Math 4564 Homework 7 1. Show that the following three polynomials

| August 30, 2017

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Math 4564 Homework 7

1. Show that the following three polynomials are orthogonal in L2 [−1, 1]
φ0 (x) = 1,

φ1 (x) = x,

1
φ2 (x) = (3×2 − 1)
2

2. Compute the norms ||φi (x)|| for the polynomials in the previous problem.
3. Compute the best approximation p2 (x) for the function f (x) = ex .
p2 (x) = c0 φ0 (x) + c1 φ1 (x) + c2 φ2 (x),

ck =

f, φk
φk , φk

You can use Integrate[…] command in Mathematica to compute c2 .
Plot both f (x) and p2 (x) in Mathematica on the same set of axes.
4. Use Mathematica to show that the functions
φk (x) = sin

2kπ
x
b−a

and ψk (x) = cos

2kπ
x ,
b−a

k = 1, 2, 3, …

form an orthogonal set in L2 [a, b].
Hint:Use Integrate[…,{x,a,b}] command to compute the inner products.

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