Let $\eta=g(\xi_1,\ldots,\xi_M)$ be a random variable expressed as a function of the random vector $\xi=(\xi_1,\ldots,\xi_M)$. Assume that $\xi_1,\ldots,\xi_M$ are absolutely continuous and independent random variables. Let $\{\phi_i(\xi)\}_{i=0}^\infty$ be a sequence of normalized orthogonal polynomials with respect to the weight function $f_\xi(\xi)=\prod_{k=1}^M f_{\xi_k}(\xi_k)$ (generalized polynomial chaos basis for $\xi$). Formally, $$ \eta=\sum_{i=0}^\infty \hat{\eta}_i \phi_i(\xi), $$ where $\hat{\eta}_i=\mathbb{E}[\eta \phi_i(\xi)]$ is the Fourier coefficient. In Theorem 3.7 from the article On the convergence of generalized polynomial chaos expansions , there are conditions to ensure that the previous series converges in $L^2$.

My question is about the convergence rate of such a series. In the book Numerical Methods for Stochastic Computations, A Spectral Method Approach, Theorem 3.6 gives a polynomial convergence rate of order $p\geq0$ for functions $g\in H_w^p[-1,1]$, where $w$ is the weight function, and the sequence of orthogonal polynomials being the Legendre family. In page 34, it is commented, but not proved, that if $g$ is $C^\infty$, then the rate of convergence is exponential.

My question is: for a general family of orthonormal polynomials $\{\phi_i(\xi)\}_{i=0}^\infty$, are there any results about the rate of convergence of the series to $\eta$ in $L^2$ ? If $g$ is $C^\infty$, is the convergence rate exponential?


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