In this paper, we prove a quantitative approximation result by orthonormal polynomials associated to an exponential weight of the form e -$\Phi$ , where $\Phi$ is an even polynomial with positive leading coefficient. This result is a consequence of a recursion relation for the orthonormal polynomials and of the strong Poincar{\'e} inequality. Simulations are provided at the end of the article, on smooth, non-smooth functions as well as in the Gaussian and the double well case.

翻译：本文证明了与形如e^{-Φ}的指数权相关的正交多项式的一种定量逼近结果，其中Φ是一个首项系数为正的偶多项式。该结果是正交多项式递推关系与强庞加莱不等式的推论。文章末尾提供了在光滑函数、非光滑函数以及高斯情形与双势阱情形下的数值模拟。