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 Gram-Schmidt process to find an orthonormal basis using the inner-product:
<f,g> = definite integral over [0,2*pi] of: f(x) * g(x) dx

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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 Gram-Schmidt 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!

It seems that you made some mistakes in calculating the inner product and the resulting orthogonal and orthonormal vectors. Let's go through the Gram-Schmidt process step by step to find the correct orthonormal basis.

Step 1: Find the first ortho-normalized vector g0. In this case, g0 = 1/sqrt(2*pi) as given.

Step 2: Find the second orthogonal vector v1.
Start with the original basis vector u1 = cos(x).
Then, subtract the projection of u1 onto g0, which is given by <u1, g0> * g0.
Since <u1, g0> = integral from 0 to 2*pi of: (cos(x)) * (1/sqrt(2*pi)) dx = 0,
we have v1 = u1 - <u1, g0> * g0 = cos(x) - 0 = cos(x).

Step 3: Normalize v1 to get g1.
The normalization formula is g1 = v1/|v1|.
The norm |v1| is given by the square root of the integral from 0 to 2*pi of: (cos(x))^2 dx.
Evaluating this integral gives sqrt(pi).

Thus, g1 = cos(x) / sqrt(pi).

It looks like your calculation for g1 is incorrect. The correct orthonormal basis vector g1 is cos(x) / sqrt(pi).

Continue the Gram-Schmidt process for the remaining basis vectors to find the orthonormal basis.