 (1982) .
Brillinger, D. R., Some asymptotics of finite Fourier transforms of a stationary p-adic process, J. Comb. In! Sys. Sci . 16: 155-169 (1991).
Brillinger, D. R., An application of statistics to seismology: dispersion and modes . In Developments in Time Series Analysis (ed. T. Subba Rao), Chapman and Hall, London, 1993, pp . 331-340.
Brillinger, D. R., Some uses of cumulants in wavelet analysis, Nonpar. Statist . 6:93-114 (1996).
Brillinger, D. R., K. H. Downing and R. M. Glaeser, Some statistical aspects of low-dose electron imaging of crystals, J. Stat . Planning In! 25: 235-259 (1990) .
Brillinger, D. R., K. H. Downing, R. M. Glaeser and G. Perkins, Combining noisy images of small crystalline domains in high resolution electron microscopy, J. App. Stat. 16: 165-175 (1989) .
Brillinger, D. R. and R. Kaiser, Fourier and likelihood analysis in NMR spectroscopy. In New Directions in Time Series I (eds . D. Brillinger, P.
Caines, J . Geweke, E. Parzen, M. Rosenblatt and M. Taqqu), Springer, New York, 1992, pp . 41-64.
Bullen, K. E. and B. A. Bolt, An Introduction to the Theory of Seismology. Cambridge Univ. Press, Cambridge, 1985.
Butzer, P. L. and R. J. Nessel, Fourier Analysis and Approximation, Academic, New York, 1971.
Cartwright, D. E., Tidal analysis-a retrospect in Time Series Methods in Hydrosciences (eds A. H. EI-Shaarawi and S. R. Esterby), Elsevier, Amsterdam, 1982, pp. 170-188.
Cooley, J. W. and J. W. Tukey, An algorithm for the machine calculation of complex Fourier series, Math. Camp. 19: 297-301 (1965) .
Copas, J . B., Regression, prediction and shrinkage, J. Roy. Statist . Soc. B 45: 3/1 -335 (1983) .
Cramer, H., On harmonic antilysis in certain function spaces, Arkiv Math. As/r. Fysik. 28: 1-7 (1942) .
Dahlhaus, R., Parameter estimation of stationary processes with spectra containing strong peaks in Robust and Nonlinea r Time Series Analysis (eds J.
Franke, W. Haerdle and D. Martin), Springer, New York, 1984, pp . 50-67.
Dahlhaus, R., Efficient parameter estimation for self-similar processes, Ann . Statist . 17: 1749-1766 (1989).
Daubechies, 1., Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.
Diaconnis, P., Group Representations in Probability and Statistics, Institute of Mathematical Statistics, Hayward, 1988.
Diaconnis, P., A generalization of spectral analysis with applications to ranked data, Ann . Statist . 17: 949-979 (1989).
Donoho, D. L. and 1. M. Johnstone, Wavelets and optimal nonlinear/unction estimates, Tech. Report 281, Statistics Dept. Univ . California, Berkeley,
1990.
Donoho, D. L., I. M. Johnstone, G . Kerkyacharian and D. Picard. Wavelet shrinkage: asymptopia? J. Roy. Statist. Soc . B 57: 301-369 (1995) .
Dzhaparidze, K., Parameter Estimation and Hypothesis Testing in Spectral Analysis of Stationary Time Series, Springer, New York, 1986.
Efron, B. and C. Morris, Stein's paradox in statistics, Scientific Amer. 236: 119-127 (1977) .
Fan, J ., Deconvolution with supersmooth distributions, Canadian J. Statist . 20: 155-169 (1992) .
Feuerverger, A., Efficiency in time series, Canadian J. Statist . 18: 155-162. (1990).
Feigin, P. D. and C. R. Heathcote, The empirical characteristic function and the Cramer-von Mises statistic, Sankhya A 38: 309-325 (1976).
Fine, N. J., On Walsh functions, Trans. Amer. Math. Soc. 65: 372-414 (1949) .
