Let V = C[0, 2] with inner product (f.g) = f f(x) g(x) dr. Also define...

  

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Let V = C[0, 2] with inner product (f.g) = f f(x) g(x) dr. Also define po(z) = = 1 integers k set 2k-1(x) sin(kr). (a) Show that (po, 1, 2,...} is an orthonormal set in V. [Hint: Use the product-to-sum identities.] (b) Let f(t) = r. Find ||f|| and (f.n) for each n 0. (You don't need to give details of the integral evaluations, just the resulting values.) = 1 1 cos(kx) and 24 (2) = 2T and for positive 1 11 (c) With f(x) = x, assuming that f(x) = 0 (f. pk) k (r), derive Leibniz's formula =1--+ 35 7 [Hint: Set x = /2.] (d) With f(x) = x, assuming that ||f||2 = Co (f.ek) (see problem 10 of the midterm for why this is a reasonable statement), find the exact value of Ek=1 42 1 Let V = C[0, 2] with inner product (f.g) = f f(x) g(x) dr. Also define po(z) = = 1 integers k set 2k-1(x) sin(kr). (a) Show that (po, 1, 2,...} is an orthonormal set in V. [Hint: Use the product-to-sum identities.] (b) Let f(t) = r. Find ||f|| and (f.n) for each n 0. (You don't need to give details of the integral evaluations, just the resulting values.) = 1 1 cos(kx) and 24 (2) = 2T and for positive 1 11 (c) With f(x) = x, assuming that f(x) = 0 (f. pk) k (r), derive Leibniz's formula =1--+ 35 7 [Hint: Set x = /2.] (d) With f(x) = x, assuming that ||f||2 = Co (f.ek) (see problem 10 of the midterm for why this is a reasonable statement), find the exact value of Ek=1 42 1