Approximation by orthonormal polynomials associated with even exponential weights

In this paper, we prove a quantitative approximation result by orthonormal polynomials associated to an exponential weight of the form e -$\mathrm{\Phi }$ , where $\mathrm{\Phi }$ 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{é} 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.