Fox, R. and M. S. Taqqu, Large-sample properties of parameter estimates
for strongly dependent stationary time series , Ann . Statist . 14: 517-532 (1986) .
Freedman, D. and D. Lane, The empirical distribution of the Fourier coefficients of a sequence of independent, identically distributed long-tailed
random variables, Z. Wahrschein, Vel'. Geb. 55: 123-132 (1981) .
Glaeser, R. M., Electron crystallography of biological macromolecules, Ann. Rev. Phys . Chem. 36: 243-275 (1985)
Good, I. J ., The interaction algorithm and practical Fourier series, J. Roy . Statist . Soc . B 20: 361-372 (1958) .
Good, I. J ., Weighted covariance for detecting the direction of a Gaussian source in Time Series Analysis (ed . M. Rosenblatt), Wiley, New York,
1963, pp. 447-470.
Goodman, J . W., Introduction to Fourier Optics, McGraw-Hili, San Francisco, 1968.
Gorgui-Naguib, R. N., p-adic transforms in digital signal processing in Mathematics in Signal Processing II (ed. J. G. McWirter), Clarendon, Oxford, 1990, pp . 43-53.
Hall, P. and P. Patil, On wavelet methods for estimating smooth functions, Bernoulli 1 (1995) , pp . 41-58.
Halvorson, c.. A. Hays, B. Kraabel, R. Wu, F. Wudl and A. J. Heeger, A 160-femtosecond optical image processor based on a conjugated polymer, Science 265: 1215-1216 (1994) .
Hannan, E. J. , Aliasing. Tech. Report 25, Statistics Dept. , Johns Hopkins University, 1965.
Hannan, E. J. , Group Representations and Applied Probability, Methuen, London, 1966.
Hannan, E. J., Fourier methods and random processes, BIIII. Internet, Statist. lnst. 42(1):475-496 (1969).
Hannan, E. J., Spectra changing over narrow bands in Statistical Models and Turbulence (eds M. Rosenblatt and C. Van Atta), Springer, New
York, 1972, pp. 460-469. Hannan, E. J. and P. J. Thomson, Spectral inference over narrow bands, J . Appl. Prob. 8: 157-169 (1971).
Hayward, S. B. and R. M. Stroud, Projected purple membrane determined to 3.7Aresolution by low temperature electron microscopy, J. Molec . BioI. 15/:491 -517 (1981).
Henderson, R., J. M. Baldwin.K. H. Downing, J . Lepault, and F. Zernlin, Structure of purple membrane from Halobacterium halobium, Ultramicroscopy 19: 147-178 (1986) .
Henderson, R., J . M. Baldwin, T. A. Ceska, F. Zemlin, E. Beckmann and K. H. Downing, Model for the structure of bacteriorhodopsin based on high-resolution electron cryo-microscopy, J. Mol. BioI.213: 899-929 (1990).
Hennel, J. W. and J. Klinowski, Fundamentals of Nuclear Magnetic Resonance, Wiley, New York, 1993.
Hewitt, E.and K. A. Ross, Abstract Harmonic Analysis I&II, Academic, New York, 1963.
Higgins, J . R., Five short stories about the cardinal series, Bull. Amer. Math. Soc. 12:45-89 (1985) .
Hirsch, M. W., The dynamical systems approach to differential equations, Bull. Amer. Math . Soc. JJ: 1-64 (1984).
Hochstadt, H., Differential Equations, A Modern Approach, Holt-Rinehart, New York, 1964.
Hovmoller, S., Structure analysis by crystallographic image processingHommage aJean Baptiste Joseph Fourier (1768-1830), Mierosc. Microanal. Microstruct. 1: 423-431 (1990).
Ihaka, R., Statistical aspects of earthquake source parameter estimation in the presence of signal generated noise, Commun. Statist. A 22: 14251440 (1993).
Katz, B. and R. Miledi, Further observations on acetylcholine noise, Nature 232: 124-126 (1971). Katznelson, Y., An Introduction to Harmonic Analysis, Dover, New York, 1976.
