find an orthonormal basis of polynomials of degree 2 over the real space, with inner product <p,q> = integral from (0,1) of p(x)q(x)dx.
if you could direct me as to how to insert an integral symbol that would be great as well!
find an orthonormal basis of polynomials of degree 2 over the real space, with inner product <p,q> = integral from (0,1) of p(x)q(x)dx.
if you could direct me as to how to insert an integral symbol that would be great as well!
You start with a known basis say $\displaystyle \{1, x, x^2\}$ then apply the Gramm-Schmidt process to generate an orthogonal basis, and then normalise the new basis.
RonL
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$\displaystyle u_1=1$ the constant function
$\displaystyle u_2=x - \frac{\langle x, u_1 \rangle}{\langle u_1, u_1 \rangle}u_1$
where $\displaystyle \langle u_1, u_1 \rangle=\int_0^1 1 ~dx =1$, and $\displaystyle \langle x, u_1 \rangle=\int_0^1 x ~dx =1/2$, so:
$\displaystyle
u_2=x-1/2
$
Now repeat to find $\displaystyle u_3$:
$\displaystyle u_2=x^2 - \frac{\langle x^2, u_1 \rangle}{\langle u_1, u_1 \rangle}u_1- \frac{\langle x^2, u_2 \rangle}{\langle u_2, u_2 \rangle}u_2$
$\displaystyle \{u_1, u_2, u_3 \}$ is an orthogonal basis, you now have to normalise them to get your orthonormal basis: $\displaystyle \{e_1, e_2, e_3 \}$, where:
$\displaystyle e_i=\frac{u_i}{\langle u_i, u_i \rangle^{1/2}}$
RonL