Projections onto Chebyshev Polynomials

[pzuraq]

Hey all, this is for a Linear Algebra class

I'm currently stuck on a VERY trying problem involving the Chebyshev Polynomials.

I currently have to find the projections of sin(x) and cos(x) onto the first four polynomials in the sequence:

T0(x) = 1
T1(x) = x
T2(x) = 2x^2 -1
T3(x) = 4x^3 - 3x

I have to do this using the inner product, which is:



The first part of the problem was checking the orthogonality of each of the polynomials (to each other). I used a trig substitution in order to solve the integral, x = cosu and dx = -sinu du.

However, if I understand the problem correctly then projecting sinx/cosx onto these polynomials means subbing them in for, say, g(x) while f(x) is polynomial in question, like so...



I'm completely stumped on how to solve these projections. Obviously I'm not seeing something, but I don't think a trig substitution would work here. Any ideas?

The final part of the problem is asking me how the projections compare to the Taylor series expansion for sinx and cosx, so perhaps they will end up similar?

 

[tkhunny]

Good call on checking orghogonality first.

Next, turn it into an Orthonormal Basis and make your life a little easier. Just divide each polynomial by its inner product with itself. The first one is \(\displaystyle \pi\).

After that, it's just an inner product with each basis vector. If \(\displaystyle U_{n}\)is the Normal polynomial assiciated with Polynomial \(\displaystyle T_{n}\), then we have:

\(\displaystyle \left<\sin(x)\cdot U_{n}\right>\cdot U_{n}\)

 

[tkhunny]

Note: Why on earth are you trying to bash out the value of that integral? Take a good close look at it! Think "odd function" and then think "limits of integration symmetric about the origin". Then just write down the answer. It takes only one symbol.