Independent polynomial-chaos tensor products are orthonormal

For $\alpha ,\beta$ in the admissible box set $g_j(y_j)={\varphi }_{j,{\alpha }_j}(y_j){\varphi }_{j,{\beta }_j}(y_j)$. Each $g_j$ is integrable because it is a polynomial of degree at most $2p_j$ and the $2p_j$ moment is finite. By [F4], the restricted measure on $\mathrm{\Gamma }$ is the product of the restricted marginal laws. Repeated Tonelli [F3] first gives ${\int }_{\mathrm{\Gamma }}{\prod }_j\mathrm{\mid }{g}_j(y_j)\mathrm{\mid }d\rho ={\prod }_{j=1}^d\int \mathrm{\mid }{g}_j\mathrm{\mid }d{\rho }_j<\mathrm{\infty }$; repeated Fubini [F3] then gives ${\int }_{\mathrm{\Gamma }}{\prod }_j{g}_j(y_j)d\rho ={\prod }_{j=1}^d\int g_{j}d{\rho }_j$. The product-law statement [F2] identifies the ambient finite product measure with the joint law, and [F4] restricts it to $\mathrm{\Gamma }$.

The factorization in step 1.1 and univariate orthonormality [F1] give ${\int }_{\mathrm{\Gamma }}{\psi }_{\alpha }{\psi }_{\beta }d\rho ={\prod }_{j=1}^d{\delta }_{{\alpha }_j{\beta }_j}$. This product is one exactly when every coordinate agrees, and zero otherwise; with $\beta =\alpha$ it also shows ${\psi }_{\alpha }\in L^2(\rho )$. Multiplying ${\varphi }_{j,0}=1$ over the finitely many coordinates gives ${\psi }_0=1$, so whenever $0\in \mathrm{\Lambda }$ the finite family includes this constant mode.