16 Polynomial vector spaces

Polynomials form finite-dimensional vector spaces on which differentiation, multiplication, and changes of basis act as linear maps.

In this exercise, we study a vector space that is not $\mathbb{F}^n$. We start out without an inner product, but supply this later.

Recall that a polynomial of degree $n$ over $\mathbb{F}$ has the form \[p(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n,\] with $a_i \in \mathbb{F}$ and $a_n \neq 0$.

Exercises

Exercise 16.1 Let $P_n$ be the set of all polynomials of degree less than or equal to $n$. Show that $P_n$ is a vector space of dimension $n+1$ over $\mathbb{C}$.

Exercise 16.2 Let $D : P_n \to P_n$ be the differentiation operator, i.e., \[(D p)(x) = p'(x).\] Show that $D$ is indeed a linear operator.

Exercise 16.3 Does there exist an inverse of $D$?

Exercise 16.4 Let $B = \{1, x, x^2, \cdots, x^n\}$ be the basis of monomials. Using bra-ket notation, we write the basis operator as \[\hat{B} = \sum_{i=1}^{n+1} \left|x^{i-1}\right\rangle\left\langle i\right|,\] where $\left|i\right\rangle$ is a standard basis vector for $\mathbb{F}^{n+1}$. Compute the matrix of $D$ relative to this basis. (Note carefully that we do not have an inner product defined.) That is, find a matrix $T$ such that \[D \hat{B} = \hat{B} T.\]

We now restrict our attention to the interval $x \in [-1,1]$. That is, \[V_n = \{ p : [-1,1] \to \mathbb{C}\mid p \text{ a polynomial of deg $\leq n$} \}.\] The above subexercises did not depend on the domain of the polynomial.

We define an inner product on $V$, \[\left\langle p|q\right\rangle := \int_{-1}^1 \overline{p(x)} q(x) \, \mathrm{d}x.\] We will now define the Legendre polynomials.

Exercise 16.5 Show that $\left\langle\cdot|\cdot\right\rangle$ is indeed an inner product by checking the axioms.

Exercise 16.6 Compute the overlap matrix $S$ of the monomial basis. Is the basis orthonormal?

Exercise 16.7 Formulate the Gram–Schmidt procedure using the bra-ket notation. Explain how the procedure generates an upper triangular matrix $R$ such that \[\hat{B} = \hat{U} R,\] where $\hat{U}$ is the basis matrix of the orthonormal set of vectors.

Exercise 16.8 Use the Gram–Schmidt decomposition by hand to compute the first three normalized Legendre polynomials $\left|\hat{P}_i\right\rangle$, $i=0,1,2$, defined using the Gram–Schmidt orthogonalization of $\{1,x,x^2\}$.

The Legendre polynomials are orthogonal polynomials $P_i : [-1,1]\to \mathbb{R}$ obtained from the monomials by Gram–Schmidt orthogonalization and normalized by $P_i(1)=1$. They are orthogonal but not normalized. The endpoint normalization gives \[\int_{-1}^1 P_i(x) P_j(x) \, \mathrm{d}x = \frac{2}{2i + 1} \delta_{ij}.\] With $P_0(x)=1$ and $P_{-1}\equiv0$, they satisfy the recurrence relation \[(i + 1) P_{i+1}(x) = (2i + 1) x P_i(x) - i P_{i-1}(x).\] From this, one can show: \[P'_{i+1}(x) = \frac{2 P_i(x)}{\|P_i\|^2} + \frac{2 P_{i-2}(x)}{\|P_{i-2}\|^2} + \frac{2 P_{i-4}(x)}{\|P_{i-4}\|^2} + \ldots \]

Exercise 16.9 Compute the matrix of the differential operator $D$ in the normalized Legendre polynomial basis, both by using matrix multiplication involving $X$, $R$, and $R^{-1}$, and also by using the derivative relation above. Compare the two explicitly using pen-and-paper calculations.

Exercise 16.10 Repeat the previous exercise with the operator $D^2$.

Orthogonal polynomials define efficient quadrature rules. Consider approximating an integral by \[\int_{-1}^1 f(x) \, \mathrm{d}x \approx \sum_{i=1}^n f(x_i) w_i,\] where the $w_i$ are weights and the $x_i$ are nodes. A unique choice of nodes and weights makes the rule exact for every polynomial of degree at most $2n-1$. The nodes are the zeros of $P_n$, and the weights are \[w_i = \frac{2}{(1-x_i^2)[P'_n(x_i)]^2}.\] The Wikipedia page on Gauss–Legendre quadrature contains more details.

The so-called Golub–Welsch algorithm is a simple algorithm based on diagonalization of a tridiagonal Hermitian matrix for computing the nodes and weights of Gaussian quadratures, see the Wikipedia page on Gaussian quadrature. The interested student should check it out!