Our objective is to calculate the derivatives of data corrupted by noise. This is a challenging task as even small amounts of noise can result in significant errors in the computation. This is mainly due to the randomness of the noise, which can result in high-frequency fluctuations. To overcome this challenge, we suggest an approach that involves approximating the data by eliminating high-frequency terms from the Fourier expansion of the given data with respect to the polynomial-exponential basis. This truncation method helps to regularize the issue, while the use of the polynomial-exponential basis ensures accuracy in the computation. We demonstrate the effectiveness of our approach through numerical examples in one and two dimensions.

翻译：我们的目标是在噪声环境下计算数据的导数。这是一项具有挑战性的任务，因为即使微小的噪声也可能导致计算结果出现显著误差。这主要源于噪声的随机性，它会引发高频波动。为克服这一难题，我们提出一种方法：通过从给定数据的傅里叶展开式中滤除高频项（基于多项式-指数基），对数据进行逼近。这种截断方法有助于对问题进行正则化，而多项式-指数基的使用则保证了计算精度。我们通过一维和二维数值算例验证了该方法的有效性。