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MATH 519 Homework Fall 2013 56. a) If f ∈ L1 (RN ) and fλ (x) = f (λx) for λ > 0, find a relationship between fbλ and fb. b) If f ∈ L1 (RN ) and fh (x) = f (x − h) for h ∈ RN , find a relationship between fbh and fb. 57. Show that Z Z φ(x)ψ(x) dx = RN b ψ(x) b dx φ(x) RN for φ and ψ in the Schwartz space. (This is also sometimes called the Plancherel formula.) 58. In this problem Jn denotes the Bessel function of the first kind and of order n. It may defined in various ways, one of which is Z i−n π iz cos θ e cos (nθ) dθ Jn (z) = π 0 Suppose that f is a radially symmetric function in L1 (R2 ), i.e. f (x) = f (r) where r = |x|. Show that Z ∞ fˆ(y) = 2π J0 (r|y|)f (r)r dr 0 It follows in particular that fˆ is also radially symmetric. Using the known d (zJ1 (z)) = zJ0 (z) compute the Fourier transform of χB(0,R) the identity dz indicator function of the ball B(0, R) in R2 . (See problem 11.16 in text for the R3 case.) 59. For α ∈ R let fα (x) = cos αx. a) Find the Fourier transform fbα . b) Find limα→0 fbα and limα→∞ fbα in the sense of distributions. 60. Compute the Fourier transform of the Heaviside function H(x) by justifying that bn Ĥ = lim H n→∞ where Hn (x) = H(x)e 11.5 in text.) − nx , and then evaluating this limit. (See exercise