Consider P2 the space of polynomials of degree at most 2. Show that the function

Question

Consider P2 the space of polynomials of degree at most 2. Show that the function = f(t)g(t)dt is an inner product on the real linear space P2. Starting from the basis (1, x, x2), use the Gram-Schmidt construction to build an orthonormal basis. The resulting polynomials are scalar multiples of what are known as the Legendre polynomials, Pj, which are conventionally normalized so that Pj(1) = 1. Computations with such polynomials form the basis of spectral methods, one of the most powerful techniques for the numerical solution of partial differential equations.

Explanation / Answer

We verify the conditions for being inner product

I will write with open brackets instead of < brackets since Cramster deletes the last.

1) (f,g)=-1..1fg dt=-1..1gf dt=(g,f)

2) (kf,g)=-1..1kfg dt =k-1..1fg dt=k(f,g) where k is a constant

3)(f+g,h)=-1..1(f+g)h dt=-1..1fh+gh dt=-1..1fh dt+-1..1 gh dt=(f,h)+(g,h)

4) (f,f)=-1..1 f2 0 because f2 0

and -1..1 f2 =0 only when f=0

To gind an orthonormal basis, then

b1=1/2

a2=x-1/2*(x,1/2)=x-1/2-1..1 x/2dt=x-x2|-1..1=x

b2=x/(x,x)=x/-1..1 x2 dt=x/(x3/3|-1..1)=x/(2/3)=3x/2

a3=x2-1/2(x2,1/2)-3x/2(x2,3x/2)=

=x2-1/3-3x/2*0=x2-2/3

b3=a3/(a3,a3)=(x2-1/3)/(2/5)=5/2(x2-1/3)

So the orthonormal basis is

1/2

3x/2

5/2(x2-1/3)

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