Kim, P. and G. R. Chapman, Group action on a lattice and an application to time series , J . Stat . Planning Inf. 34: 183-195 (1993).
King, N., An alternative for the linear regression equation when the predictor variable is uncontrolled and the sample size is small, J . Amer. Statist. Assoc. 67: 217-219 (1972).
Korner, T. W., Fourier Analysis . Cambridge Univ. Press, Cambridge, 1989.
Lanczos, c., Discourse on Fourier Series, Hafner, New York, 1966.
Leonov, V. P. and A. N. Shiryaev, Some problems in the spectral theory of higher moments, Theory Prob. Appl. 5: 460-464 (1960).
Lillestol, J., Improved estimates for multivariate complex normal regression with application to analysis of linear time-invariant relationships. J . Mult . Anal. 7: 512-524 (1977) .
Loomis, L., An Introduction to Abstract Harmonic Analysis, Van Nostrand, New York, 1953.
Malik, F., D. R. Brillinger, and R. D. Vale, High resolution tracking of microtubule motility driven by a single kinesin motor, Proc. Natl. Acad. Sci. USA 91, pp. 4584-4588 (1994) .
Michaelson, A. A., On the application of interference methods to spectroscopic methods-I. Phil. Mag. 33: 338-346 (189Ia).
Michaelson, A. A., On the application of interference methods to spectroscopic methods-II. Phil . Mag. 34:280-299 (189Ib) .
Moloney, J . V. and A. C. Newell, Nonlinear optics. Tech. Report 574. 1MA, University of Minnesota, Minneapolis, 1989.
Ott, J. and R. A. Kronmall, Some classification procedures for multivariate binary data using orthogonal functions, J. ArneI'. Statist . Assoc. 71: 391399 (1976).
Picinbono, M. B., Tendance vers Ie caractere gaussien par filtrage selectif, Compte s Rendus Acad. Sci., 1174-1176 (1960) .
Priestley, M. B., Evolutionary spectra and non stationary processes, J. Roy. Statist. Soc. B 27: 204-237 (1965) .
Richards, F. S. G., A method of maximum-likelihood estimation, J. Roy. Statist. Soc . B 23: 469-475 (1961) .
Rockmore, D., Fast Fourier analysis for abelian group extensions, Adv. Appl. Math. ll : 164-204 (1990).
Rosenberg, J . R., A. M. Amjad, P. Breeze, D. R. Brillinger and D. M. Haliday, The Fourier approach to the identification of functional coupling between neuronal spike trains, Prog . Biophys. Molec. Bioi. 53: 1-31 (1989).
Rosenblatt, M., Some comments on narrow band-pass filters , Quart. Appl. Math . 18: 387-394 (1961)
Rosenblatt, M. , Probability limit theorems and some questions in fluid mechanics in Statistical Models and Turbulence (eds M. Rosenblatt and C. Van AUa) , Springer, New York, 1972, pp. 27-40.
Ro senblatt, M., Limit theorems for Fourier transforms of functionals of Gaussian sequences, Z . Wahrsch . VerII'. Gebiete. 55: 123-132 (\981).
Rudin, W., Fourier Analy sis on Groups, Wiley, New York, 1962.
Ruelle, D., Cha otic Evolut ion and Strange Attractors, Cambridge Univ. Press , Cambridge, 1989.
Saleh , A. K., Contributions to Preliminary Test and Shrinkage Est imation , Department of Mathematics and Statistics, Carleton Uni versity, Canada, 1992.
Shao, M. and C. L. Nikias, Signal processing with fracti onal lower order moments: stable processes and their applications, Proc. IEEE 8/: 9861010 (\993).
Shankar, P. M., S. N. Gupta, and H. M. Gupta, Applications of coherent optics and holography in biomedical engineering, IEEE Trans. Biomed. Eng . BME-29:8-15 (\982).
