If you are reading this post, I assume you have learned or at least heard about Fourier Transform. As a matter of fact, most engineering student should have some background of it. But you might not know exactly how important it is and why. I went through an interesting book “A First Course in Fourier Analysis” by David W. Kammler (it is pretty involving but give me much more appreciation on the Fourier). At one appendix of the book, the author listed some interesting facts about the impact of Fourier and I would like to quote here.

Is it possible to prove that the definitions, equations, patterns, theorems,…that comprise the body of mathematics known as Fourier analysis have *value* as well as *validity*? Consider the following:

- The function concept; the Riemann and Lebesgue integrals; the concepts of pointwise, uniform, and mean square convergence; Cantor’s theory of sets; the Schwartz theory of distributions;…were all invented in an attempt to answer the question: When is Fourier’s representation valid?, see
- E.B.VanVleck, The influence of Fourier’s series on the development of mathematics,
*Science***39**(1914), 113–124; - W.A.Coppel, J.B.Fourier—On the occasion of his two hundredth birthday,
*Amer. Math. Monthly***76**(1969), 468–483; - S.Bochner, Fourier series came first,
*Amer. Math. Monthly***86**(1979), 197–199; - E.A.González-Velasco, Connections in mathematical analysis: The case of Fourier series,
*Amer. Math. Monthly***99**(1992), 427–441.

- E.B.VanVleck, The influence of Fourier’s series on the development of mathematics,

- The
**most frequently cited mathematics paper ever written**,- J.W.Cooley and J.W.Tukey,
*Math. Comp.***19**(1965), 297–301,

- J.W.Cooley and J.W.Tukey,

- describes a clever scheme for doing Fourier analysis on a computer. The FFT, which Gilbert Strang calls the most important algorithm of the 20th century, initiated a revolution in scientific computation. The bibliography in
- E.O.Brigham,
*The Fast Fourier Transform and Its Applications**,*Prentice Hall, Englewood Cliffs, NJ, 1988

will point you to some of the applications.

- E.O.Brigham,

- The
**first U.S. patent for a mathematical algorithm**was assigned to Stanford University for R.Bracewell’s FHT, a variation of the FFT. After reading- E.N.Zalta, Are algorithms patentable?
*Notices AMS***35**(1988), 796–799,

see if you can figure out why knowledgeable individuals would invest thousands of dollars seeking patent protection for this particular algorithm.

- E.N.Zalta, Are algorithms patentable?

- The power and flexibility of Fourier analysis have facilitated an incredibly diverse range of applications to modern mathematics, science, and engineering. It is easy to verify this assertion: Feed the key word “Fourier,” to any internet search engine and explore some of the links you are given.
**Approximately 3/4 of the Nobel prizes in physics**were awarded for work done using the tools and concepts of Fourier analysis. Examine the abstracts from- F.N.Magill, ed.,
*The Nobel Prize Winners—Physics, Vols. 1–3**,*Salem Press, Englewood Cliffs, NJ, 1989

- F.N.Magill, ed.,

- with a knowledgeable physicist and see how close you come to this estimate!

- Herbert Hauptman (a mathematician) and Jerome Karle shared the 1985
**Nobel prize in chemistry**for showing how to use Fourier analysis to determine the structure of large molecules from X-ray diffraction data, see- W.A.Hendrickson, The 1985 Nobel Prize in Chemistry,
*Science***231**(1986), 362–364; - Mathematics: The unifying thread in science,
*Notices AMS***33**(1986), 716–733.

- W.A.Hendrickson, The 1985 Nobel Prize in Chemistry,

- Francis Crick, James Watson, and Maurice Wilkins won the 1962
**Nobel prize in medicine and physiology**for discovering the molecular structure of DNA. Fourier analysis of X-ray diffraction data played an essential role in this work, see- F.Crick,
*What Mad Pursuit**,*Basic Books, Inc., New York, 1988, pp. 39–61.

- F.Crick,

- Alan Cormack and Godfrey Hounsfield won the 1979 Nobel prize in medicine for the development of computer assisted tomography. Paul Lauterbur and Peter Mansfield won the 2003 Nobel prize in Medicine for their discoveries concerning magnetic resonance imaging. Today detailed medical images are routinely produced by using Fourier analysis to process X-ray and nuclear spin signals, see
- C.L.Epstein,
*Introduction to the Mathematics of Medical Imaging**,*Pearson Education, Upper Saddle River, NJ, 2003.

- C.L.Epstein,

- The sophisticated instruments of modern science now produce
*signals*instead of*numbers*as the basic data for scientific research. It is impossible to describe the function of an FT-NMR spectrum analyzer, an X-ray diffraction machine, a seismic recorder, …without using the vocabulary and concepts of Fourier analysis, see- A.G.Marshall and F.R.Verdun,
*Fourier Transforms in NMR, Optical, and Mass Spectroscopy**,*Elsevier, New York, 1990

to develop some appreciation for what is involved in learning to use an FT-NMR spectrum analyzer.

- A.G.Marshall and F.R.Verdun,
- The amazing technology and consumer products associated with digital signal processing (compact disk players, high-definition TV, digital phones,…) rest on the mathematical base of Fourier analysis. You can confirm this by examining the
*sampling*chapter from an introductory text such as- A.V.Oppenheim, A.S.Willsky, and I.T.Young,
*Signals and Systems**,*Prentice Hall, Englewood Cliffs, NJ, 1983.

- A.V.Oppenheim, A.S.Willsky, and I.T.Young,

- During the past half-century Fourier analysis has transformed the study of optics (with
**Fourier optics**being the name now used for the new discipline). A lens acts by changing phase quadratically; Fresnel diffraction is “factored” into succesive operations of phase transformation, Fourier transformation, dilation, and phase transformation; and any converging lens has the inherent ability to take a Fourier transform*optically,*see- J.W.Goodman,
*Introduction to Fourier Optics**,*3rd ed., Roberts & Company, Englewood, CO, 2005.

- J.W.Goodman,

- There is an extraordinary high rate of return on your investment of time in learning Fourier analysis. You will
*experience*the Fourier advantage as you “speed learn” the applications chapters of this text and when you take subsequent courses in PDEs, Quantum Mechanics, Signals and Systems, Fourier Optics, … !