Math 115 Spring 2015 Problem Set 6 Prof. Bernoff Orthogonal Functions, L2 Convergence and Fourier Series Due date: Wednesday, March 4th in class. For these problems, you may use MAPLE when appropriate. F1: Gram-Schmidt orthogonalization. We know from elementary linear algebra that any set of linearly independent vectors may be turned into a set of orthogonal vectors by the Gram-Schmidt orthogonalization process. The same process works for functions defined on an interval a < x < b (a mathematician would say on L2 [a, b]). Let {fn } be a linearly independent set of functions. Define an inner-product Z b hp(x), q(x)i = p(x) · q(x) dx x=a Define the sequence of functions gn is defined inductively by g1 = f1 , g2 = f2 − hf2 , g1 i g1 , ||g1 ||2 g3 = f3 − hf3 , g1 i hf3 , g2 i g1 − g2 , . . . . 2 ||g1 || ||g2 ||2 where ||g||2 = hg, gi. Use induction to show that gn is an orthogonal sequence. F2: Approximating a function by Legendre Polynomials. This is a great problem for you to hone your MAPLE skills on. (a) The functions {1, x, x2 , x3 , . . . }, are linearly independent on the interval x ∈ [−1, 1]. Use the previous problem and the functions {1, x, x2 , x3 } to generate a sequence of four orthogonal polynomials on x ∈ [−1, 1]. Denote the polynomials by P0 (x), P1 (x), P2 (x), P3 (x) (they are called the Legendre Polynomials). (b) Find an approximation of ex on x ∈ [−1, 1] of the form ex ≈ c0 P0 (x) + c1 P1 (x) + c2 P2 (x) + c3 P3 (x) that is best in the root-mean-square sense, and graph ex and the approximation on a set of coordinate axes. (c) For your approximation what is the maximum of the pointwise error on [−1, 1]? What is the L2 error? Editorial Note: The pointwise error of an approximation F ∗ (x) to a function F (x) is E(x) = |F (x) − F ∗ (x)|. The root-mean-square or L2 error here is sZ 1 [E(x)]2 dx ||E(x)|| = −1 (please turn over) Math 115 Spring 2015 Problem Set 6 Prof. Bernoff F3: Consider a periodic function, g(t), with period 2π and g(t) = t2 −π <t<π (a) Find a Fourier cosine series for g(t). Check your answer by graphing the partial sums with MAPLE. (b) Differentiate the cosine series and the function to obtain a sine series for f (t) = 1 dg =t 2 dt −π <t<π Check your answer by graphing the partial sums with MAPLE. What happens at t = ±π? (c) Show by consider g(0) that ∞ π 2 X (−1)n+1 = . 12 n=1 n2 What do you need to assume for this identity to be true? (d) Prove the two identities ∞ π2 X 1 = 6 n2 n=1 ∞ π4 X 1 = 90 n=1 n4 by applying Parseval’s Theorem to f (t) and g(t) respectively. What have you assumed here and how does it differ from part (c)? F4: Consider a function, h(x), defined for −π < x ≤ π ( 2x 0 ≤ x ≤ π h(x) = 0 −π < x ≤ 0 (a) Draw the 2π-periodic extension of h(x). (b) Compute the Fourier expansion for h(x), ∞ a0 X + an cos(nx) + bn sin(nx) . h(x) = 2 n=1 Graph some partial sums and show that it converges to the periodic extension of h(x). What happens at π? (please turn over) Math 115 Spring 2015 Problem Set 6 Prof. Bernoff (c) Note that the even part of h(x) is given by E(x) = 1 [h(x) + h(−x)] = |x| 2 −π ≤x≤π . Use this identity to compute the series for E(x) = |x| on this interval. Show the series reduces to a cosine series (why is this true?). You can verify this result with the calculation we did in class. (d) Note that the odd part of h(x) is given by O(x) = 1 [h(x) − h(−x)] = x 2 −π <x<π . Use this identity to compute the series for O(x) = x on this interval. Show the series reduces to a sine series (why is this true?). You can verify this result with H3(b).