We equip $\mathbb{R}_{n}[X]$ with the inner product defined by $$\langle P, Q \rangle = \int_{-1}^{1} P(x)Q(x)\,dx$$ and the associated norm $\|P\|_{2} = \sqrt{\langle P, P \rangle}$. For $j \in \mathbb{N}$, we define the polynomial $$P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$$ By convention, $P_{0} = 1$. (a) What is the degree of $P_{j}$? (b) Show that $P_{j}$ is an even or odd polynomial, depending on the value of $j$. (c) Show that $P_{j}(1) = 1$ and $P_{j}(-1) = (-1)^{j}$.
We equip $\mathbb{R}_{n}[X]$ with the inner product defined by
$$\langle P, Q \rangle = \int_{-1}^{1} P(x)Q(x)\,dx$$
and the associated norm $\|P\|_{2} = \sqrt{\langle P, P \rangle}$.
For $j \in \mathbb{N}$, we define the polynomial
$$P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$$
By convention, $P_{0} = 1$.
(a) What is the degree of $P_{j}$?
(b) Show that $P_{j}$ is an even or odd polynomial, depending on the value of $j$.
(c) Show that $P_{j}(1) = 1$ and $P_{j}(-1) = (-1)^{j}$.