We prove lower bounds on the error incurred when approximating any oscillating function using piecewise polynomial spaces. The estimates are explicit in the polynomial degree and have optimal dependence on the meshwidth and frequency when the polynomial degree is fixed. These lower bounds, for example, apply when approximating solutions to Helmholtz plane wave scattering problem.

翻译：我们证明了使用分段多项式空间逼近任意振荡函数时误差的下界。这些估计在多项式次数上是显式的，并且在固定多项式次数时对网格宽度和频率具有最优依赖关系。例如，这些下界适用于逼近亥姆霍兹平面波散射问题的解。