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6. Let V = C[0, 2] with inner product (f.g) = f f(x)g(x) dr. Also define po(r) 1 integers k set 2k-1(2) = cos(kx) and 24 (2) = √ sin(kx). = (c) With f(x) = x, assuming that f(x) = Eko (f. 9k) 4k (1), derive Leibniz's formula [Hint: Set r= r = n/2.] √2T (a) Show that {90, 9₁, 92,...} is an orthonormal set in V. [Hint: Use the product-to-sum identities.] (b) Let f(x) = 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.) and for positive π 1 1 1 3 5 =1--+

6. Let V = C[0, 2] with inner product (f.g) = f f(x)g(x) dr. Also define po(r) 1 integers k set 2k-1(2) = cos(kx) and 24 (2) = √ sin(kx). = (c) With f(x) = x, assuming that f(x) = Eko (f. 9k) 4k (1), derive Leibniz's formula [Hint: Set r= r = n/2.] √2T (a) Show that {90, 9₁, 92,...} is an orthonormal set in V. [Hint: Use the product-to-sum identities.] (b) Let f(x) = 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.) and for positive π 1 1 1 3 5 =1--+

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 78E
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6. Let V = C[0, 2] with inner product (f.g) = f = cos(kx) and (21 (2) = √ si 1 integers k set 2k-1(2) = co f(x)g(x) dr. Also define po(z) = sin(kx). /2π (a) Show that (90, 91, 92,...} is an orthonormal set in V. [Hint: Use the product-to-sum identities.] (b) Let f(x) = 1. Find ||f|| and (f.n) for each n 20. (You don't need to give details of the integral evaluations, just the resulting values.) and for positive 0 (f. pk) (a), derive Leibniz's formula =1--+ 1 1 1 3 5 7 k=0 π 4 ΠΑΣ1 (1) 5) and then comparing the power series coefficients of both sides. n² (c) With f(x) = x, assuming that f(x) = [Hint: Set = π/2.] (d) With f(x) = x, assuming that ||f||² = Exo (f.k)² (see problem 10 of the midterm for why this is a reasonable statement), find the exact value of Σk=1 42 1 Remarks: The identity ||f||² = x (f.9k) is known as Parseval's identity. The problem of computing the value of the infinite sum k=1 is known as the Basel problem. The correct value was (famously) sin(x) as the infinite product k² first found by Euler, who evaluated the sum by decomposing the function TX
Transcribed Image Text:6. Let V = C[0, 2] with inner product (f.g) = f = cos(kx) and (21 (2) = √ si 1 integers k set 2k-1(2) = co f(x)g(x) dr. Also define po(z) = sin(kx). /2π (a) Show that (90, 91, 92,...} is an orthonormal set in V. [Hint: Use the product-to-sum identities.] (b) Let f(x) = 1. Find ||f|| and (f.n) for each n 20. (You don't need to give details of the integral evaluations, just the resulting values.) and for positive 0 (f. pk) (a), derive Leibniz's formula =1--+ 1 1 1 3 5 7 k=0 π 4 ΠΑΣ1 (1) 5) and then comparing the power series coefficients of both sides. n² (c) With f(x) = x, assuming that f(x) = [Hint: Set = π/2.] (d) With f(x) = x, assuming that ||f||² = Exo (f.k)² (see problem 10 of the midterm for why this is a reasonable statement), find the exact value of Σk=1 42 1 Remarks: The identity ||f||² = x (f.9k) is known as Parseval's identity. The problem of computing the value of the infinite sum k=1 is known as the Basel problem. The correct value was (famously) sin(x) as the infinite product k² first found by Euler, who evaluated the sum by decomposing the function TX
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