Chapter 5.5, Problem 34E

In the space $C\left[-1,1\right]$ , we define the inner product $\langle f,g\rangle ={\displaystyle {\int }_{-1}^1\frac{2}{\pi }\sqrt{1-t^2}f\left(t\right)g\left(t\right)dt}=\frac{2}{\pi }{\displaystyle {\int }_{-1}^1\sqrt{1-t^2}f\left(t\right)g\left(t\right)dt}$ . See Exercise 33; here we let $w\left(t\right)=\frac{2}{\pi }\sqrt{1-t^2}$ . [This function $w\left(t\right)$ is called a Wigner semicircle distribution, after the Hungarian physicist and mathematician E. P. Wigner (1902-1995), who won the 1963 Nobel Prize in Physics.] Since this is not a course in calculus, here are some inner products that will turn out to be useful: $\langle 1,t^2\rangle =1/4,\langle t,t^3\rangle =1/8$ , and $\langle t^3,t^3\rangle =5/64$ .
a. Find ${\displaystyle {\int }_{-1}^{1}w\left(t\right)dt}$ . Sketch a rough graph of the weight function $w\left(t\right)$ .
b. Find the norm of the constant function $f\left(t\right)=1$ .
c. Find $\langle t^2,t^3\rangle$ ; explain. More generally, find $\langle t^n,t^m\rangle$ for positive integers n and in whose sum is odd.
d. Find $\langle t,t\rangle$ and $\langle t^2,t^2\rangle$ . Also, find the norms of the functions t and $t^2$ .
e. Applying the Gram—Schmidt process to the standard basis $1,t,t^2,t^3$ of $P_3$ , construct an orthonormal basis $g_0\left(t\right),\mathrm{...},g_3\left(t\right)$ of $P_3$ for the given inner product. [The polynomials $g_0\left(t\right),\mathrm{...},g_3\left(t\right)$ are the first few Chebyshev Polynomials of the second kind, named after the Russian mathematician PafnutyChebyshev (1821-1894). They have a wide range of applications in math, physics, and engineering.]
f. Find the polynomial $g\left(t\right)$ in $P_3$ that best approximates the function $f\left(t\right)=t^4$ on the interval $\left[-1,1\right]$ , for the inner product introduced in this exercise.