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：有理函数逼近提供了一种简单但灵活的替代多项式逼近的方法，能够捕捉复杂的非线性特性而不产生振荡伪影。然而，由于可能产生虚假奇点，使用有理函数处理含噪声数据的方法鲜有尝试。为避免奇点的产生，我们采用伯恩斯坦多项式并对其系数施加适当约束，强制分母严格为正。虽然这缩小了可表达的有理多项式范围，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。我们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。与之形成对比的是，我们的方法无法在逼近域中产生极点，并且对于多变量函数，其拟合效果优于多项式逼近甚至带惩罚的样条方法。此外，在利用谱方法数值求解微分方程时，保证区间内无极点对于估计非常数系数至关重要。这提供了原始微分方程的紧凑表示，使数值求解器能够快速达到高精度，正如我们的实验所示。