This work is concerned with the kernel-based approximation of a complex-valued function from data, where the frequency response function of a partial differential equation in the frequency domain is of particular interest. In this setting, kernel methods are employed more and more frequently, however, standard kernels do not perform well. Moreover, the role and mathematical implications of the underlying pair of kernels, which arises naturally in the complex-valued case, remain to be addressed. We introduce new reproducing kernel Hilbert spaces of complex-valued functions, and formulate the problem of complex-valued interpolation with a kernel pair as minimum norm interpolation in these spaces. Moreover, we combine the interpolant with a low-order rational function, where the order is adaptively selected based on a new model selection criterion. Numerical results on examples from different fields, including electromagnetics and acoustic examples, illustrate the performance of the method, also in comparison to available rational approximation methods.

翻译：本文关注基于核函数对复数值函数数据进行近似的问题，其中频域中偏微分方程的频率响应函数具有特殊意义。在此背景下，核方法的使用日益频繁，但标准核函数的表现并不理想。此外，复数值情形下自然出现的核函数对的作用及其数学内涵仍有待阐明。我们引入了新的复数值函数再生核希尔伯特空间，并将核函数对的复数值插值问题表述为这些空间中的最小范数插值问题。同时，我们将插值函数与低阶有理函数相结合，基于新的模型选择准则自适应地选择阶数。来自不同领域（包括电磁学与声学实例）的数值结果展示了该方法的性能，并与现有有理逼近方法进行了对比。