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4), or otherwise, prove the following results. (i) If f : T → C is continuous, then fˆ(n) → 0 as |n| → ∞. ) (ii) If f : T → R is continuous, then 2π 0 f (t)| sin nt| dt → 2 π 2π f (t) dt 0 as n → ∞. 17. Let R be a rectangle cut up into smaller rectangles R(1), R(2), . . , R(k) each of which has sides parallel to the sides of R. Then, if each R(j) has at least one pair of sides of integer length, it follows that R has at least one pair of sides of integer length. First try and prove this without using Fourier analysis.
8. (i) If φ, ψ ∈ D and λ, µ ∈ C then Tλφ+µψ = λTφ + µTψ . (ii) If F, φ ∈ D then TF φ = F T φ . 37 An even clearer example of the use of condition (C) occurs when we seek to define the derivative of a distribution. Observe that if φ, f ∈ D then π 1 φ (t)f (t) dt 2π −π 1 1 = [φ(t)f (t)]π−π − 2π 2π π 1 =− φ(t)f (t) dt 2π −π = − φ, f . φ ,f = π φ(t)f (t) dt −π This fixes the form of our definition. 10. If T ∈ D then T , f = − T, f for all f ∈ D. 11. (i) If T ∈ D then T ∈ D . (ii) If T, S ∈ D and λ, µ ∈ C then (λT + µS) = λT + µS .
If fˆ(λ) = 0 for |λ| ≥ K then then f is determined by its values at points of the form nπK −1 with n ∈ Z. We call πK −1 the ‘Nyquist rate’. Since electronic equipment can only generate, transmit and receive in a certain band of frequencies and sampling more frequently than the Nyquist rate produces, in principle, no further information it is reasonable to suppose that the rate of transmission of information is is proportional to the Nyquist rate. We thus have rate of transmission of information ≤ constant band width of signal 34 where the constant can be improved a little by elegant engineering but must remain of the same order of magnitude.