The main topic of the paper is an adaptation of the trigonometric Fourier series to set-valued functions of bounded variation with compact images. The authors also try to gain error bounds under minimal regularity requirements on the approximated multifunctions. This analysis exploits properties of maps of bounded variation from [V. V. Chistyakov, J. Dyn. Control Syst. 3, No. 2, 261–289 (1997; Zbl 0940.26009)]. The authors define a metric analogue of the partial sums of the Fourier series of a multifunction via convolutions with the Dirichlet kernel. To define these convolutions the authors introduce a new weighted metric integral. The study of the error bounds of the approximations leads to the introduction of a new one-sided local moduli of continuity and a one-sided local quasi-moduli of continuity. The main result of the paper is an analogy of the classical Dirichlet-Jordan Theorem for real functions. In particular, if a multifunction \(F\) of bounded variation is continuous at a point \(x\), then its metric Fourier approximants at \(x\) converge to \(F(x)\). This convergence is uniform on bounded closed intervals in which \(F\) is continuous. At a point of discontinuity the limit set is determined by the values of the metric selections of \(F\).