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Lecture 06 - Orthogonal Basis

Definition

We define the following spaces for future reference:

Lp(Rn)={f:RnC:Rn|f(x)|pdx<}||f||pp=Rn|f(x)|pdxL(Rn)={f:RnC:f is essentially bounded}||f||=ess supRn|f(x)|CB(Rn)={fL(Rn:f is bounded and countinuous)}||f||=supRn|f(x)|C0(Rn)={fCB(Rn):lim||x||f(x)=0}Cc(Rn)={fCB(Rn):f has compact support}

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

CB(Rn)={fCB(Rn):f is infinitely differentiable & f(m)(x)L(Rn)}S(R)={fCB(Rn):xmf(n)(x)L(Rn) m,n0}

Orthogonal Basis

In finite dimension, an euclidean space is a vector space equipped with an inner product. The space En of polynomials of degree less than or equal n is of dimension n+1. We can identify it with Rn+1, and on En we can define a scalar product  , k:

P,Q1=i=0naibiP,Q2=11P(x)Q(x) dxP,Q3=Rex2P(x)Q(x) dx

Note that the standard basis 1,x,...,xn 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)Pn+1(x)=(2n+1)xPn(x)nPn1(x),P0(x)=1,P1(x)=x

For  , 3, it would be Hermite polynomials.

Gram-Schmidt Method

Let EsubspaceX be a normed linear space with E=span{x1,x2,...,xn} which are pairwise . We define the projection to be:

(*)projE(v)=j=1nxj,vxj,xjxj

Now if x1,...,xn are not pairwise we can still use () to compute profE(v) if we can find a set of vectors {y1,...,yn} such that E=span{y1,...,yn}. This may be always done by adopting the GS process.

Definition

The procedure starts with a set {x1,...,xn} linearly independent vectors. Then:

y1=x1,yi=xiprojEi1(xi),

where Ei=span{x1,...,xi}.