Prince - A First Look at Fourier Analysis T W Krner o...

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A First Look at Fourier Analysis T. W. K¨ orner August 2, 2003 These are the skeleton notes of an undergraduate course given at the PCMI conference in 2003. I should like to thank the organisers and my audience for an extremely enjoyable three weeks. The document is written in L A T E X2e and should be available in tex, ps, pdf and dvi format from my home page http://www.dpmms.cam.ac.uk/˜twk/ Corrections sent to [email protected] would be extremely welcome. Mihai Stoiciu who was TA for the course has kindly written out solutions to some of the exercises. These are also accessible via my home page. Contents 1 Waves in strings 2 2 Approximation by polynomials 5 3 Cathode ray tubes and cellars 7 4 Radars and such-like 10 5 Towards rigour 12 6 Why is there a problem? 13 7 Fej´ er’s theorem 16 8 The trigonometric polynomials are uniformly dense 18 9 First thoughts on Fourier transforms 22 10 Fourier transforms 23 1
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11 Signals and such-like 25 12 Heisenberg 29 13 Poisson’s formula 31 14 Shannon’s theorem 32 15 Distributions on T 35 16 Distributions and Fourier series 39 17 Distributions on R 43 18 Further reading 45 19 Exercises 45 1 Waves in strings It is said that Pythagoras was the first to realise that the notes emitted by struck strings of lengths l . l/ 2, l/ 3 and so on formed particularly attractive harmonies for the human ear. From this he concluded, it is said, that all is number and the universe is best understood in terms of mathematics — one of the most outrageous and most important leaps of faith in human history. Two millennia later the new theory of mechanics and the new method of mechanics enabled mathematicians to write down a model for a vibrating string. Our discussion will be exploratory with no attempt at rigour. Suppose that the string is in tension T and has constant density ρ . If the graph of the position of the string at time t is given by y = Y ( x, t ) where Y ( x, t ) is always very small then, working to the first order in δx , the portion of the string between x and x + δx experiences a force parallel to the y -axis of T ∂Y ∂x ( x + δx, t ) - ∂Y ∂x ( x, t ) = Tδx 2 Y ∂x 2 . Applying Newton’s second law we obtain (still working to first order) ρδx 2 Y ∂t 2 = Tδx 2 Y ∂x 2 . Thus we have the exact equation ρ 2 Y ∂t 2 = T 2 Y ∂x 2 . 2
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For reasons which will become apparent later, it is usual to write c for the positive square root of T/ρ giving our equation in the form 2 Y ∂t 2 = c 2 2 Y ∂x 2 . F Equation F is often called ‘the wave equation’. Let us try and solve the wave equation for a string fixed at 0 and l (that is, with Y (0 , t ) = Y (0 , l ) = 0 for all t ). Since it is rather ambitious to try and find all solutions let us try and find some solutions. A natural approach is to seek solutions of the particular form Y ( x, t ) = X ( x ) T ( t ). Substitution in F gives X ( x ) T 00 ( t ) = c 2 X 00 ( x ) T ( t ) which we can rewrite as T 00 ( t ) T ( t ) = c 2 X 00 ( x ) X ( x ) .
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