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Unformatted text preview: Topics in Fourier and Complex Analysis Part III, Autumn 2009 T. W. K¨orner July 31, 2009 Small print This is just a first draft of the first part of the course. I suspect these notes will cover the first 16 hours but I will not be unduly surprised if it takes the entire course to cover the material. The content of the course will be what I say, not what these notes say. Experience shows that skeleton notes (at least when I write them) are very error prone so use these notes with care. I should very much appreciate being told of any corrections or possible improvements and might even part with a small reward to the first finder of particular errors. This course definitely requires a first course in complex variable and enough analysis to be happy with terms like norm, complete metric space and compact. I am happy to give classes on any topics that people request. At at least one point, the course requires measure theory, but you need only quote the required results in examination. Contents 1 Non-existence of functions of several variables 2 2 Fourier series on the circle 5 3 Jackson’s theorems 12 4 Vistuˇ skin’s theorem 16 5 Simple connectedness and the logarithm 18 6 The Riemann mapping theorem 23 7 Equicontinuity 27 8 Boundary behaviour of conformal maps 29 1 9 Picard’s little theorem 32 10 Picard’s great theorem 33 11 References and further reading 35 1 Non-existence of functions of several vari- ables Theorem 1. Let λ be irrational We can find increasing continuous functions φ j : [0 , 1] → R [1 ≤ j ≤ 5] with the following property. Given any continuous function f : [0 , 1] 2 → R we can find a function g : R → R such that f ( x,y ) = 5 summationdisplay j =1 g ( φ j ( x ) + λφ j ( y )) . The main point of Theorem 1 may be expressed as follows. Theorem 2. Any continuous function of two variables can be written in terms of continuous functions of one variable and addition. That is, there are no true functions of two variables! For the moment we merely observe that the result is due in successively more exact forms to Kolmogorov, Arnol’d and a succession of mathematicians ending with Kahane whose proof we use here. It is, of course, much easier to prove a specific result like Theorem 1 than one like Theorem 2. Our first step is to observe that Theorem 1 follows from the apparently simpler result that follows. Lemma 3. Let λ be irrational. We can find increasing continuous functions φ j : [0 , 1] → R [1 ≤ j ≤ 5] with the following property. Given any continuous function F : [0 , 1] 2 → R we can find a function G : R → R such that bardbl G bardbl ∞ ≤ bardbl F bardbl ∞ and sup ( x,y ) ∈ [0 , 1] 2 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle F ( x,y ) − 5 summationdisplay j =1 G ( φ j ( x ) + λφ j ( y )) vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle ≤ 999 1000 bardbl F bardbl ∞ ....
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