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MATH 519 Homework Fall 2013 61. Where is the mistake in the following argument? If u(x) = e−x then u0 + u = 0 so by Fourier transformation iyû(y) + û(y) = (1 + iy)û(y) = 0 y∈R Since 1 + iy 6= 0 for real y, it follows that û(y) = 0 for all real y and hence u(x) = 0. N 62. Let h ∈ L1 (RN ) and assume that there exists L < (2π)− 2 such that |b h(y)| ≤ L a.e. Use the contraction mapping theorem to show that for any g ∈ L2 (RN ) the convolution equation Z f (x) + f (y)h(x − y) dy = g(x) RN has a unique solution f ∈ L2 (RN ). 63. Let f ∈ C(R) be 2π periodic. It then has a Fourier series in the classical sense, but it also has a Fourier transform since f is a tempered distribution. What is the relationship between the Fourier series and the Fourier transform? 64. Show that E(x, t) = 21 H(t − |x|) is a fundamental solution for the wave operator Lu = utt − uxx . 65. The fourth order operator Lu = uxxxx + 2uxxyy + uyyyy in R2 is the biharmonic operator which arises in the theory of deformation of elastic plates. a) Show that L = ∆2 , i.e. Lu = ∆(∆u) where ∆ is the Laplacian. b) Find a fundamental solution of L. (Suggestions: To solve LE = δ, first p solve ∆F = δ and then ∆E = F . Since F will depend on r = x2 + y 2 only, you can look for a solution E = E(r) also.)