Consider monic orthogonal polynomials ${ \{\phi_{0}, \phi_{1},\dots, \phi_{n},\dots \}}$. We want to show that $||\phi_{k}||\leq||q||$ for all monic polynomials $q \in \Pi_{k}$ which are of exact degree $k$.
The norm here refers to the norm arising from the inner product: $ \langle f,g \rangle =\int_{a}^{b} w(x)f(x)g(x) \,\mathrm dx $ where $w$ is a positive weight function.
I was wondering how to approach this problem. I tried applying the definition of the norm and the recurrence relation on orthogonal polynomials, but I can't seem to get anywhere.
Best Answer
We have $$q=\phi_k+\sum_{j=0}^{k-1}a_j\phi_j.$$ By the Pythagoras theorem $$\|q\|^2=\|\phi_k\|^2+\sum_{j=0}^{k-1}|a_j|^2\|\phi_j\|^2\ge \|\phi_k\|^2.$$