In the space C[ 1, 1], we introduce the inner product{/g) = \ / , fU)g{t)dt a. Find (tn

Chapter 5, Problem 32

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In the space C[ 1, 1], we introduce the inner product{/g) = \ / , fU)g{t)dt a. Find (tn, fm), where n and m are positive integers.b. Find the norm of /(f) = frt, where n is a positive integer. c. Applying the Gram-Schmidt process to the standard basis 1, f, f2, f3 of P3, construct an orthonormal basis go(t), .., 3(f) of P3 for the given inner product.d. Find the polynomials ,..., (Those are goO) 3(1) the first few Legendre Polynomials, named after the great French mathematician Adrien-Marie Legen- dre, 1752-1833. These polynomials have a wide range of applications in math, physics, and engineering. Note that the Legendre polynomials are normalized so that their value at 1 is 1.) e. Find the polynomial g(t) in P3 that best approximates the function f(t) = ------^ on the inter1 + f2 val [ - 1, 1], for the inner product introduced in this exercise. Draw a sketch.