We address the problem of constructing approximations based on orthogonal polynomials that preserve an arbitrary set of moments of a given function without loosing the spectral convergence property. To this aim, we compute the constrained polynomial of best approximation for a generic basis of orthogonal polynomials. The construction is entirely general and allows us to derive structure preserving numerical methods for partial differential equations that require the conservation of some moments of the solution, typically representing relevant physical quantities of the problem. These properties are essential to capture with high accuracy the long-time behavior of the solution. We illustrate with the aid of several numerical applications to Fokker-Planck equations the generality and the performances of the present approach.

翻译：本文研究基于正交多项式构造逼近的问题，要求在保持谱收敛特性的同时，保留给定函数的任意指定矩集。为此，我们针对正交多项式的一般基函数，求解了带约束的最佳逼近多项式。该构造方法具有完全普适性，可推导出适用于偏微分方程的结构保持数值方法，这类方法需确保解的部分矩（通常代表问题的相关物理量）的守恒性。这些性质对于高精度捕捉解的长时间行为至关重要。通过多个福克-普朗克方程的数值应用实例，我们阐释了本方法的普适性与性能表现。