We introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.

我们介绍并研究了傅立叶级数对有界变化的集值函数（多功能，SVF）的适应性。在我们的方法中，借助于狄利克雷核，使用新定义的加权度量积分，我们定义了傅里叶级数的部分和的类似物。我们导出这些近似值的误差范围。结果，我们证明了部分和序列在Hausdorff度量中连续地点收敛到近似集合值函数的值，或者收敛到根据近似度量的选择描述的某个集合不连续点的多功能。