The first and second moments properties of an orthonormal basis $\{u_j\}_{j \geq 1}$ are well known: $$ \int_{t\in \mathcal{T}} u_j(t) F(dt)=0 $$ $$ \int_{t\in \mathcal{T}} u^2_j(t) F(dt)=1 $$ and $$ \int_{t\in \mathcal{T}} u_j(t) u_i(t) F(dt)=0,\quad i\neq j $$
I am curious, what are fourth moment properties of $\{u_j\}_{j \geq 1}$, i.e., $$ \int_{t\in \mathcal{T}} u^4_j(t) F(dt) = ? $$ and $$ \int_{t\in \mathcal{T}} u^2_j(t) u^2_i(t) F(dt) = ? $$