math
posted by mathstudent on .
A trigonmetric polynomial of order n is t(x) =
c0 + c1 * cos x + c2 * cos 2x + ... + cn * cos nx
+ d1 * sin x + d2 * sin 2x + ... + dn * sin nx
The output vector space of such a function has the vector basis:
{ 1, cos x, cos 2x, ..., cos nx, sin x, sin 2x, ..., sin nx }
Use the GramSchmidt process to find an orthonormal basis using the innerproduct:
<f,g> = definite integral over [0,2*pi] of: f(x) * g(x) dx

The book gives the following answer for the orthonormal basis.
g0 = 1/sqrt(2*pi)
g1 = 1/sqrt(pi) * cos x
gn = 1/sqrt(pi) * cos nx
I've done the first two vectors using GramSchmidt and my answers don't match:
Original basis vector u0 = 1
Orthogonal basis vector v0 = 1
Orthonormal basis vector g0 = v0/v0 = sqrt(2/pi)
Original basis vector u1 = cos x
Orthogonal basis vector v1 = u1  <u1, g0> * g0
= cos x  sqrt(2/pi) * sqrt(2/pi) = cos x  2/pi
Orthonormal basis vector g1 = v1/v1 = (cos x  2/pi) / sqrt((pi^2  8)/(4*pi))
= (cos x  2/pi) * sqrt((4*pi)/(pi^2  8))
What am I doing wrong?

Original basis vector u0 = 1
Orthogonal basis vector v0 = 1
Orthonormal basis vector g0 = v0/v0
V0^2 = 2 pi >
g0 = 1/sqrt(2 pi)
Original basis vector u1 = cos x
Orthogonal basis vector v1 = u1  <u1, g0> * g0 = cos x because
<cos(x), 1> = 0 
Thanks Count Iblis!
I was mistakenly integrating with pi/2 instead of 2*pi and every time I redid the problem, I just remade the same mistake without noticing it.
Your help pointed out the issue. Thanks so much!