I have the Legendre polynomials base $\{p_0 = 1, p_1 = x, p_2 = x^2 - \frac{1}{3}\}$ which is known to be orthogonal on $[-1,1]$ with respect to $\int_{-1}^1f(x)g(x)\text{ }dx$ and I want a base $\{\phi_1, \phi_2,\phi_3\}$, orthogonal on $[a,b]$ with respect to $\int_a^bf(x)g(x)\text{ }dx$, given that $$\langle p_0,p_0\rangle=2, \langle p_1,p_1\rangle=\frac{2}{3}\text{ and } \langle p_2,p_2\rangle=\frac{8}{45}$$ for inner product on $[-1,1]$
I know I could straighly use Gram-Schmidt orthogonalization process, but I want to get $\phi_1, \phi_2, \phi_3$ with the info given above to minimize calculations (as I would do in a test).
Define $$\phi_1(x) = 1$$ $$ \phi_2(x)=\frac{2(x-a)}{b-a}-1$$ $$ \phi_3(x)=\left(\frac{2(x-a)}{b-a}-1\right)^2-\frac{1}{3}$$
Which we obtained by the substitution $t\to \frac{2(t-a)}{b-a}-1$ which maps $[a,b] \to [-1,1]$
Check that this preserves the inner product.