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By Korner T.A.

Those are the skeleton notes of an undergraduate direction given on the PCMI convention in 2003. I should still wish to thank the organisers and my viewers for a very relaxing 3 weeks. The record is written in LATEX2e and may be on hand in tex, playstation , pdf and clvi layout from my domestic web page

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0 Kelvin once asked his class if they knew what a mathematician was. He wrote the formula ∞ e−x 2 /2 dx = √ π ∞ and the board and said. ‘A mathematician is one to whom that is as obvious as that twice two makes four is to you. 20. (i) Explain why j=−N 1/2 N N |aj bj | ≤ j=−N |aj | for all aj , bj ∈ C. 51 1/2 N 2 j=−N |bj | 2 ∞ j=−∞ (ii) Use (i) to show that, if |aj bj | converges and ∞ j=−∞ 1/2 ∞ ∞ j=−∞ |aj bj | ≤ j=−∞ ∞ j=−∞ |aj |2 and |aj | |bj |2 converges, then 1/2 ∞ 2 j=−∞ |bj | 2 . (iii) If f : T → C is continuously differentiable, explain why ∞ 1 j 2 |fˆ(j)|2 ≤ 2π j=−∞ (iv) Use (ii) to show that converges uniformly to f (t).

Consider the heat equation on the circle. In other words consider well behaved functions θ : T × [0, ∞) → C satisfying the partial differential equation ∂2θ ∂θ = K 2. ∂t ∂x for all t > 0 and all x ∈ T. Try to find solutions using separation of variables and then use the same kind of arguments as we used for the vibrating string to suggest that the general solution is ∞ 2 an einx e−Kn t . θ(x, t) = n=−∞ What happens as t → ∞? 11. Suppose Ln ; T → R is continuous and 1 (A) 2π L (t) dt = 1, T n (B) If η > 0 then Ln → 0 uniformly for |t| ≥ η as n → ∞, (C) Ln (t) ≥ 0 for all t.

27. 23. There we discussed the behaviour of Sn (F, t) when t is small but did not show that Sn (F, t) behaves well when t is far from 0. This follows from general theorems but we shall prove it directly. This brings us into direct contact with Fourier since he used F as a test case for his statement that any function1 had a Fourier expansion. (i) Show that t n Sn (F, t) = cos rx dx 0 r=1 and n cos rx = r=1 sin(n + 12 )x . 2 sin x2 (ii) Deduce that t Sn (F, t) = 2 0 sin(n + 12 )x dx − t + x 1 t 0 g(x) sin(n + 12 )x dx We would now say any ‘reasonable function’ but Fourier and his contemporaries had a narrower view of what constituted a function.

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A First Look at Fourier Analysis by Korner T.A.


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