Let $\Phi_j(j=0,1,2,...,n+1)$ be a system of orthogonal polynomials on [a,b].
Prove: $\Phi_{n+1}$ is orthogonal to all p $\in P_n[a,b]$
I'm going to solve like following:
I set $p_n(x)=\sum_{k=0}^n a_k\Phi_k$
$$A= <\Phi_{n+1},p_n(x)> = \int_a^b\Phi_{n+1}p_n(x)dx = \int_a^b\Phi_{n+1}\sum_{k=0}^n a_k\Phi_kdx = \sum_{k=0}^n a_k\int_a^b\Phi_{n+1}\Phi_kdx $$
Since $\Phi_k \neq \Phi_{n+1}$, A = 0
--> QED
Is it true that we need to check whether $<\Phi_{n+1},p_n(x)>$ = 0? And I think the way I set $p_n(x)$ is incorrect.
Could someone help me please? Thank you.