Orthogonal polynomials and functions

Due by Mon, 22 Jan 2018

Let $V$ be the vector space $V = \{f: [-1, 1] \to \mathbb{R} \vert \text{ $f$ is continuous}\},$ with inner product given by $\langle f, g\rangle = \frac{1}{2}\int\limits_{-1}^1 f(t)g(t)dt.$

1. Show that polynomials $x^n$ and $x^m$ (as elements of $V$) are orthogonal if and only if $n-m$ is odd.
2. Show that $f(x) = 1$ and $g(x) = \sqrt{3}x$ form an orthonormal basis for the subspace $L \subset V$ consisting of linear functions (you may use the fact that the constant function $1$ and the function $x$ form a basis for the space $L$).
3. Find the linear function $h(x) = ax + b$ that minimizes $\int\limits_{-1}^1 (e^t - (at+b))^2 dt.$
4. Find an orthonormal basis for the subspace $Q \subset V$ consisting of polynomials of degree at most $2$.
5. Let $S$ be the subspace of $V$ consisting of symmetric continuous functions, i.e., functions $f$ such that $f(-x) = f(x)$ for all $x \in [-1, 1]$. Let $g\in V$ be an anti-symmetric function, i.e., $g(-x) = -g(x)$ for all $x \in [-1, 1]$. Show that $\forall f\in S$, $\langle f, g \rangle = 0$.
6. For a function $f \in V$, let $\hat{f}$ be given by $\hat{f}(x) = \frac{f(x) + f(-x)}{2}$. Show that $f - \hat{f}$ is anti-symmetric, and deduce that $\hat{f}$ is the orthogonal projection onto $S$ of $f$.

All assignments