Lecture 06 - Orthogonal Basis

#math

Definition

We define the following spaces for future reference:

L^{p} (R^{n}) = {f : R^{n} \to C : \int_{R^{n}} | f (x) |^{p} d x < \infty} | | f | |_{p}^{p} = \int_{R^{n}} | f (x) |^{p} d x

L^{\infty} (R^{n}) = {f : R^{n} \to C : f is essentially bounded} | | f | |_{\infty} = e s s \underset{R^{n}}{s u p} | f (x) |

C_{B} (R^{n}) = {f \in L^{\infty} (R^{n} : f is bounded and countinuous)} | | f | |_{\infty} = \underset{R^{n}}{s u p} | f (x) |

C_{0} (R^{n}) = {f \in C_{B} (R^{n}) : l i m_{| | x | | \to \infty} f (x) = 0}

C_{c} (R^{n}) = {f \in C_{B} (R^{n}) : f has compact support}

Where the support of f, denoted by $s u p p f$ , is defined to be: $$supp\ f = {x\in \mathbb{R}^{n}: f(x) \neq 0}$$

C_{B}^{\infty} (R^{n}) = {f \in C_{B} (R^{n}) : f is infinitely differentiable & f^{(m)} (x) \in L^{\infty} (R^{n})}

S (R) = {f \in C_{B}^{\infty} (R^{n}) : x^{m} f^{(n)} (x) \in L^{\infty} (R^{n}) \forall m, n \geq 0}

Orthogonal Basis

In finite dimension, an euclidean space is a vector space equipped with an inner product. The space $E_{n}$ of polynomials of degree less than or equal $n$ is of dimension $n + 1$ . We can identify it with $R^{n + 1}$ , and on $E_{n}$ we can define a scalar product $⟨, ⟩_{k}$ :

\begin{aligned} ⟨ P, Q ⟩_{1} & = \sum_{i = 0}^{n} a_{i} b_{i} \\ ⟨ P, Q ⟩_{2} & = \int_{- 1}^{1} P (x) Q (x) d x \\ ⟨ P, Q ⟩_{3} & = \int_{R} e^{- x^{2}} P (x) Q (x) d x \end{aligned}

Note that the standard basis $1, x, . . ., x^{n}$ are a pairwise $⊥$ for the scalar product $⟨, ⟩_{1}$ but not for the other two products. Orthogonal polynomials for $⟨, ⟩_{2}$ are for example the Legendre polynomials which satisfy the following recurrence relation:

(n + 1) P_{n + 1} (x) = (2 n + 1) x P_{n} (x) - n P_{n - 1} (x), P_{0} (x) = 1, P_{1} (x) = x

For $⟨, ⟩_{3}$ , it would be Hermite polynomials.

Gram-Schmidt Method

Let $E \underset{subspace}{\subset} X$ be a normed linear space with $E = s p a n {x_{1}, x_{2}, . . ., x_{n}}$ which are pairwise $⊥$ . We define the projection to be:

\begin{matrix} (*) & {proj}_{E} (v) = \sum_{j = 1}^{n} \frac{⟨ x_{j}, v ⟩}{⟨ x_{j}, x_{j} ⟩} x_{j} \end{matrix}

Now if $x_{1}, . . ., x_{n}$ are not pairwise $⊥$ we can still use $(*)$ to compute ${prof}_{E} (v)$ if we can find a set of $⊥$ vectors ${y_{1}, . . ., y_{n}}$ such that $E = s p a n {y_{1}, . . ., y_{n}}$ . This may be always done by adopting the $G - S$ process.

Definition

The procedure starts with a set ${x_{1}, . . ., x_{n}}$ linearly independent vectors. Then:

\begin{aligned} y_{1} & = x_{1}, \\ y_{i} & = x_{i} - {proj}_{E_{i - 1}} (x_{i}), \end{aligned}

where $E_{i} = s p a n {x_{1}, . . ., x_{i}}$ .