0 like 0 dislike
75 views
Let $w(x)$ be a positive continuous function on the interval $0 \leq x \leq 1, n$ a positive integer, and $\mathcal{P}_{n}$ the vector space of polynomials $p(x)$ whose degrees are at most $n$ equipped with the inner product
$\langle p, q\rangle=\int_{0}^{1} p(x) q(x) w(x) d x .$
a) Prove that $\mathcal{P}_{n}$ has an orthonormal basis $p_{0}, p_{1}, \ldots, p_{n}$ with the degree of $p_{k}$ is $k$ for each $k$.
b) Prove that $\left\langle p_{k}, p_{k}^{\prime}\right\rangle=0$ for each $k$.
| 75 views

0 like 0 dislike
0 like 0 dislike
0 like 0 dislike
0 like 0 dislike
1 like 0 dislike
0 like 0 dislike
0 like 0 dislike
0 like 0 dislike
0 like 0 dislike