Slutsky, E., Alcuni applicazioni di coefficienti di Fourier al analizo di sequenze eventuali coherenti stazionari, Giorn. d. Instituto Italiano degli Atuari 5:435-482 (1934) .
Smith, K. T. , The uncertainty principle on groups, S IAM J. Appl. Math. 50 :876-882 (1990).
Stein , C; Inadmissibilit y of the usual estimator for the mean of a multivariate normal di st rib ution in Proc. Third Berk. Symp. Math. S tatist .Prob. Vol. t , Univ. Calif. Press, Berkeley, 1955, pp. 197-206.
Stoffer, D. S., Walsh-Fourier analysis and its statistica l application s, 1. Amer. Statist . Assoc. 86: 481-482 (1991).
Strang, G., Wavelet transforms versus Fourier transforms, Bull. Amer . Math. So c. 28: 288-305 (1993) .
Strichartz, R. S., How to make wavelet s, Amer. Math. Monthly /00: 539-556 (1990).
Tarter, M. E. and M. D. Lock , Model-free Curv e Estimation, Chapman and Hall , New York, 1993.
Terras, A., Harmonic Analysis on Symmetric Spaces and Applications I & II, Springer, New York, 1988.
Thompson, J. R., Some shrinkage techniques for estimating the mean, 1. Amer. Statist . Assoc. 63: 113-122 (1968).
Timan, A. F., Theory of Approximation of Functions of a Real Variable. Pergamon, Oxford, 1963.
Tukey, J.W., An introduction to the frequency analysis of time series in The Collected Works ofJohn W. Tukey 1(1984). (ed . D. R. Brillinger), Wadsworth, Pacific Grove, 1963, pp. 503-650.
Tukey, J. W., Equalization and pulse shaping techniques applied to determination of initial sense of Rayleigh waves in The Collected Works of John W. Tukey 1(1984). (ed . D.R. Brillinger), Wadsworth, Pacific Grove, 1959, pp . 309-358.
Tukey, J . W., Introduction to the dilemmas and difficulties of regression . Unpublished, 1979.
Walter, G. G., Approximation of the delta funct ion by wavelets, J . Approx. Theory 7l: 329-343 (1992).
Walter, G. G. (1994) . Wavelets and Other Orthogonal Systems with Applications. CRC Press, Boca Raton.
Wenk H. R., K. H. Downing, M. S. Hu and M. A. Okeefe, 3D structure determination from electron-microscope images-electron crystallography of staurolite, Acta Crystallographica AI. 48: 700-716 (1992).
Whittle, P., Estimation and information in time series analysis, Skand. Aktuar. 35: 48-60 (1952).
Whittle, P., Discussion of C. M. Stein "Confidence sets for the mean of a multivariate normal distribution", J . Roy. Statist. Soc . B 24: 294 (1962).
Wiener, N., The Fourier Integral and Certain of Its Applications, Dover, New York, 1933.
Yaglorn , A. M. , Second-order homogeneous random fields in Proc . Fourth Berkeley Symp, Math. Statist. Prob. 2, Univ. Calif. Press, Berkeley, 1961, pp. 593-622.
Yajima, Y., A central limit theorem of Fourier transforms of strongly dependent stationary processes, J. Time Series Analysis 10: 375-384 (1989).
Yariv, A., -Quantum Electronics, Second Edition, Wiley, New York, 1975.
Zidek, J., Discussion of Copas (1983), Regression, prediction and shrinkage, pp . 347-48. J. Roy. Statist . Soc. B 45: 311-335 (1983).
Zygmund, A., Trigonometric Series, Cambridge Univ. Press, Cambridge, 1968.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Brillinger, D.R. (2012). Some Examples of Empirical Fourier Analysis in Scientific Problems. In: Guttorp, P., Brillinger, D. (eds) Selected Works of David Brillinger. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1344-8_16
Download citation
DOI: https://doi.org/10.1007/978-1-4614-1344-8_16
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-1343-1
Online ISBN: 978-1-4614-1344-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)