  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.
