Analysis

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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 prefer to thank the organisers and my viewers for a really stress-free 3 weeks. The record is written in LATEX2e and may be to be had in tex, playstation , pdf and clvi layout from my domestic web page

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26, that t 2 0 sin(n + 12 )x dx = 2 x 1 (n+ )t 2 0 sin x dx → π x and deduce that Sn (F, t) → F (t) as n → ∞ whenever 0 < |t| < π. Show directly that the result is true when t = 0 and t = π and so holds for all t. 23? 28. Suppose that hn : T → R be a sequence of continuous functions such that (i) hn (t) ≥ 0 for all t ∈ T, (ii) hn (t) dt = 1, T hn (t) → 0 uniformly (iii) If K > 0, let us write for all η ≤ |t| ≤ π whenever η > 0. En = {x ∈ T : hn (x) ≥ K}. 57 Show that En hn (t) dt → 1 as n → ∞. Deduce that there exists an N (K) such that En hn (t)2 dt ≥ K 2 for all n ≥ N (K).

Iii) Show by induction, using the definition of differentiation, that E is infinitely differentiable at 0 with E (n) (0) = 0 for all n. ] (iv) Show that ∞ E(x) = j=0 E (j) (0) j x j! if and only if x = 0. ) (v) If you know some version of Taylor’s theorem examine why it does not apply to E. 31. 30. 30. Show that F is infinitely differentiable. (ii) Sketch the functions f1 , f2 : R → R given by f1 (x) = F (1 − x)F (x) x and f2 (x) = 0 f1 (t) dt. (iii) Show that given a < α < β < b we can find an infinitely differentiable function f : R → R with 1 ≥ f (x) ≥ 0 for all x, f (x) = 1 for all x ∈ [α, β], f (x) > 0 for x ∈ (a, b) and f (x) = 0 for all x ∈ / [a, b].

N n=−N f (n) sinc(t − n) − f (t) N = n=−N π 1 = 2π 1 ≤ 2π = 1 2π  ≤ = 1 1 ˆ F (−n) 2π 2π 1 2π −π π −π 1 2π 1 2π  n=−N 1  2π 1 2π π F (λ) exp(iλt) dλ −π Fˆ (−n) exp(iλ(t − n)) − F (λ) exp(iλt) dλ N n=−N F (λ) − Fˆ (−n) exp(−iλn) − F (λ) dλ 1 2π N Fˆ (n) exp(iλn) dλ n=−N N π −π −π exp(i(t − n)λ) dλ − N π −π π F (λ) − |n|≥N +1 1 Fˆ (n) exp(iλn) 2π n=−N 1/2 |Fˆ (n)|2  2 1/2 dλ →0 as N → ∞. 2 in a very slightly generalised form. 6. (Shannon’s Theorem) Suppose f : R → C is a contin∞ uous function with −∞ |f (t)| dt < ∞ and that K > 0.

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