  Many high-dimensional uncertainty quantification problems are solved by
polynomial dimensional decomposition (PDD), which represents Fourier-like
series expansion in terms of random orthonormal polynomials with increasing
dimensions. This study constructs dimension-wise and orthogonal splitting of
polynomial spaces, proves completeness of polynomial orthogonal basis for
prescribed assumptions, and demonstrates mean-square convergence to the correct
limit -- all associated with PDD. A second-moment error analysis reveals that
PDD cannot commit larger error than polynomial chaos expansion (PCE) for the
appropriately chosen truncation parameters. From the comparison of
computational efforts, required to estimate with the same precision the
variance of an output function involving exponentially attenuating expansion
coefficients, the PDD approximation can be markedly more efficient than the PCE
approximation.
