Thursday, April 7, 2011

Joey's Kids II: A Nice Prose Overview of Fourier Series

What a DAY yesterday was, huh? The Z-prime announcement at Tevatron, more Multiverse madness, LISA cancelled (at least the USA contribution, not ESA's, but can ESA do it by itself?), AND Glenn Beck of the US's Tea Party was fired by uber-Conservative FOX News! I'm not sure I can take too many days like that. Hopefully today will be quieter. :-)

Today I share a beautiful prose overview description of Fourier Series, as this is April and in my own personal journey towards wherever I am going, I have decided that April 2011 is my own personal "Fourier and Fractals" month. I'm not saying the two are interrelated. I don't know, maybe they are, but for the moment I like each irrespective of the other and I have been noted to study 2 different subjects at once, to be able to walk and chew gum at the same time, so to speak.

So, today we talk Fourier.

The following is from The MaTH bOOK, which gives a wonderful prose overview description. We gave a brief mathematical example previously in Joey's Kids 1 : From Sines to Sawteeth.


Fourier series are useful in countless applications today, ranging from vibration analysis to image processing -  virtually any field in which a frequency analysis is important. For example, Fourier series helps scientists characterize and better understand the chemical composition of stars or how the vocal tract produces speech. 

Before French mathematician Joesph Fourier discovered his famous series, he accompanied Napoleon on his 1789 expedition of Egypt, where Fourier spent several years studying Egyptian artifacts. Fourier's research on the mathematical theory of heat began around 1804 when he was back in France, and in 1807 he had completed his important memoir On the Propagation of Heat in Solid Bodies. One of his interests  was heat diffusion in different shapes. For these problems, researchers are usually given the temperatures of points on a surface, as well as at its edges, at time t = 0. Fourier introduced a series with sine and cosine terms in order to find solutions to these kinds of problems. More generally, he found that any differentiable function can be represented to arbitrary accuracy by a sum of sine and cosine functions, no matter how bizarre the function may look when graphed. 

Biographers Jerome Ravetz and I. Gratten-Guiness note, "Fourier's achievement can be understood by [considering] the powerful mathematical tools he invented for the solutions of the equations, which yielded a long series of descendants and raised problems in mathematical analysis that motivated much of the leading work in that field for the rest of the century and beyond." British physicist Sir James Jeans (1877-1946) remarked, "Fourier's theorem tells us that every curve, no matter what its nature may be, or in what way it was originally obtained, can be exactly reproduced by superposing a sufficient number of simple harmonic curves - in brief, every curve can be built up by piling up waves."

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