Rational function approximations provide a simple but flexible alternative to polynomial approximation, allowing one to capture complex non-linearities without oscillatory artifacts. However, there have been few attempts to use rational functions on noisy data due to the likelihood of creating spurious singularities. To avoid the creation of singularities, we use Bernstein polynomials and appropriate conditions on their coefficients to force the denominator to be strictly positive. While this reduces the range of rational polynomials that can be expressed, it keeps all the benefits of rational functions while maintaining the robustness of polynomial approximation in noisy data scenarios. Our numerical experiments on noisy data show that existing rational approximation methods continually produce spurious poles inside the approximation domain. This contrasts our method, which cannot create poles in the approximation domain and provides better fits than a polynomial approximation and even penalized splines on functions with multiple variables. Moreover, guaranteeing pole-free in an interval is critical for estimating non-constant coefficients when numerically solving differential equations using spectral methods. This provides a compact representation of the original differential equation, allowing numeric solvers to achieve high accuracy quickly, as seen in our experiments.

翻译：有理函数逼近为多项式逼近提供了一种简单而灵活的替代方案，能够在不产生振荡伪影的情况下捕获复杂的非线性特征。然而，由于可能产生虚假奇点，有理函数在含噪声数据上的应用尝试较少。为避免奇点的产生，我们利用Bernstein多项式及其系数上的适当约束条件，迫使分母严格为正。尽管这缩小了可表示的有理多项式范围，但保留了有理函数的所有优势，同时保持了多项式逼近在含噪声数据场景中的鲁棒性。在含噪声数据上的数值实验表明，现有有理逼近方法持续在逼近域内产生虚假极点。相比之下，我们的方法既不会在逼近域内产生极点，又能为多变量函数提供优于多项式逼近甚至惩罚样条的拟合效果。此外，使用谱方法数值求解微分方程时，确保区间内无极点对估计非恒定系数至关重要。这为原始微分方程提供了紧凑表示，使数值求解器能够像实验所示般快速达到高精度。