By Korner T.A.
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Extra resources for A First Look at Fourier Analysis
Ii) Explain why x−1 Pn (1/x) exp(−1/x2 ) → 0 as x → 0. (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.
33. ) (i) Show that, if we set ∞ d(f, g) = r=0 2−r f (r) − g (r) 1 + f (r) − g (r) ∞ , ∞ then (D, d) is a metric space. (ii) Show that fn → f if and only if d(fn , f ) → 0. ) Show that (D, d) is complete. 34. Show that the following equality holds in the space of tempered distributions ∞ ∞ em δ2πn = 2π m=−∞ n=−∞ where δ2πn is the delta function at 2πn and en is the exponential function given by en (t) = exp(int). What formula results if we take the Fourier transform of both sides?
1 (iii)). (i) If > 0, show that ∞ sin λx dx → x − −∞ sin x dx, x as λ → ∞. (ii) If π ≥ > 0, show, by using the estimates from the alternating series test, or otherwise, that − sin (n + 21 )x dx → sin x2 π −π sin (n + 12 )x dx = 2π sin x2 as n → ∞. (iii) Show that 1 2 − →0 x sin 12 x as x → 0. and deduce that ∞ 0 π sin x dx = . 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.