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Name: CSU ID: Homework 11 April 24, 2015 1. S6.3 ]2(b),(d) 2. S6.3 ]4 3. S6.3 ]10,14 HW12 will include the following problems. 1. S6.3 ]22 2. S6.3 ]26 3. S6.3 ]34 4. S6.3 ]43 5. S6.3 ]44 6. (It is likely this problem will be on HW 12) Let V be the space of piecewise continuous functions on −π ≤ x ≤ π. For functions u, v ∈ V , Rπ let the inner product be hu, vi = −π u(x)v(x)dx. Consider the basis functions on [−π, π] defined by 1 1 g0 (x) = √ , gk (x) = √ cos(kx) π 2π and 1 fk (x) = √ sin(kx) π (a) Show that the basis functions are orthonormal. Use integral tables. (b) Find the projection of F (x) = x − 1 onto g0 , g1 , g2 , f1 , f2 . Use integral tables. (c) Find the seond order Fourier expansion for F (x) = x − 1. Use integral tables.