1. Orthonormal basis

Assume that $\{ P_n(x) \} ^{ \infty }_{n=0}$ is an orthonormal basis with respect to $ = \int _{-1} ^{1} f(t)g(t)dt$

Find an orthonormal basis $\{ Q_n (x) \} ^ { \infty } _{n=0}$ with respect to $ = \int ^1 _0 f(t)g(t)dt$

Assume that $\{ P_n(x) \} ^{ \infty }_{n=0}$ is an orthonormal basis with respect to $ = \int _{-1} ^{1} f(t)g(t)dt$
Find an orthonormal basis $\{ Q_n (x) \} ^ { \infty } _{n=0}$ with respect to $ = \int ^1 _0 f(t)g(t)dt$
In fact, let $Q_n(x) = \frac1{\sqrt2}P_n(2x-1)$. Then you just have to check that $\int_0^1Q_m(x)Q_n(x)\,dx$ is 1 if m=n and 0 otherwise.